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What contributions do quantum mechanics and general relativity make to our comprehension of the universe's genesis in physics?

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Could it be possible to harness dark matter and dark energy? For example, could their gravitational interactions with baryonic matter be used to generate power? How might this be done?

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This is me. don't steal my colour

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What Kind of Technology Could Dark Matter Research Lead To?

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What Kind of Technology Could Dark Matter Research Lead To?

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Jia Liu, a physicist made a paper in 2009 about utilizing dark matter to power spacecrafts

The concept of using dark matter to power spacecrafts is assuming dark matter is made of neutralinos

Neutralinos are particles with no electrical charge that can collide into each other in certain circumstances and convert their mass into energy

Assuming dark matter is truly made of neutralinos, a pound of it may potentially produce 5 billion times more energy than a pound of dynamite

The idea of harnessing dark matter for spacecraft uses a box in the rocket that opens in the direction of motion to take in dark matter. After that, the box is sealed and compresses itself so for a higher chance of dark matter annihilation. Once the dark matter annihilates itself the energy itself, the particles become energy which is used to propel the spacecraft

There are many advantages of using dark matter since it's all over the universe

It is believed that a 100-ton spacecraft can reach the speed of light in only a couple of days if this use of dark matter works.

Conclusion:

Thus, assuming that dark matter is made of neutralinos we may be able to harness its explosive properties to convert it to energy so we can power spacecrafts

Next Question:

How does Dark Energy affect things differently from normal energy and dark matter?

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Beneath the Black Hills of South Dakota, scientists at the Sanford Underground Research Facility are using a device called a Large Underground Xenon (LUX) detector to hunt for particles of dark matter, the mysterious substance that's believed to account for most of the matter in the universe. Inside the massive device, which contains a third of a ton of liquid xenon inside a titanium vessel, an array of sensitive light detectors wait for the moment when a dark matter particle will collide with a xenon atom and emit a tiny flash of light.

In hopes of capturing the faint signal, LUX has been placed under a mile-thick layer of rock, which will help shield it from cosmic rays and other radiation that might interfere with the signal.

So far, LUX hasn't yet detected dark matter. But with a new set of calibration techniques that improve the detector's sensitivity, researchers hope to soon, finally, spot dark matter. "It is vital that we continue to push the capacity of our detector," says Brown University physics professor Rick Gaitskell in a press release.

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What Kind of Technology Could Dark Matter Research Lead To?

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Scientists are still trying to figure out the essence of dark matter. If they do, will it lead only to greater understanding, or can we develop new technologies?

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A possible timeline of the universe's expansion, beginning with a quantum tunneling event.

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Searching for the Quantum Beginning of the Universe

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Searching for the Quantum Beginning of the Universe

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General

General relativity works well to describe events that took place following the beginning of the universe, but falls flat when trying to explain its birth. The Big Bang’s singularity would have infinite density (causing infinite curvature of space), which isn’t possible - at least according to general relativity.

Therefore, theories of quantum mechanics are also required to sufficiently explain the beginning of the universe. Two of the prevailing theories are as follows:

The “no boundary” proposal

Attempts to explain how the universe expanded at the very beginning of its life while avoiding the singularity that broke general relativity

The theory path uses integrals to perform calculations in quantum theory. These path integrals essentially find the probability of an event by performing a sum over all ways that event could have occurred

By using this math, Stephen Hawking and James Hartle hypothesized the initial expansion of the universe

According to the creators of this proposal, the universe started with no size, but expanded smoothly - without a singularity in the first place.

This removes the initial singularity and clears up the original issue with general relativity’s explanation of the universe’s genesis

The tunneling proposal

Unlike the no boundary proposal, this theory attempts to rationalize the initial singularity with quantum tunneling instead of removing it

According to this theory, the singularity has zero size but finite density. This is a state that, according to classical physics, cannot happen

Luckily, quantum tunneling allows particles to do things deemed impossible by classical physics

According to this proposal, the universe could have started with zero size but finite density thanks to a quantum tunneling event

Next question: How does quantum tunneling occur in the first place?

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Understanding the Unique Effects of Dark Energy

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Understanding the Unique Effects of Dark Energy

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Dark energy, unlike normal energy and light matter, behaves uniquely and has distinct effects on the universe's expansion dynamics.

Accelerated Expansion: One of the most significant differences is that dark energy is believed to be responsible for the accelerated expansion of the universe. While normal matter and energy, including light, are subject to gravitational attraction, dark energy exerts a repulsive force, driving galaxies apart at an accelerating rate.

Counterintuitive Gravitational Effect: Dark energy's repulsive nature counteracts the gravitational pull of matter, including dark matter. While gravity tends to pull objects closer together, dark energy's influence becomes dominant on large cosmic scales, causing the universe's expansion to accelerate.

Effects on Cosmic Microwave Background: Dark energy's presence alters the cosmic microwave background (CMB), the remnant radiation from the early universe. It affects the large-scale structure and geometry of the universe, leaving characteristic imprints on the CMB that cosmologists study to understand dark energy's properties.

Expansion Rate: Unlike ordinary matter and energy, which tend to clump together under gravity's influence, dark energy's repulsive force becomes more dominant as the universe expands. This results in an increasing expansion rate over time, leading to the eventual "heat death" scenario where the universe's expansion continues indefinitely.

Dark energy's unique properties challenge conventional understanding of fundamental forces and have profound implications for the fate and structure of the universe.

Next Question: What observational evidence supports the existence of dark energy, and how do astronomers measure its effects on the universe's expansion?

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Everything scientists can observe in the universe, from people to planets, is made of matter. Matter is defined as any substance that has mass and occupies space. But there's more to the universe than the matter we can see. Dark matter and dark energy are mysterious substances that affect and shape the cosmos, and scientists are still trying to figure them out.

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Dark Matter & Dark Energy - NASA Science

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Overview Everything scientists can observe in the universe, from people to planets, is made of matter. Matter is defined as any substance that has mass and occupies space. But there's more to the universe than the matter we can see. Dark matter and dark energy are mysterious substances that affect and shape the cosmos, and [...]

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Explaining the Mechanism of Quantum Tunneling

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Explaining the Mechanism of Quantum Tunneling

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Wave-Particle Duality: Quantum tunneling arises from the wave-particle duality inherent in quantum mechanics. Particles, such as electrons, are described by wave functions that represent the probability amplitudes of finding the particle at different positions. These wave functions can extend beyond classical barriers due to their wave-like nature.

Uncertainty Principle: The Heisenberg Uncertainty Principle states that there is an inherent limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. This principle allows particles to have a finite probability of being found in regions where their classical energy would not permit them to be.

Barrier Penetration: When a particle encounters a potential energy barrier, classical mechanics would dictate that the particle would be reflected back if its energy is insufficient to overcome the barrier. However, in quantum mechanics, there is always a probability that the particle's wave function extends into the classically forbidden region, allowing it to tunnel through the barrier.

Exponential Decay: The probability of tunneling through a barrier decreases exponentially with increasing barrier width and height. However, even for relatively thick barriers, there is still a non-zero probability of tunneling, albeit very small. This probability is described by the tunneling probability, which depends on factors such as the particle's energy and the characteristics of the barrier.

Quantum tunneling plays a crucial role in various phenomena, including nuclear fusion in stars, radioactive decay, and the operation of tunnel diodes in electronics.

Next question: What are the practical applications of quantum tunneling, and how do scientists utilize this phenomenon in technology and research?

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Quantum-tunnelling time is measured using ultracold atoms – Physics World

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Spin rotation provides a stopwatch that measures millisecond journey

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Quantum Tunnelling to the Origin and Evolution of Life

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Quantum Tunnelling to the Origin and Evolution of Life

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Quantum tunnelling becomes relevant at a nanoscale

Wave-particle duality can explain how particles manage potential barriers while lacking sufficient energy to surpass them

Schrodinger equation

The behaviour of particles in a particular space and time is shown through the Schrodinger equation using waves for material particles

This equation helped to explain quantum tunnelling

The Schrodinger equation provides qualitative reports to understand quantum tunnelling and is crucial for figures of tunnelling which are advanced

Quantum tunneling can also be explained using The Uncertainty Principle

The Uncertainty Principle

This principle claims pairs of properties in particles like position/momentum or energy/time are not able to be known at the same time.

It also states that measuring conjugated variables simultaneously cannot be done with complete accuracy

The higher accuracy is known for the momentum of a particle, the less certain the position is and vice versa

Thus, tunnelling takes place when uncertainties of momentum and kinetic energy in a particle are big enough to surpass the energy barrier

This phenomenon can be understood using various wave-particle dualism interpretation

Conclusion

There are multiple ways to find out how quantum tunnelling occurs with The Uncertainty Principle and Schrodinger Equation using wave-particle dualism. Using The Uncertainty Principle, however, we know that tunnelling happens if the uncertainty of momentum and kinetic energy is large enough to surpass the energy barrier

Next Question:

What concepts in quantum physics challenge our understanding of how physics generally works and how do they do so?

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Abstract

Quantum tunnelling is a phenomenon which becomes relevant at the nanoscale and below. It is a paradox from the classical point of view as it enables elementary particles and atoms to permeate an energetic barrier without the need for sufficient energy to overcome it. Tunnelling might seem to be an exotic process only important for special physical effects and applications such as the Tunnel Diode, Scanning Tunnelling Microscopy (electron tunnelling) or Near-field Optical Microscopy operating in photon tunnelling mode. However, this review demonstrates that tunnelling can do far more, being of vital importance for life: physical and chemical processes which are crucial in theories about the origin and evolution of life can be traced directly back to the effects of quantum tunnelling. These processes include the chemical evolution in stellar interiors and within the cold interstellar medium, prebiotic chemistry in the atmosphere and subsurface of planetary bodies, planetary habitability via insolation and geothermal heat as well as the function of biomolecular nanomachines. This review shows that quantum tunnelling has many highly important implications to the field of molecular and biological evolution, prebiotic chemistry and astrobiology.

Keywords: Biomolecular nanomachines, Interstellar, Life, Nucleosynthesis, Planetary habitability, Prebiotic, quantum tunnelling, Radioactive decay.

1. INTRODUCTION

Quantum mechanical phenomena are not comprehensible via concepts and assumptions derived from the macroscopic world. For example, the daily experience that things have a definitive property, are located at a certain time on a definite place and have a definite energy doesn't help when we zoom in to the scale of nanometres and below. At this scale, when trying to understand why reactions between elementary particles, nucleons or atoms occur at energies which would classically not allow these reactions to occur, we arrive at a paradox. Quantum mechanics, however, resolves this paradox. The way it does this guides us to a fascinating phenomenon: tunnelling. This review sheds light on the importance of the tunnelling phenomenon for many processes essential in theories of the origin and evolution of life such as prebiotic chemistry, chemical evolution, biochemical reactions and the formation of habitable planetary environments.

Now, what is quantum tunnelling? A brief introduction into this strange phenomenon becomes more illustrative when referring to one of its technological applications. A good example is Scanning Tunnelling Microscopy (STM) [ 1- 3] as it illustrates both the preconditions and the nature of quantum tunnelling. STM enables us to directly obtain images of surfaces with atomic or sub- molecular resolution with comparatively low instrumental effort (Fig. ). In order to obtain such images, a sharp metallic tip is placed within a distance of about one nanometre or less above an electrically conductive sample surface and a voltage is applied between the tip and the surface. The tip is moved above the surface by a piezoelectric actuator in a raster scanning pattern in order to map tiny variations of the resulting electrical current. These variations are caused by the local density of electronic states in the substrate surface and/or by the topography of the surface. Atomic resolution of crystalline surfaces is easily possible due to the extreme sensitivity of the measured current on the tip-sample distance. This sensitivity is so high that it causes a detectable difference in current when moving the tip from a position above the valley between adjacent atoms to a position exactly above an atom. The trick to achieving such an enormous sensitivity is the phenomenon of quantum tunnelling.

The central characteristic of quantum tunnelling is the fact that under certain conditions elementary particles, nucleons or atoms are able to negotiate the obstacle of a potential barrier (which is, from the classical point of view, a forbidden area for a particle) without having the energy to overcome it. The STM setup gives an illustrative example of potential barrier width: the nanometre-wide gap between the tip and the sample. The benefit of creating such a gap becomes clear when considering the nature of quantum tunnelling.

According to the standard interpretation of quantum mechanics, the quantum state of material particles such as electrons can be described by a wave function Ψ. It seems strange to describe the state of a material particle by a wave function. However, this refers to one of the central postulates of quantum mechanics: wave-particle duality. It has been postulated that particles also have a wave character based on the observation that the detected character of a material particle (corpuscle or wave) depends solely on the experimental design. Therefore a particle can be described as a matter-wave [ 4]. A matter-wave can mathematically be represented by a wave function which is the solution of a wave equation. The appropriate wave equation for material particles is the Schrödinger equation [ 5- 8] whose solution describes the behaviour of a particle in space and time (e.g. with respect to a potential barrier). It is therefore this equation which helps to explain the quantum tunnelling phenomenon.

In the case of an electron faced with a gap between a STM tip and a sample we can use a simple model of a one-dimensional rectangular potential energy barrier with limited height. This model is sufficient to explain the nature of quantum tunnelling in a qualitative way. Based on this model we can write the time-independent, one-dimensional Schrödinger equation [ 9, 10] as

where m is the particle mass (in our case the mass of an electron), ћ is the reduced Planck constant, V is the potential energy and E is the total energy of a particle. The differential operator in this equation is connected with the kinetic energy. Within the rectangular potential barrier of height extending from x=0 to x= s (Fig. ) the solution of this equation reads

for a high ( > E) and wide barrier ( ks > 1). Here s denotes the barrier width (the tip-sample distance in STM) and k denotes the decay constant of a particle state in the potential barrier region (in our example: an electron state near the Fermi level E F of the metallic sample), where

with Φ as the local tunnelling barrier height (). In the case of STM this tunnelling barrier height is the effective local work function or, in other words, the minimum energy required to remove an electron from the metal electrode (e.g. the STM sample) and to bring it to the vacuum level [ 1, 3].

Equation (2) tells us that within a potential barrier there is an exponential damping of the wave function along the positive x direction. (Fig. ) illustrates this damping for the example of a STM situation. The fact that the wave function within the potential barrier ( 0 ≤ x ≤ s) is just damped and not erased altogether has an important consequence: as long as the barrier width is not too large there is still a small amplitude of the wave function behind the barrier. What does that mean? Although the wave function Ψ reflects the wave nature of a particle, it is complex-valued and therefore doesn't provide a measurement parameter. However, the square of the absolute value of the wave function (|Ψ(x)| ²) can statistically be interpreted as the probability density for a particle to be detected at a location point x [e.g. 10]. Therefore a remaining amplitude of the wave function behind the barrier means that there is a small but nonetheless existing probability for a particle to ''transmit'' or ''tunnel'' through the barrier even if there is no sufficient energy to overcome the barrier classically. The transmission probability ( P) can be calculated via the equation

and provides a clear measurement parameter. In fact a STM measures a current which is proportional to the exponential e-2ks. The value of the current depends not only on the local electron density of states in the sample and the tip but also corresponds to the transmission probability of an electron passing the tip-sample barrier. It is this exponential dependence of the transmission probability on the width and height of a potential barrier which causes the extreme sensitivity of the resulting current to the tip-sample distance. The exponential dependence enables gaining images with atomic resolution because tiny changes in the tip-sample distance cause large changes in the resulting tunnelling current. This is why a nanometre-wide tip-sample gap in STM is so important: it creates a situation characteristic of quantum tunnelling by measuring the result of quantum mechanical transmission probabilities.

The Schrödinger equation not only enables a qualitative description of quantum tunnelling, but is also essential for more advanced, quantitative models of tunnelling (for a most recent overview of such tunnelling models for STM see [ 11]). However, it is not the only possible way to explain and describe the tunnelling phenomenon. Let us remember what the Schrödinger equation is based on: the wave-particle dualism. As this dualism is among the central postulates of quantum mechanics there are other important principles which are also related to it. One of these principles is the Uncertainty Principle (Heisenberg principle) [ 12, 13]. A relation between the Uncertainty Principle and the wave-particle dualism arises by the fact that this principle applies to all wave-like systems, including matter waves.

The Uncertainty Principle refers to complementary properties of a particle such as position/momentum or energy/time and states that the values of such conjugated variables cannot be known simultaneously to arbitrary accuracy. What are the consequences of this fundamental limit of precision and how can this principle explain that particles are able to negotiate a potential barrier without having the energy to overcome it? Let us focus on the example of energy-time uncertainty. It can be written as

with Δ E as the uncertainty in energy and Δ t as the uncertainty in time. As the right part of (5) is a constant, the relation implies that e.g. the shorter the time Δ t is, the more the energy of a particle becomes uncertain (Δ E) and vice versa. Based on this relation the tunnelling phenomenon can be explained as follows: if the time Δ t is sufficiently short, the energy of a particle E + Δ E can exceed the height of the potential barrier and thus the particle can overcome it. In other words, quantum tunnelling occurs if a particle is able to negotiate a potential barrier within the time Δ t. With respect to the model of a rectangular potential barrier of height and width s this time can be estimated via the following relation [ 3, 14]:

Another possible point of view on quantum tunnelling is the position/momentum uncertainty:

It states that the more accurate the momentum of a particle is known, the more the position becomes uncertain and vice versa. As a consequence, tunnelling occurs if the momentum uncertainty Δ p and thus the uncertainty in kinetic energy of a particle is so large that it can overcome the energy barrier [ 14]. We see that quantum tunnelling can be explained in multiple ways based on the wave-particle dualism.

The central messages about the tunnelling phenomenon can be summarized as follows: Quantum tunnelling enables particles to manage a potential barrier when there is insufficient energy to overcome it. This phenomenon can be explained by referring to wave-particle duality and can be described on an exact mathematical basis. It is of probabilistic nature and shows an exponential probability decay with increasing barrier width.

Quantum tunnelling is not such a rare phenomenon as it might seem when looking at its preconditions and characteristics. The following sections will demonstrate that it is of vital importance for life in many aspects.

2. THE IMPORTANCE OF QUANTUM TUNNELLING FOR LIFE IN INSOLATION HABITABLE ZONES

Chemical and biological evolution requires a steady stream of energy. Free energy is necessary to generate self-organized, adaptable molecular and living systems and to keep them away from thermodynamic equilibrium [ 15]. The most important source of free energy for life on earth is sunlight as it is the key factor in oxygenic photosynthesis. This type of photosynthesis fuels most ecosystems directly (phototrophy) as well as indirectly (chemotrophy based on energy from organic matter produced by oxygenic photosynthesis). It also enriches the atmosphere and hydrosphere with its byproduct O ₂. The enrichment with O 2 forced the development of aerobic cellular respiration, which was the major precondition for the evolution of complex, metazoan life (multicellular animals). This is because the complexity, size and mobility of metazoans requires a very high amount of free energy compared to other lifeforms, and the reduction of O ₂ by aerobic cellular respiration releases about an order of magnitude more energy per intake of food than other types of energy metabolism [ 16]. Therefore, it is the input of solar photon energy (insolation) used by oxygenic photosynthesis which energetically enabled the evolution of complex, multicelluar life on earth.

A long period of time is also required for chemical and subsequent biological evolution of complex life. It took billions of years to achieve major steps in the origin and evolution of life on earth. According to the geological and paleontological record and recent age determinations, the complete accretion and major early differentiation of the earth occurred around 4.5 - 4.4 billion years (Ga) ago [ 17]. The oldest evidences for bacterial life date back to 3.4 Ga [ 18, 19], the oldest unambiguous eucaryotic microfossils to 1.8 Ga [ 20], the oldest fossils of metazoa to 0.63 Ga [ 21] and the oldest known fossils of anatomically modern Homo sapiens to 0.00016 Ga [ 22]. Thus, a constant solar energy flux over billions of years was a precondition for the biological evolution towards complex life on earth. But what has this all to do with quantum tunnelling?

Solar energy production is based on the fusion of lighter atomic nuclei to heavier ones [ 23]. A fusion reaction releases nuclear binding energy. The released binding energy is equivalent to a mass deficit according to Einsteins energy-mass equivalence [ 24] and therefore the mass of the fusion product (the heavier nucleus) is smaller than the sum of the individual masses of the lighter nuclei. But what enables a nuclear fusion reaction within a star? As each nucleus of the stellar plasma has a positive nuclear charge, there is a high repulsive electrostatic force between the nuclei. Thus, for a nuclear fusion, a high amount of energy is required to bring atomic nuclei close enough to each other (distance ≤ 10-15 m) to overcome the repulsive Coulomb force so that they are tied up by the strong interaction. The only energy available in a star to potentially overcome the Coulomb barrier is the kinetic energy of the nuclei due to thermal motion [ 25]. The central temperature of stars is within the regime of 10 7 K [ 26] - a value for which the assumption of an ideal gas is justified [ 23]. Therefore, the average kinetic energy of a nucleus can be calculated via

where k is the Boltzmann constant and T is the temperature. For a temperature of 10 7 K within the center of of a star is within the regime of 1 keV. But is this average kinetic energy sufficient to exceed the Coulomb barrier height? The energy which is needed to overcome the Coulomb barrier height is

where is the positive nuclear charge of each nucleus i [ 23, 25]. Since the values of are in the regime of MeV [ 27], the average kinetic energies available by the solar core temperature are more than a factor of 1000 below the necessary energy to overcome the Coulomb barrier. This difference is so high that even though there are nuclei with a larger than average kinetic energy (according to the Maxwell-Boltzmann distribution) the probability of having even one nucleus in the whole solar plasma with a kinetic energy 1000 times above average is virtually zero [ 25]. As there is no way for atomic nuclei in the stellar interior during quiescent burning to overcome the Coulomb barrier, there must be something additional to thermal motion to make a thermonuclear reaction probable.

This is where quantum tunnelling comes in. Although thermal motion is by far too low in stellar interiors during quiescent burning to directly induce thermonuclear reactions, it is sufficiently high to bring colliding nuclei so close to each other that a significant probability density behind the Coulomb barrier arises: quantum tunnelling occurs with some probability and enables nuclei to penetrate the Coulomb barrier far below the necessary kinetic energy needed to overcome it [ 28]. Thus, tunnelling is essential for thermonuclear reactions which would otherwise not occur in stars [ 23, 29, 30]. This implies that insolation habitable worlds exist due to quantum tunnelling because stellar energy production during quiescent burning can only occur via this phenomenon.

But there is an additional story. We have seen that the evolution of complex eucaryotic, multicellular life on earth required a steady insolation over billions of years. What makes such a long duration of sunlight production possible? Let´s have a closer look at the tunnelling probability It is proportional to

with v as the velocity of the nuclei. This formula [ 25] shows that the tunnelling probability decreases steeply with lower thermal velocities and with a higher product of the charges . One important consequence is that at relatively low stellar core temperatures (as in the case of the sun) a significant tunnelling probability arises only for nuclei pairs with a low value of . The lowest value of is given for reactions between hydrogen nuclei (protons). For proton-proton reactions (pp-reaction) at solar core conditions is of the order of 10⁻²⁰ [ 23]. Such a low value of the tunnelling probability gives an indication of the long lifetime which the sun and lower mass stars have on the "main sequence" (quiescent hydrogen burning within the stellar core). However, the tunnelling probability is not equal to the reaction probability. Other factors which are of nuclear origin are also relevant for a reaction to occur, such as the conversion of protons into neutrons via the β⁺-decay.

The reaction probability is mainly determined by the nuclear reaction cross section and the velocity distribution of the atomic nuclei [ 29]. The reaction cross section σ(E) for nonresonant reactions is given by

with G as the Gamow factor, S(E) as the Astrophysical S-Factor and E as the energy [ 23]. S(E) is of nuclear origin, and the relevant fact for this discussion is that S(E) depends only very weak on energy. In contrast, exp(-2G) represents the strongly energy dependent part of the cross section because this part is proportional to the tunnelling probability [ 29].

The astrophysical S-factor S(E) influences the value of reaction rates, but is it the tunnelling probability which causes the extreme temperature dependence of the reaction probability. This has an important consequence for low-mass stars, which are the most frequent stars in our galaxy [ 31, 32]: with masses below 1.5 times of the solar mass these stars mainly burn via pp-chain reactions. This is due to the relatively low pressure and thus relatively "low" temperatures within the core. As a consequence of their "low" temperatures, low-mass stars have much smaller reaction cross sections compared to more massive stars because of the strongly energy dependent tunnelling probability. Calculations reveal that at solar core temperature and pressure the reaction cross section of the pp-chain is so small that the mean reaction time for a given proton to react with another proton is 1.4 x 10 ¹⁰ years [ 25]. However, the number of nuclei within the solar interior is so high that there is a sufficient amount of nuclear fusion reactions between protons at any given time to enable a continuous thermonuclear burning. The hydrogen burning continues until all protons in the solar core have been reacted. This will be the case after 1.4 x 10 ¹⁰ years of quiescent hydrogen burning - the lifetime of the sun on the main sequence.

In summary, the analysis of the nuclear reaction cross section and the reaction rate reveals that the strong energy dependence of the tunnelling probability causes very small reaction rates within the core of low-mass stars and thus enables a quiescent hydrogen burning over billions of years - a time span which has proven to be necessary for the evolution of complex life on earth.

The importance of quantum tunnelling for life on planets in insolation habitable zones can thus be summarized as follows:

without quantum tunnelling, thermonuclear reactions would not occur and insolation habitable worlds would not exist

the low tunnelling probability at the conditions of low-mass stars (such as the sun) "generates" the long duration of sunlight required for the evolution of complex multicellular life on earth and probably on other insolation habitable worlds.

3. QUANTUM TUNNELLING AT THE BASIS OF CHEMICAL EVOLUTION

The set of chemical elements which life on Earth uses to build its molecules and to run its biochemistry is relatively small compared to the sum of known elements listed in the periodic table [ 33, 34]. Besides hydrogen, this small set of elements includes carbon, nitrogen, oxygen, phosphorous, sulphur, sodium, potassium, magnesium and iron. These quantitatively most important elements for life are also among the most abundant heavier elements in the universe and the most common heavier elements in the solar system and sea [ 35, 36]. Their high abundance points to a process of element production which is very prevalent in the universe. This process is nucleosynthesis via thermonuclear reactions in stars [ 25, 28]. As we have seen in section 2 that quantum tunnelling is essential for thermonuclear reactions in stars, the origin of the elements heavier than hydrogen that are most important for life is closely linked to the tunnelling phenomenon.

Despite the hot environment of stellar interiors, the nucleosynthesis process requires quantum tunnelling in order to occur because of insufficient thermal energy for overbarrier fusion reactions. But what is the case when there is a general lack in thermal energy such as in the cold regions of the interstellar medium? What kind of reactions and chemical evolution can occur in these regions - the birthplaces of stars and planetary systems - if the chance of overbarrier reactions is low? Let us shed light on dark clouds and other forms of cold interstellar matter.

Interstellar matter is the matter between star systems and is composed mainly of hydrogen and helium. The most abundant elements heavier than helium such as carbon, nitrogen, oxygen, magnesium and iron are injected into the interstellar medium mainly by stellar outbursts, stellar winds and by red super-giant stars. These elements account for about 0.1% of the interstellar matter [ 37]. Dark clouds and neutral diffuse clouds represent the cold part of the interstellar matter with temperatures between 10-100 K and contain micrometer-sized dust grains mixed with the gas [ 37, 38]. The dust content and the density especially of dark clouds is so high (~10 ⁶ atoms cm⁻³) that electromagnetic radiation is unable to reach the inner region of dense clouds (Fig. ) and thus the temperature can drop below 30 K [ 39].

A rich chemistry takes place in such cold interstellar environments producing a large variety of molecules [ 40]. Among these molecules H ₂ (molecular hydrogen) is by far the most abundant one [ 41]. This molecule is very important for various astrochemical and astrophysical processes such as the synthesis of more complex interstellar molecules or the beginning of star formation [ 42, 43]. However, the generation of H ₂ within the interstellar medium has been a long standing problem: the high abundance of H 2 in neutral diffuse and dark clouds is in contrast to the inefficient gas-phase synthesis routes and destruction by ultraviolet photons and cosmic rays [ 43, 44].

What is enhancing the formation of H ₂ in the interstellar clouds so efficiently? Laboratory experiments and astrochemical models point to the importance of the dust grains mixed with interstellar gas for an efficient interstellar H ₂ synthesis. The importance of dust grains is due to their solid surface which is considered to catalyse the formation of H ₂ [ 42, 45, 46]. The first step in this catalytic process is the landing of a hydrogen atom from the gas phase to the morphologically complex surface of a dust grain (adsorption). According to the most familiar route of surface catalysed molecule synthesis, an adsorbed hydrogen atom exchanges energy with the surface and diffuses over the surface until it combines with an already adsorbed hydrogen atom, forming a molecule, which then returns to the gas phase [ 43].

In dark clouds where the temperature can drop below 20 K the mobility of adsorbed hydrogen atoms is low. On the other hand, the high efficiency of molecular hydrogen production observed in dark clouds indicates a relatively high mobility for adsorbed atoms to come close to each other on the surface via diffusion and to react. What helps to solve this dilemma is quantum tunnelling: thermally assisted tunnelling adds to thermal hopping as a route to enable the required mobility of hydrogen atoms. That means that hydrogen atoms diffuse via tunnelling on the dust grain surfaces in order to find a reaction partner after they have been transported by thermal activation in energy sites where quantum tunnelling is efficient [ 37, 43, 45]. This process requires weak interaction forces between the surface and the hydrogen atoms (physisorption) to allow a significant diffusion.

The weak physisorption forces lead to a decrease of efficiency of the described process at higher temperatures as they cannot prevent a fast desorption of H atoms from the surface [ 42]. However, it has been observed that H ₂ production is also very efficient in neutral diffuse clouds and photon-dominated regions where temperatures are higher (30-200 K). How can this be explained? Observations indicate a quantitative correlation between H ₂ and polycyclic aromatic hydrocarbons (PAHs) especially in photon-dominated regions [ 47]. According to recent astrochemical models, H ₂ formation in such an interstellar environment can be catalysed by PAHs via chemisorbed H atoms. A chemisorption of H atoms on PAHs means that there are strong interaction forces between the adsorbed H atom and the substrate which significantly reduces the chance of H desorption due to thermal motion and UV irradiation. But the activation barrier for hydrogen atoms to chemisorb on graphitic surfaces is by far too high to occur classically even at temperatures around 200 K. So how is it possible for H atoms to add to interstellar PAHs? Calculations show that the activation barrier for this reaction is clearly reduced at the edge sites of PAHs compared with the binding of H atoms to central PAH carbon atoms [ 48]. A further help for the addition of H atoms to PAHs can be given by quantum tunnelling of H atoms through the remaining activation barrier. The resulting hydrogen tunnelling rates under the conditions of photon-dominated regions and neutral diffuse clouds have to be included in a full surface reaction network [ 44, 49].

So far we have seen that quantum tunnelling is deeply involved in the astrochemical synthesis of H ₂. The next big step of chemical evolution towards prebiotic chemistry is the formation of water (H ₂ O) as one of the most important molecules for life. Is tunnelling also involved in this step? In order to get an answer we have to go back to the coldest regions of the interstellar medium and ask the dark clouds.

Under ultracold conditions within dark clouds (5-20 K) icy mantles are formed on interstellar dust particles. Observations revealed that H ₂O is the most abundant constituent of such icy mantles [ 50]. As gas-phase synthesis of H ₂O turned out to be too inefficient with respect to the detected abundances it is proposed that H ₂O is formed directly on dust grains via surface reactions [e.g., 51]. Among different surface reactions the following reaction based on already formed H ₂ (see above) is considered to be one of the most common routes [ 52]:

This reaction, however, has a large activation barrier and thus at ultracold temperatures there is a very low probability that OH and H ₂ react via thermal activation. Experiments running under simulated conditions of dark clouds revealed that the efficiency of this reaction is highly influenced by isotope effects. This means that using molecular deuterium (D ₂) instead of H 2 reduces the efficiency of the discussed reaction by about one order of magnitude [ 52]. Such observations point to quantum tunnelling due to the strong dependence of the tunnelling rate on the particle mass. Other relevant reaction pathways for the synthesis of solid H ₂O in dark clouds such as the hydrogen peroxide pathway also depend on quantum tunnelling in order to circumvent the low probability of overbarrier reactions [ 53]. This shows that quantum tunnelling is of high importance for the formation of H ₂ O within dark clouds.

But does tunnelling also play a role in the astrochemical formation of larger, more complex molecules of prebiotic relevance? To answer this question, we must consider carbon monoxide and formaldehyde. Carbon monoxide (CO) is the second most abundant molecule condensed on icy dust grains in dark clouds [ 54], while formaldehyde (H ₂CO) is among the most important prebiotic molecules detected so far in dark clouds. The importance of H ₂CO is mainly due to its role as a precursor for the interstellar formation of amino acids (the building blocks of proteins) and sugars (e.g. ribose, one of the building blocks of DNA) [ 55- 57]. But how is H ₂ CO formed?

CO ice grows on top of a water-rich layer on interstellar dust grains when the density increases in a dark cloud [ 50]. This CO ice can be hydrogenated via the successive addition of hydrogen atoms according to the following sequence [ 58]:

The sequence shows that the consecutive hydrogenation of CO leads to the formation of H ₂CO. But the first addition reaction of the sequence, the CO → HCO reaction, is faced with a huge activation barrier. As a consequence, the low probability of overbarrier reactions under the ultracold conditions of dark clouds would lead to very low addition reaction rates. Such low reaction rates are not consistent with the observed interstellar abundance of H ₂CO. Experimental and theoretical results revealed that under the ultracold conditions of dark clouds quantum tunnelling becomes the dominant transition mechanism for the H + CO addition, leading to substantial reaction rates of H ₂CO [ 59, 60]. The same case of high activation barrier and the importance of quantum tunnelling is also true for the third addition reaction in this sequence as a precursor to the formation of methanol [ 59]. From these hydrogenation examples we see that quantum tunnelling also plays an important role for surface reactions towards more complex molecules with high prebiotic relevance.

In summary, it can be stated that quantum tunnelling is a key process at the basis of chemical evolution towards prebiotic chemistry. The tunnelling phenomenon does not only play a crucial role for astrochemistry in stellar interiors but also for cold regions of the interstellar medium. In neutral diffuse clouds and especially in dark clouds quantum tunnelling boosts various surface reactions on interstellar dust grains towards the synthesis of important prebiotic molecules.

4. SUB-SURFACE HABITABILITY, GEOTHERMAL ENERGY AND QUANTUM TUNNELLING

We have seen in the previous section that interstellar dust particles of micrometer size are effective quantum tunnelling factories for the astrochemical synthesis of molecules which are relevant for prebiotic chemistry. Interstellar dust particles can aggregate within a few million years to the extend that they build large objects such as asteroids, moons or planets. If we zoom out from the scale of micrometers and look at such celestial bodies with sizes from hundreds to thousands of kilometres, will we still see quantum tunnelling effects relevant for prebiotic chemistry and habitability?

Enceladus, one of the icy moons of Saturn, is a very extraordinary moon in the solar system. Despite its relatively small mean diameter of 504 km [ 61] it shows very high endogenic geologic activity. This activity leads to complex surface features and even to plumes of ice particles and water vapour escaping from a system of surface fractures [ 62] (Fig. ). The composition of the ejected particles strongly suggests a liquid water reservoir below the solid surface [ 63]. Such enormous geologic activity could not be caused by gravitational energy because of the small size of this moon [ 64]. The activity is even more surprising if we consider tidal heating as a source of energy and compare the conditions for Enceladus with its adjacent moon Mimas. This icy moon is also a few hundreds of kilometres in diameter. With a smaller distance to Saturn and a higher eccentricity of its orbit than Enceladus the effect of heating via tidal deformation should be higher for Mimas. However, there has been no current geologic or thermal activity detected so far on Mimas [ 65]. So why does Enceladus have such a high activity and probably a liquid water reservoir, making this world potentially habitable? Where does this high amount of internal energy come from?

Recent space probe based investigations and numerical simulations imply that about 60 % of the total mass of Enceladus is rock rather than ice [ 66]. This rock mass fraction is much higher than that of Mimas whose value is only about 20 % [ 67] - a fact which becomes important for our question when we look at radioactive isotopes. Such isotopes are concentrated in the rock mass fraction of planetary bodies and release decay energy. The decay energy causes internal radiogenic heating with an intensity directly proportional to the rock mass fraction of icy satellites [ 68]. Thus, due to the high rock mass fraction of Enceladus, internal radiogenic heating is by far more important for the thermal evolution of this moon than for Mimas. In current models of the thermal evolution of Enceladus, radioactive isotopes with long half-lives play a prominent role. According to these models the decay energy of such isotopes could melt the ice fraction after the formation of Enceladus and lead to planetary differentiation (the creation of a rocky core and a liquid/ice mantle) if favourable conditions were given. But even if the conditions were not so favourable, radiogenic heating of long-lived isotopes could have raised the temperature of the ice fraction up to a level that tidal deformation - and thus tidal heating - became effective to differentiate Enceladus. The long half-lives have kept the internal temperature of this moon above freezing so far by the combination of long-term radiogenic heating and tidal friction [ 64, 67, 69, 70]. This model explains the difference in geologic activity between Mimas and Enceladus: the much smaller rock fraction of Mimas restricts the value of long-term radiogenic heating to a level which cannot facilitate tidal heating [ 67]. The described model for the thermal evolution of Enceladus still leaves several questions open [ 71]. However, the higher rock mass fraction and thus the important role of radiogenic heating by long-lived radioactive isotopes has to be included in all alternative models.

The most important long-lived radioactive isotopes (in terms of their abundance within the rock fraction and amount of released decay energy) are ²³⁵U, ²³⁸U and ²³²Th [ 68]. It is these important heavy isotopes which bring quantum tunnelling back into our discussion. We have seen in section 2 that the fusion of lighter atomic nuclei to form heavier ones is exothermic and releases nuclear binding energy that is equivalent to a mass deficit according to The released binding energy per nucleon increases with increasing mass numbers, but the steps become smaller with higher mass numbers until a maximum is reached at ⁵⁶Fe [ 23]. Heavier isotopes cannot be produced via exothermic fusion reactions (such as thermonuclear fusion) because this would consume energy rather than releasing it. But this fact has its flipside: radioactive alpha-decay. This decay releases energy and thus occurs spontaneously. Alpha decay works with heavy isotopes such as uranium and thorium and can be described with the general formula

where X is the parent isotope, Y the daughter isotope, Z the atomic number and A the mass number [ 72]. The formula shows that alpha-decay transforms the parent nucleus into a daughter isotope by emitting a ⁴He nucleus (He ²⁺). However, when considering at the kinetic energy of emitted He 2+ a paradox becomes evident: the potential barrier between the initial state of the parent nucleus and the final state determined by the separated He ²⁺ and the daughter nucleus is too high for He ²⁺ to overcome the barrier classically [ 73]. This nuclear force barrier which He ²⁺ nucleons feel when trying to separate and escape from a parent nucleus corresponds to the Coulomb barrier for particles which are forced close to a nucleus. The paradox during alpha-decay can thus be resolved similarly to that of the thermonuclear reactions: quantum tunnelling [ 74, 75]. This means that during alpha-decay He ²⁺ particles tunnel through the nuclear force barrier out of a heavy nucleus. By comparing the mass of a parent nucleus with the sum of the masses of its daughter nucleus and the separated He ²⁺ we see a mass deficit of the decay products. The energy-mass equivalence allows us to calculate the released energy during alpha-decay via

where E is the released energy (referred to as the Q-value in particle physics), the mass of the parent nucleus, the mass of the daughter nucleus, the mass of the emitted He ²⁺ and c the velocity of light [ 72]. It is this high amount of released energy (e.g. 4.25 MeV for the decay of ²³⁸ U) which causes the above mentioned internal radiogenic heating of icy satellites.

Our understanding of decaying uranium and thorium nucleons reveals that quantum tunnelling plays an important role in making Enceladus a geologically active and potentially habitable world: He ²⁺ tunnelling causes internal radiogenic heating via the alpha-decay of heavy, long-lived isotopes concentrated in its large rock fraction. This type of heating is high enough that tidal heating becomes effective for Enceladus and contributes to the ongoing high geologic and hydrologic activity which makes this small moon a potentially habitable world.

But is Enceladus one of a kind or could quantum tunnelling turn other solar system bodies into hot spots for our topic?

Let us remain in the Saturnian system for the moment and look at another highly dynamic and enigmatic world - Titan. This moon is outstanding in the solar system due to its similarities with Earth: it has a nitrogen (N ₂) atmosphere like our own planet with a comparable atmospheric surface pressure [ 76]. Abundant evidence for clouds, episodic rainfall, rivers and even lakes have recently been found by the Cassini/Huygens mission [ 77, 78]. However, this world is by far too cold at its surface (about -180 °C) for water to be available in a liquid state for a hydrologic cycle [ 76]. In fact, instead of a water cycle we observe a methane cycle on Titan [ 77]. Methane is a key component in the complex organic chemistry taking place in Titan's upper atmosphere: here, solar ultraviolet radiation and highly energetic particles in Saturn's magnetosphere dissociate methane and initiate the production of larger organic hydrocarbon units such as acetylene, ethane and ethylene [ 79, 80]. These units as well as nitrogenated hydrocarbon species are subsequently coupled into more complex organic compounds which results in the generation of hydrocarbon-nitrile aerosols (tholins) [ 80, 81] (see also Trainer, 2013, in this issue). Such tholin aerosols slowly sink into the deeper atmospheric layers and finally rain to the surface [ 82]. Experiments revealed that complex organic aerosols produced under simulated atmospheric conditions of Titan contain nitrogenated aromatic hydrocarbons in the form of Nucleobases and Amino Acids [ 83]. All these described processes and observations make Titan a hot topic for prebiotic chemistry.

After we have seen how important methane is for initiating the complex organic and possibly prebiotic chemistry in Titan's atmosphere we should search for a reliable source of methane on this moon. The demand is high: methane is constantly consumed by photodissociation and conversion into larger, more complex hydrocarbons as well as by atmospheric escape. The level of consumption is such that the amount of methane in the present day atmosphere would be removed within 30-100 million years [ 84]. This time frame is short compared to Titan's age of about 4500 million years and raised the question about the source of methane supply capable of maintaining its atmospheric content of several percent until the present day. The observed lakes on Titan's surface are no serious suppliers because their stock of liquid methane is too small [ 85]. Now, where does the replenishment of atmospheric methane come from? Could it possibly come from Titan's interior? Let's have a closer look at the internal structure of this moon.

Current models about Titan's internal composition based on space probe measurements describe this icy satellite as a differentiated planetary body with a rock mass fraction of about 50-70 % [ 71]. This high rock mass fraction gives radiogenic heating via the decay of long-lived heavy isotopes a very prominent role in driving the planetary differentiation of this moon and especially in causing and supporting a liquid subsurface water ocean [ 71, 85, 86]. It has recently been shown that most of the methane which was initially incorporated in Titan could still be present by having been basically dissolved in the liquid water ocean: although the solubility of methane in water is low, Titan's proposed subsurface ocean is very massive, taking up more than 50 % of the moon's total volume. Therefore the absolute amount of dissolved methane is high. Outgassing of a moderate level of methane from this massive water ocean could thus be sufficient to support the present-day amount of atmospheric methane [ 85].

When looking at Titan's complex organic chemistry from our quantum point of view we see that tunnelling initiates and supports the chemical evolution in its atmosphere in two different ways: the endogenic support is caused by He ²⁺ tunnelling due to the energy release of radioactive alpha-decay. This radiogenic heat has created and mainly supports the proposed subsurface liquid water ocean which stores methane by dissolution such that moderate outgassing can compensate its consumption through prebiotic atmospheric chemistry. The exogenic support can be traced back to nuclear tunnelling which enables thermonuclear reactions in our sun and therefore causes the UV irradation required for the photolysis of atmospheric methane. Thus, both the quantum tunnelling of a particle out of a nucleus (the source of internal radiogenic heat) and the tunnelling of a particle into a nucleus or forming a nucleus (the source of solar UV irradation) are important factors which turn Titan into a planetary reaction vessel for prebiotic chemistry.

Leaving the Saturnian system, we examine several other planetary bodies which demonstrate He ²⁺ tunnelling. Further prominent examples of worlds with possible subsurface water oceans such as the asteroid Ceres [ 87], the large icy satellites of Jupiter [ 88- 90], Neptune's moon Triton [ 91] and some large Kuiper belt objects [ 92] tell us that internal long-lived radionuclide heating via quantum tunnelling is important throughout the solar system.

What about our own planet? Could He ²⁺ tunnelling be of any importance for prebiotic chemistry and life on Earth? In the first chapter we have seen that sunlight drives most ecosystems either directly or indirectly via oxygenic photosynthesis or its products. But there is also a dark side of life: hidden deep below the photosphere in places such as the deep sea or the crustal lithosphere some species and ecosystems exist completely independent from sunlight. How do they do that? As life needs free energy to avoid thermodynamic equilibrium and because sunlight as a source of free energy is not available, alternative sources must be tapped.

One strategy pursued by anaerobic green sulphur bacteria is to run anoxygenic photosynthesis using geothermal light [ 93]. Deep-sea geothermal light is caused basically by thermal radiation from high-temperature hydrothermal vents which transfer geothermal heat from the lithosphere to the surface [ 94]. Systems of hydrothermal vents can be found at various deep sea locations such as mid-ocean ridges, hotspot volcanoes and seamounts [ 95]. The hot fluids emanating from such submarine hydrothermal vents create several gradients between the vents and the surrounding sea water ranging from centimetres to metres. Such gradients are described in terms of temperature, redox and pH values and also by chemical gradients caused by large amounts of reduced elements and gases (e.g. Fe ²⁺, H ₂, H ₂S, CO ₂ and CH ₄) dissolved in the hot fluids [ 95- 97].

These far-from-equilibrium conditions are very attractive for theories about prebiotic chemistry and the origin of life at hydrothermal vents [ 97] (see also Lemke, 2013, and Mast et al., 2013, in this issue) and point to another strategy life uses to tap resources of free energy other than sunlight: transferring energy from the geothermal source of hydrothermal vents to higher trophic levels by anaerobic chemolithoautotrophy [ 95]. With this strategy organic material is produced from CO ₂ and energy is gained by redox reactions between inorganic chemicals supplied by hydrothermal fluids.

Now what is the source of geothermal energy that fuels the described submarine habitats with anaerobic chemolithoautotrophic micro-organisms at the base of the food chain? Recent measurements of the flux of geoneutrinos (electron antineutrinos produced by the decay of radioactive isotopes in Earth's mantle) made with the KamLAND detector of the Kamioka Neutrino Observatory in Japan allow the contribution of radiogenic heat to the total heat flux from Earth's interior to the surface to be estimated. The results indicate that about 50% of Earth's heat flux is currently caused by the decay of ²³⁸U and ²³²Th [ 98, 99]. This means that a very substantial part of the geothermal energy that generates thermodynamic disequilibrium conditions at hydrothermal vents and thus allows micro-organisms to live completely independently from sunlight can be traced back to He ²⁺ tunnelling.

But settling down in deep-sea hydrothermal vent environments is not the only way for life to turn its back on the photosphere: fracture zones within Earth's crust extending to depths of several kilometres below the land surface have been proven to be lithospheric habitats that are completely decoupled from the photosphere and its products [ 100]. Here, bacterial life has found a way to link anaerobic chemolithoautotrophy directly with the radioactive decay of uranium: the radiolysis of water and bicarbonate molecules induced by uranium oxides generates protons, molecular hydrogen and highly reactive molecular species which interact with various minerals in the surrounding. The products of these interactions are consumed by the bacteria for sulfate reduction, carbon and nitrogen fixation, ATP synthesis and other metabolic processes [ 101].

In summary, when looking at He ²⁺ tunnelling, we see a very important factor for Earth's habitability outside the photosphere. This fact has a vital consequence: the potential subsurface oceans of various planetary bodies all around the solar system - oceans created and/or supported by the impact of He ²⁺ tunnelling - become highly interesting in terms of chemical evolution, prebiotic chemistry and exobiology.

5. QUANTUM TUNNELLING IN MOLECULAR BIOLOGY

In the previous sections we have seen how quantum tunnelling is important for the topics environment and synthesis. In this section I want to focus on the aspect of "function". This aspect is involved in the origin of organized complexity of organic molecules such as DNA, proteins and cofactors. Unlike arbitrary complexity (e.g. the chaotic assembly of small crystals to a stone), organized complexity means that a certain molecular structure is formed via specific information and correlates to a specific function within a network (cell, organism). Could quantum tunnelling have a role in these functional molecules?

DNA is a functional polymer with an extraordinary property: the polymerization of nucleotides to DNA is not arbitrary but contains specific information. The information is the result of evolution and codes for protein synthesis as well as other functions required to build an organism. This genetic information is written as specific sequences of nucleic acid base pairs and is transferred via autocatalytic replication to new nucleotide polymers. However, the genetic code in DNA is not completely stable even in the absence of extrinsic damaging factors: a universal spontaneous point mutation bias can be detected with respect to transitions of the base pairs G:C to A:T and A:T to G:C which cannot be sufficiently explained by well-known mutation models such as UV irradiation, oxidative damage or CpG hypermutation [ 102]. This implies that this kind of point mutation has to be regarded as an inherent characteristic of DNA. How is this possible?

Recent experiments with deuterated bacterial spores revealed that the rates of spontaneous point mutations in cells which are grown in culture media with deuterium oxide (D ₂O) are lower as compared to spontaneous mutations rates in water (H ₂O) based culture media [ 103]. This means that we see a dependence of the mutation bias on the involved particle mass. In section 1 we have seen that quantum tunnelling rates strongly depend on the particle mass. We have also seen in section 3 that this dependence is used to check if quantum tunnelling is important for certain chemical reactions (isotope effect). Could the detection of an isotope effect on spontaneous mutations rates tell us that quantum tunnelling plays a significant role in the universal point mutation bias in DNA?

The Löwdin DNA mutation model [ 104] pays attention to the quantum nature of protons involved in the hydrogen bonds between complementary base pairs. Löwdin's model is based on the fact that stable Watson-Crick base paring and thus stable genetic information requires the precondition that protons don't change their position in the base pairs. The position of protons in a base pair is defined by the canonical form of the involved nucleic base molecules. However, the Löwdin model states that due to quantum tunnelling through proton transfer barriers, the position of protons within a nucleic base can change with a small probability when the protons are in a quantum mechanically nonstationary state immediately after a DNA replication. As a consequence of quantum tunnelling, the canonical form of a nucleic base is transformed into its tautomeric form which doesn't bind to the initial base pair partner. But the tautomeric form can bind to a different base. This process and subsequent steps ultimately leads to point mutations in DNA.

The Lödwin model describes the universal point mutation bias as a result of quantum tunnelling of protons in DNA. This means that proton tunnelling influences the main function of DNA - the reliable storage of genetic information. Publications referring to the Löwdin mutation model have become numerous in recent times. A most recent collection of theoretical studies is given by Cooper [ 105], and a collection of genomic analysis reports supporting the Löwdin model has been made by Lu [ 103].

Ironically quantum tunnelling is not only discussed as an intrinsic damaging factor to DNA but is also involved in DNA repair [ 106]. The key factor which enables DNA repair is electron tunnelling. How does such a repair mechanism work? Ultraviolet irradiation can damage DNA by inducing the formation of bulges in DNA strands. These bulges are caused by the dimerisation of adjacent pyrimidines on a DNA strand in reaction to UV irradiation. The flavoprotein photolyase is able to repair such deformed DNA by splitting the covalentely linked pyrimidines via electron transfer so they can return to their normal monomeric form and no longer form a dimer. To do this repair job, photolyase needs visible light.

It sounds strange that light is used to repair a light induced damage, but the situation becomes clear if we look at the wavelengths: one of the cofactors of photolyase is an antenna complex which absorbs light with longer wavelength (blue) than the UV radiation. The absorbed light is important as it leads to an excitation of the antenna complex. The excitation is then transferred to the other cofactor of photolyase, the redox cofactor FADH (Flavin Adenine Dinucleotide). The function of FADH⁻ is to transfer the electron to the antibonding molecular orbital of the DNA pyrimidine dimer. After the transferred electron has done its repair job to split the dimer it returns to the redox cofactor. But how is the electron transferred from FADH⁻ to the pyrimidine dimer? A closer look at the coupling between photolyase and DNA reveals an unexpected situation: FADH- couples most strongly to the dimer with its adenine part. The coupling parameter influences the redox reaction rate which shows that adenine is indeed involved in the electron transfer process. But, however, it is the flavin part of the FADH⁻ where the electron is located prior to its transfer. When considering the intramolecular redox conditions of FADH⁻ and experimental results on the structural behaviour of FADH⁻, the electron transfer cannot be explained by a jump of the electron from the flavin part to the adenine part and then to the dimer (multistep hopping mechanism).

So, how does the redox reaction take place in such a situation? The answer is superexchange-mediated tunnelling over long distances [ 106- 108]. This process makes it possible that quantum tunnelling of electrons in proteins can occur over distances of up to three nanometres [ 109, 110] although such a potential barrier width is much larger than possible widths in vacuum (see section 1). But low-lying electronic states provided by the protein via a superexchange-mediated mechanism boost electron transfer rates so they are still relatively high over such large distances [ 108]. Applied to our example, this means that an electron is directly transferred from the flavin part to the dimer via single-step long-range quantum tunnelling. The repair of UV-damaged DNA by the protein photolyase is thus essentially based on long-range electron tunnelling.

The above mentioned quantum tunnelling processes in molecular biology have been chosen as examples for a detailed discussion as they clearly demonstrate that both nuclear tunnelling and electron tunnelling can influence and enable biomolecular functions and that functions of both DNA and proteins/cofactors are involved.

But there are more such cases of quantum tunnelling in biology. Long-range electron tunnelling is also crucial for many other types of proteins with functions based on redox-reactions [ 111]. Important examples are cellular respiration [ 112] and photosynthesis [ 113]. In enzyme-catalysed reactions hydrogen tunnelling is widely discussed [ 114]. Enzymes are primarily proteins with specific catalytic functions. This means that they enhance the rate of specific reactions in biomolecular syntheses relative to reaction rates in their absence [ 115, 116]. How are they doing this is still not well understood and a comprehensive model of catalysis is one of the big challenges in chemical biology [ 117, 118]. The lack of a comprehensive model causes intense debates about e.g. the role of protein dynamics [ 115, 116] and the role of hydrogen tunnelling in enzyme-catalysed reactions [ 118- 120]. Now, what can be stated so far about hydrogen tunnelling in enzyme-catalysed reactions? Large isotope effects are observed in many enzymatic reactions that involve hydrogen transfer [ 117, 121, 122]. These observations indicate that tunnelling occurs in enzymatic reactions. But what is its role? A most recent and thorough review [ 118] concludes that in enzyme-catalysed reactions tunnelling is probably only weakly catalytic. This means that it is not significantly enhancing reaction rates. But hydrogen tunnelling is an important component of reaction rates so the reactions would otherwise be much slower.

In summary tunnelling has guided us via prominent examples to biochemistry where quantum tunnels are important infrastructures in the world of functional molecules: spontaneous mutations in DNA, enzymatic reactions and especially energy transduction pathways are highly influenced, supported and even enabled by quantum tunnelling. The transition probability of the tunnelling concept from the field of organic chemistry [ 123] to biochemistry [ 124] has significantly increased within the past years while, at the same time, the conceptual barrier between chemistry and biology has become smaller and has begun to disappear [ 125] (see also Pross, 2013, in this issue).

CONCLUSION

This review was a round trip from hot stellar interiors to cold interstellar medium, from deep lithospheres and subsurface oceans to planetary upper atmospheres and from the microcosm of biomolecular nanomachines to the evolution of macroscopic multicellular life. At each stop we have seen that quantum tunnelling is of vital importance for life and its origin. These stops are not isolated but rather closely connected with each other. The following example further clarifies this concept by focusing on photosynthesis as an example.

Photosynthesis is basically performed by functional biomolecules whose redox reactions rely on long-range electron tunnelling. All elements which build these biomolecules (except hydrogen) were produced by thermonuclear reactions in stars via nuclear tunnelling. Photosynthesis in deep sea environments uses geothermal light where half of its energy can be traced back to the effect of He ²⁺ tunnelling during radioactive alpha decay of heavy elements within the lithosphere. In contrast, the energy source of oxygenic photosynthesis is sunlight which is a product of nuclear tunnelling in thermonuclear reactions. Our sun itself was formed by the collapse of dark interstellar clouds. In this process, molecular hydrogen plays an important role and is formed in dark clouds by the help of quantum tunnelling. Oxygen, the important byproduct of oxygenic photosynthesis, is used by multicellular lifeforms for cellular respiration - a process which relies on long-range electron tunnelling. The evolution of all multicellular lifeforms is based on DNA mutations and energy input over billions of years. A source of mutations is proton tunnelling in DNA while the large period of time of solar energy input is due to specific characteristics of quantum tunnelling taking place in thermonuclear reactions.

Although this example does not cover every single topic previously discussed, it illustrates the main message of this review: there is a highly multidisciplinary network of quantum tunnels essential for the origin and evolution of life. Tunnelling is essential because it

provides different sources of constant energy flux over a long period of time and thus makes highly advanced complexification in molecular and biological evolution possible,

opens synthesis pathways for prebiotic astrochemical reactions which would otherwise not occur,

enables or influences specific functions of biomolecular nanomachines maintaining the process of life.

ACKNOWLEDGEMENTS

I thank Brian Silver from the Florida Institute of Technology for proofreading the manuscript.

CONFLICT OF INTEREST

The author confirms that this article content has no conflicts of interest.

REFERENCES

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Quantum Tunnelling to the Origin and Evolution of Life

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Quantum tunnelling is a phenomenon which becomes relevant at the nanoscale and below. It is a paradox from the classical point of view as it enables elementary particles and atoms to permeate an energetic barrier without the need for sufficient energy ...

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This Is How Mastering Dark Matter Could Take Us To The Stars

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This Is How Mastering Dark Matter Could Take Us To The Stars

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To accelerate a spacecraft, there needs to be an impulse imparted on it, which at this point in time has been most successful using combustion reactions, however, this is a very limiting form of travel

Some forms of creating energy for spacecrafts are: chemical, nuclear, nuclear fission, and antimatter

Chemical reactions used to thrust spacecrafts forward are most commonly used, however it is only 0.0001% efficient, and the fuel has to be carried on board and moved along with the spacecraft, adding more mass

Matter-Antimatter annihilation is 100% efficient, which is what we look for in the ideal fuel, but it is incredibly complicated and resource heavy to create antimatter, the world having only ever made less than a microgram in total

Using dark matter to fuel spacecrafts, there would be an infinite amount of fuel with 100% efficiency because of its abundance

There would be a finite probability 2 dark matter particles will annihilate if you make them interact with each other

If dark matter is a bosonic particle (particles with integer spins rather than half-integer spins), it would be its own antiparticle → It annihilates with itself

Dark matter is abundant wherever you go in the universe, meaning theoretically, you would never run out of fuel and could infinitely travel through the universe

Next Question:

What are the limitations of using dark matter as rocket fuel? Could there be limitations with mixing with normal matter?

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Dark matter is one of the biggest mysteries in all of modern science. Everywhere we look on large cosmic scales - from low-mass galaxies to the largest galaxy clusters, from the cosmic microwave background to the cosmic web that traces the Universe's structure - we can see the imprints and effects of its presence. For every proton's mass worth of normal matter, there's five times as much dark matter, out-massing and out-gravitating the conventional stuff that makes up everything we've ever directly detected.

Even though we have yet to directly detect it, and even though we aren't sure exactly what its true properties are, dark matter holds a tremendous promise for the future of humanity. Ubiquitously located all throughout the galaxy and far beyond, dark matter could be the perfect fuel that makes our interstellar dreams come true. Here's the story of how.

Whenever humanity sets our sights on exploring the depths of space, there are constraints we can't avoid: the laws of physics. In order to accelerate a spacecraft - or any mass - you have to impart an impulse to it in order to change its momentum. The larger the impulse, the more you can change an object's speed. All that determines the magnitude of an impulse is how much force you apply and how long you apply it for.

In a conventional rocket, that impulse is provided by rocket fuel that undergoes a combustion reaction, which produces impulse in the form of thrust. Although this is the best method humanity has come up with for space travel so far, it's incredibly limiting. All of our past and current rockets are chemical-based, unfortunately, and that places tremendous constraints on how far we've been able to go.

The reason for this is simple: in order to produce thrust - i.e., in order to impart an impulse to your spacecraft - you have to convert that stored chemical energy in the fuel into kinetic energy that pushes your spacecraft. In order to generate that energy, however, you have to use up some of that fuel you're carrying with you.

The key to getting lots of thrust, and therefore lots of acceleration, is fuel efficiency. Certain types of fuel are more energy-efficient than others, meaning that we can get more energy (and thrust, and acceleration) out of, say, 1 kilogram of some types of fuel. An easy way to think about this is through Einstein's most famous equation: . If you had a perfect, ideal fuel, it would convert 100% of your fuel's mass into energy, enabling you to make the most efficient fuel imaginable.

At most, though, chemical-based reactions are somewhere around 0.0001% efficient. The reason is as follows: chemical reactions rely on electron transitions between atoms and molecules. Most of an atom's mass is in the form of protons and neutrons, each of which have a mass that contains around 10 ⁹ eV worth of energy. Electron transitions, however, are on the order of a few (typically 1-10) eV of energy. Even with all the chemical-based tricks we can perform, there are no known reactions that allow us to improve on this.

Sure, we can go for some type of nuclear fuel, but that's only marginally better, achieving efficiencies of around 0.1%. It's a huge improvement, if we can realize it, but there's still a fundamental problem with accelerating to speeds that will carry you interstellar distances on reasonable timescales.

The key problem is as follows: whenever you burn fuel, you have to accelerate the entire mass of your spacecraft, including any fuel that's still on board.

Read that again: including any fuel that's still on board.

In other words, let's imagine you can shoot exhaust out of your vehicle at an incredible rate: 100,000 mph (about 160,000 kph), relative to the rocket itself. If you start off with a rocket where 99% of your initial mass is fuel, and you assume that your fuel is perfectly 100% efficient (as though it were pure matter-antimatter annihilation), you'd wind up with a final speed of 460,000 mph (740,000 kph). Even at this record-setting rate, it would still take thousands of years to reach the nearest star.

On the other hand, there's another approach to interstellar travel that could - in principle - make our science-fiction dreams come true. Instead of bringing your fuel with you, what if you collected it as you went? Typically, ideas like this involve enormous magnetic fields that funnel charged particles into some sort of "trap" in your spaceship, allowing you to put nuclei and electrons together where you can then extract energy and perform further reactions with them.

But dark matter offers a tremendous advantage over normal matter in this regard. Why? Because you don't have to do anything special to collect it. It's literally everywhere, distributed in an enormous halo surrounding and encompassing every large galaxy we know of, including the Milky Way. If we find ourselves anywhere in the galaxy, there's bound to be dark matter lying around.

The second tremendous advantage comes from the progression away from chemical-based rockets and towards the idea of a perfect fuel. For chemical-based rockets, 0.0001% energy efficiency is the best we can hope for. For nuclear-based rockets, fission power might get us up to 0.1% efficiency, and nuclear fusion might get us a little further: perhaps up to 0.7%.

The ideal configuration would be to use matter-antimatter annihilation, which is 100% energy efficient. The downside of matter-antimatter annihilation comes with a terrible cost, though: it takes a tremendous amount of work, energy, and effort to create an extraordinarily small amount of antimatter. If you took all the particle physics laboratories ever constructed on Earth and added up all the antimatter humanity has ever created, from Fermilab to CERN, you'd wind up with less than a microgram of antimatter.

Sure, might be the most efficient way to extract energy from mass in the entire Universe, as it represents perfect efficiency. But even if you manage to contain and store your antimatter successfully and annihilate it only at the proper moment, you'll still have a finite supply of fuel that required an incredible amount of energy to collect. Once you use up this perfect fuel, you're all out, and all you can do is travel at a constant velocity through space for an indefinite duration of time. Even if we could generate an arbitrary amount, we'd still be fundamentally limited with an antimatter rocket.

That's why the promise of a dark matter fuel source is so alluring. Not only might dark matter be an unlimited fuel source (in terms of abundance) that we don't have to carry on board with us, but it might have that perfect, 100% efficient matter-to-energy conversion potential we so strongly desire.

There are a multitude of experiments looking for the collisions of dark matter with both normal matter and itself. In general, there are two types of particles in the Universe: fermions (with half-integer spins) and bosons (with integer spins). If dark matter is a bosonic particle with no electric, color, or weak charge, that would mean it behaves as its own antiparticle.

If you can collect two dark matter particles and make them interact with one another, there's a finite probability that they'll annihilate. When an annihilation occurs, they'll produce pure energy in a 100% efficient fashion: via Einstein's . In other words, if we understand dark matter correctly, there's a free, unlimited source of energy everywhere humanity dreams of going.

Because dark matter is everywhere, we wouldn't even need to carry it with us as we traversed the Universe. As far as we understand it - and admittedly, we need to understand it a lot farther - dark matter could truly deliver our dream of the ultimate fuel. It's abundant all throughout our galaxy and beyond; it should have a non-zero annihilation cross-section with itself; and when it does annihilate, it should produce energy with 100% efficiency.

Perhaps, then, most of us have been thinking about experiments seeking to directly detect dark matter all wrong. Yes, we want to know what makes up the Universe, and what the physical properties of its various abundant components truly are. But there's a science-fiction dream that could come true if nature is kind to us: unlimited, free energy just waiting there for us to harness, no matter where in the galaxy we go.

Mastering dark matter is the endeavor that just might make it so.

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This Is How Mastering Dark Matter Could Take Us To The Stars

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Dark matter is one of the biggest mysteries in all of modern science. Everywhere we look on large cosmic scales - from low-mass galaxies to the largest galaxy clusters, from the cosmic microwave background to the cosmic web that traces the Universe's structure - we can see the imprints and effects of its presence.

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We have never seen dark matter and dark energy. Why do we think they exist?

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We have never seen dark matter and dark energy. Why do we think they exist?

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5% of the universe is made of known matter and energy, meaning 95% is made up of invisible energy and matter

The new Euclid telescope is being used to map out dark matter and energy

Dark Matter

The formation of galaxies since the big bang wouldn’t have been possible with only gravity from visible matter

In the 1930s, the existence of dark matter was first observed and they have since been predicted to make up 85% of the universe

Dark matter is believed to consist of particles similar to neutrinos and don’t have much interaction with visible matter

Dark Energy

Dark energy was discovered in 1998, following the discovery of the acceleration of the expansion of the universe

The presence of dark energy was not obvious earlier in time because the original energy from the big bang was the cause of most of the universe’s expansion

Dark energy has a constant density as the universe expands, meaning is continues to create energy as the energy is scattered throughout more space

The current models which we are basing predictions and observations on are based on Einstein’s theory of general relativity, so future discoveries on the existence of dark energy (or rather its lack of presence) could prove this theory wrong

What new technology is being made to learn more about dark energy? How is it different from the current technology and telescopes being used?

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Scientists think that only about 5% of the stuff that makes up the universe is known to us.

Yes, you are reading that right. Cosmological models show that invisible energy and matter must make up a combined 95% of the universe for it to work according to existing theories.

The invisible part of the consists of two components, dark energy and dark matter. The latter is somewhat less mysterious and scientists are fairly certain of its existence: The rules of simply wouldn't work without it. as we see them wouldn't have been able to assemble to their current sizes in the time they've had since the if visible matter was all there was. In fact, many galaxies wouldn't even hold together as they do, if the gravity of visible matter was the only force pulling them together.

Europe's new space telescope Euclid will attempt to map this invisible stuff, but what if its findings don't fit expectations? A lot is at stake, including Einstein's famed and widely accepted theory of general relativity.
Related: How will Europe's Euclid space telescope see into the dark universe?
The first observations indicating that dark matter must be prevalent in the to prevent things from flying apart were made in the 1930s. Since then, astronomers have found that this invisible dark matter must make up at least 85% of all matter in the universe. Scientists have been trying for decades to detect this invisible matter, which, they think, must consist of some sort of particles, maybe akin to the inert that barely interact with the visible world.

The situation with dark energy is, however, more complicated. The discovery of this elusive force dates back to 1998 when astronomers found that the expansion of the universe is accelerating. The expansion itself was initially driven by the energy from the Big Bang, but since the Big Bang happened so long ago (13.8 billion years ago, to be exact), this expansion should be slowing down.

To make up for this conundrum, theorists postulated that some mysterious force, dark energy, must exist that acts against gravity and drives matter apart. Cosmological models show that dark energy accounts for 68% of all energy in the cosmos, according to NASA. But astronomers admit that the evidence for its existence is a bit vague.

"[The acceleration of the expansion] of the universe doesn't make sense when you think that there's just gravity there," Isobel Hook, a professor of astrophysics at Lancaster University in the U.K. and a scientist, told Space.com. "It shouldn't be slowing down. So the fact that we observe it getting faster means that there must be something else. And we just call that thing dark energy because we don't really know at all what it is."

Hook was a member of the team that made that 1998 discovery and since then has been wondering, just like many other scientists, what this dark energy actually might be.

"We really don't know," she said. "The best theory is that it's probably some kind of property of space itself, like a kind of energy that's everywhere, at all times, but there's no explanation for what that should be."

Models show that this energy must be distributed evenly across the universe and that it has always had the same strength. In the universe's early days, the effects of dark energy were not as visible, because the original 'kick' from the Big Bang was driving most of the universe's expansion. Some 5 to 6 billion years ago, however, dark energy became "a predominant force, acting against the gravity of both, the normal and dark matter," Giuseppe Racca, Euclid project manager at the European Space Agency (ESA), told Space.com

"It's a little bit unsettling, that this energy appears to have a constant density as the universe expands," Racca said. "It means that [additional] energy is created because the density is constant. It is not diluted by the universe becoming bigger. So it means that there is additional energy."

There is, however, also the possibility that there is no such thing as dark energy after all, and that the cosmological models that require its existence are wrong. These models are based on Albert Einstein's theory of general relativity that describes the physical 'rules' of the universe in a series of equations. If future observations by the Euclid telescope (and other upcoming telescopes designed to study dark energy) suggest that dark energy is not such a constant, omnipresent force, that would mean that this famed theory is not entirely correct.

"We know that general relativity works very well at the level of a system," said Racca. "But there is a possibility that it doesn't work all that well at cosmic scales and therefore dark energy is not needed."

Euclid will look for the evidence of dark energy by mapping the distribution of galaxies 10 billion years back in time and comparing how this distribution has changed throughout the evolution of the universe. The observations will reveal whether there really appears to be a constant omnipresent force in action or whether perhaps something else is going on.

"We might find that we don't see [the effects of] this energy constant everywhere, but we see some differences somewhere or in time," said Hook. "Then things will get really interesting and it really would be revolutionary, because we'd have to completely change our theories."

And that would likely mean Nobel Prizes for the discoverers. The Euclid telescope, launched on July 1 by a SpaceX Falcon 9 rocket, will take six years to map one third of the sky outside our galaxy, the Milky Way. Only then will astronomers have enough data to know for sure.

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We have never seen dark matter and dark energy. Why do we think they exist?

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We have never seen dark matter and dark energy, so why do we think they exist? And what would happen if Europe's new space telescope Euclid finds evidence that they don't exist after all?

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What is Dark Energy? Why is it a Necessary Part of the Universe and What Are It's Properties?

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What is Dark Energy? Why is it a Necessary Part of the Universe and What Are It's Properties?

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Dark Energy

Dark energy is a hypothetical reason for why the universe is expanding at such a fast rate and is thought to work similarly to "anti-gravity". This means rather than pulling objects closer together, it stretches them apart and expands the fabric of the universe

Dark energy is between 68% and 72% of the universe's mass and energy and doesn't seem to affect galaxies and stars, other than physically drawing objects apart from one another

An object 100 million light years away will be traveling 21500 km/second away, this rate increases significantly as objects are farther and farther away

Interestingly enough, Gravity acts stronger on objects than dark energy at a smaller scale

Galaxies and stars do not become thinner as the distance between them expands because of dark matter

Dark Matter

What is dark matter? Candidates for what it can be are: a fifth force that causes negative pressure which expands the universe, particles that instantly come in and out of existence, and a vacuum energy

We don't know what dark matter is, we can only make guesses based on what its effects are on the universe

Without dark matter, most believe the universe would have imploded as large galaxies would grow closer to one another

Next Question:

How might Dark Matter and Dark Energy be connected? What are we currently doing to better our understanding of what Dark Energy is and come to a solid conclusion?

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The dominating form of energy in the cosmos, dark energy is driving the accelerating expansion of the universe, but its nature remains a complete mystery.

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Dark energy remains a mystery. Maybe AI can help crack the code

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Dark energy remains a mystery. Maybe AI can help crack the code

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Research Conducted by the University College London

Artificial Intelligence (AI) allowed them to study dark energy with greater precision and efficiency than traditional methods of research while having less data

Created a supercomputer simulation of the universe using ratios of dark matter to regular matter to understand how dark energy influences the universe

AI was employed to showcase the effects of dark energy in the past 7 billion years on a specific area of the universe by creating a map

Using AI makes finding observations and conclusions quicker, often drawing our attention to patterns within data we regularly would not notice

This AI map created represented data for 100 million galaxies, something that would require a ton of data (

The map was then used to better align cosmic evolution models according to dark energy dynamics

This project would require 4 times as much data to be executed with as much accurateness

Researchers at ETH Zurich (Other Article)

Used Machine Learning models that use facial recognition to estimate the amount of dark matter there is in a particular location of the universe. This is done by looking for "signs of dark matter".

Dark matter is invisible to telescopes, however, it bends light rays, distorting images of galaxies. This and the mass of the sky are used to build maps and compare to predictions previously made regarding how much dark matter is present.

Maps are then related to one another and AI is used to find patterns among the maps.

Deep neural networks are then used to extract as much information as possible and optimize for “dark energy signs” to make maps as accurate as possible.

Question:

How are scientists ensuring that the technology we use to research Dark Energy is accurate? As we do not have any solid theory concerning dark energy and are currently in the early stages of using Machine Learning in astrophysics, how do we measure the accuracy of our conclusions?

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Dark energy remains a mystery. Maybe AI can help crack the code

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"If you wanted to get this level of precision and understanding of dark energy without AI, you'd have to collect the same data three more times in different patches of the sky."

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Practical Applications of Quantum Tunneling

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Practical Applications of Quantum Tunneling

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Quantum tunneling, despite being a fundamental aspect of quantum mechanics, has found numerous practical applications across various fields. Some notable applications include:

Scanning Tunneling Microscopy (STM): STM utilizes the quantum tunneling phenomenon to image surfaces at the atomic level. A sharp metal tip is brought very close to a sample surface, and a small voltage bias is applied. Electrons can tunnel through the vacuum between the tip and the sample, allowing researchers to map out the surface topography with incredible resolution.

Tunnel Diodes: Tunnel diodes are electronic devices that exploit quantum tunneling to create a high-speed switching action. By carefully engineering the materials and geometry of the diode, engineers can control the tunneling process to create devices with applications in high-frequency oscillators, microwave amplifiers, and low-noise amplifiers.

Quantum Computing: Quantum tunneling plays a crucial role in quantum computing architectures. Quantum bits, or qubits, can exist in superpositions of states, allowing them to tunnel through potential energy barriers. This property enables quantum algorithms to explore multiple possibilities simultaneously, potentially leading to significant speedups in certain computational tasks.

Nuclear Fusion: Quantum tunneling is essential for nuclear fusion processes in stars and potential fusion reactors. In nuclear fusion reactions, positively charged atomic nuclei must overcome a repulsive Coulomb barrier to fuse together. Quantum tunneling allows some nuclei to overcome this barrier by tunneling through it, facilitating the fusion process.

Enzymatic Reactions: Quantum tunneling has been implicated in enzymatic reactions in biochemistry. Certain chemical reactions catalyzed by enzymes involve the transfer of electrons or protons through potential energy barriers. Quantum tunneling can enhance the efficiency of these reactions by allowing particles to tunnel through barriers, reducing the need for high activation energies.

These applications demonstrate the versatility and importance of quantum tunneling in various scientific and technological endeavors.

Question: How do quantum computers utilize quantum tunneling to perform computational tasks, and what advantages does this offer over classical computing?

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A lion uses quantum tunneling to pass through a barrier - a feat impossible in classical physics

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What is the Difference Between Classical and Quantum Physics?

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What is the Difference Between Classical and Quantum Physics?

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Many of quantum physics’ concepts challenge the norm set by classical physics, but the main points are as follows:

In classical physics, objects must be particles or waves. In quantum physics, however, objects are neither

Unlike in classical physics, it is essentially impossible to predict actions in the quantum realm

Thus, the trajectory of a quantum object can no longer by found using normal physics equations, but instead using the Schrodinger equation

In quantum physics, every measurement is random and probabilistic to some degree

In addition to this, quantum objects have a number of properties that would be impossible according to classical physics:

Quantum Entanglement:

Two quantum particles become “tangled”, gaining the ability to transfer information between them

This information transfer doesn’t happen through communication, however. The two particles essentially act as the same object, existing in two places at the same time

Quantum Tunneling

A quantum particle performs an action without the energy required to perform it

For example: a quantum physics ball could (with a small chance) roll through a hill without the energy required to push it over, whereas a classical physics ball would never reach the other side.

This violates the classical law of energy conservation

Next Question: What were the earliest experiments that led to the discovery of quantum mechanics?

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Energy distribution of the universe, showing the difference in prevalence between dark energy and dark matter

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The short answer: They really aren’t connected.

The longer answer:

Despite sharing similar names, dark matter and dark energy are not connected besides their shared invisibility and difficulty in observation.

Dark matter is, well, matter. It is responsible for the much of the mass of the universe at large and helps keep galaxies organized. Due to its mass, it increases forces of gravity at a universal scale, drawing the universe closer together and slowing its expansion.

So, with all these forces keeping the universe together, how is its expansion accelerating?

Interestingly, this is where dark energy comes in. Not only is it a different type of thing altogether (energy, not matter), but it actually has the opposite effect on the universe. In addition to this, dark energy is a lot stronger and more prevalent, making up about 68 percent of the energy that exists. Overpowering the force of gravity and dark matter, it pushes objects apart and expands the universe.

In conclusion: Despite their similar names, dark matter and dark energy are actually comically different. One is matter, one is energy. One vastly overshadows the other in prevalence in power. And, of course, they exert completely opposite forces on the universe!

Next Question: Will dark energy continue to accelerate the expansion of the universe forever? Will it eventually slow down, stop, or even reverse? What would happen to the universe in these cases?

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Quantum Tunnelling: Learn its Definition & Applications

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Quantum Tunnelling: Learn its Definition & Applications

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Quantum Tunneling

Quantum tunneling is a phenomenon which breaks the rules of classical mechanics, involving the penetration of potential energy barriers

Subatomic particles are found on the other side of physical obstacles, able to reconnect there

Transmission through a barrier is dependent on the barrier’s height and width, only able to have a thickness of 1-3 nm

Because of the uncertainty regarding the exact location of an electromagnetic particle, they are able to dey the laws of classical physics

Applications

Scanning Tunneling Microscope: allows view at atomic level by using a conductor tip at the surface of the object

Nuclear Fusion: the temperature at the centre of the site of fusion is not enough to pass the Coulomb barrier and carry out the reaction. The use of tunneling increases the probability of the particles getting through the barrier

Quantum Biology: tunneling plays a key part in many biochemical processes such as, photosynthesis, cell respiration, and DNA mutation

Next Question:

How has quantum tunneling changed our knowledge of classical mechanics? What further development is being studied on the use of tunneling?

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Quantum Tunnelling: Know Definition and Applications with FAQs

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Learn about quantum tunnelling, its definition and its applications like scanning tunnelling microscope, Nuclear Fusion, Electronics, Quantum biology & FAQs

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Scientists make surprising new discovery about dark energy

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Scientists make surprising new discovery about dark energy

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First observations and theories

Edwin Hubble was the first to observe distant galaxies in the 1920s and discovered that the Universe was expanding

Afterwards, in 1988 studies showed that galaxies began a phase of accelerating expansion

The source of this expansion was referred to as “dark energy” which acts as the opposite of gravity, repulsing things from it

Dark energy suppresses the merging of large cosmic objects which would’ve formed because of gravity

New data

Later on, the Max Planck Institute used the eROSITA X-ray space telescope in 2019 to check galaxy clusters

The telescope found approximately 500 galaxy clusters, making it one of the biggest cluster samples in the past 10 billion years

Researchers I-Non Chiu (from National Cheng Kung University, Taiwan), Matthias Klein, Sebastian Bocquet and Joe Mohr (from Loyola Marymount University, USA) teamed up to use the data formerly mentioned and others from across the globe to characterize the galaxy clusters

The researchers also measured the masses using gravitational lensing

The first cosmological study of dark energy using galaxy clusters was done thanks to combining the data

Calculations and Theories Using the New Data

It was discovered that 76% of the total energy density in the Universe was made of dark energy when comparing all the datasets and predictions

Calculations have been done to point out that dark energy remains uniform in space and constant in time using its energy density

Using all this observational evidence, it is suggested that dark energy is described as a cosmological constant

Next Question:

How can we visualize/model what dark matter or dark energy is? How does this help our understanding?

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Using the eROSITA space telescope, scientists have discovered that dark energy is uniformly distributed in space and time.

When Edwin Hubble observed distant galaxies in the 1920s, he made the groundbreaking discovery that the Universe was expanding. However, in 1988, further studies indicated that it had also begun a phase of accelerating expansion.

"To explain this acceleration, we need a source," explained Joe Mohr, an astrophysicist at Ludwig-Maximilians University (LMU). "We refer to this source as 'dark energy,' which provides an 'anti-gravity' to speed up cosmic expansion."

The existence of dark energy and cosmic acceleration is a surprise. This indicates that our current understanding of physics is either incomplete or incorrect.

Now, I-Non Chiu from National Cheng Kung University in Taiwan, in collaboration with LMU astrophysicists Matthias Klein, Sebastian Bocquet, and Joe Mohr, has published a first-of-its-kind study. To do this, they used the eROSITA X-ray telescope, which focuses on galaxy clusters.

The work, titled ' Cosmological constraints from galaxy clusters and groups in the eROSITA final equatorial depth survey,' is published in Monthly Notices of the Royal Astronomical Society.

What impact does dark energy have on the Universe?

The anti-gravity, possibly caused by dark energy, pushes objects away from each other and suppresses the formation of large cosmic objects that would otherwise form due to gravity.

Therefore, it affects where and how the largest objects in the Universe form, including galaxy clusters with total masses ranging from 10 ¹³ to 10 ¹⁵ solar masses.

Klein said: "We can learn a great deal about the nature of dark energy by counting the number of galaxy clusters formed in the Universe as a function of time, or in the observational world as a function of redshift."

Galaxy clusters are extremely rare

One issue the team needed to overcome was the rarity of galaxy clusters.

To discover them, surveys of a large portion of the sky using the most sensitive telescopes in the world are required. The eROSITA X-ray space telescope - a project led by the Max Planck Institute for Extraterrestrial Physics (MPE) in Munich - was launched in 2019 to carry out an all-sky survey to search for galaxy clusters.

In the eROSITA Final Equatorial-Depth Survey (eFEDS), a mini-survey designed for performance verification of the subsequent all-sky survey, about 500 galaxy clusters were found. This represents one of the largest samples of low-mass galaxy clusters to date and spans the past 10 billion years of cosmic evolution.

Chiu and his colleagues used an additional dataset on top of the eFEDS data - optical data from the Hyper Suprime-Cam Subaru Strategic Program. This is led by the astronomical communities of Japan and Taiwan, and Princeton University.

The researchers used this data to characterise the galaxy clusters in eFEDS and measure their masses using the process of weak gravitational lensing. The combination of the two datasets enabled the first cosmological study using galaxy clusters to study dark energy.

A surprising discovery

Through comparison between the data and theoretical predictions, the results show that dark energy makes up around 76% of total energy density in the Universe.

Moreover, the calculations indicated that the energy density of dark energy appears to be uniform in space and constant in time.

Bocquet commented: "Our results also agree well with other independent approaches, such as previous galaxy cluster studies as well as those using weak gravitational lensing and the cosmic microwave background."

All pieces of observational evidence, including the latest results from eFEDS, suggest that dark energy can be described by a simple constant, usually referred to as the 'cosmological constant.'

"Although the current errors on these constraints are still larger than we would wish, this research employs a sample from eFEDS that occupies an area less than 1% of the full sky," Mohr concluded.

This first analysis has therefore laid a solid foundation for future studies of the full-sky eROSITA sample, as well as other cluster samples.

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Scientists make surprising new discovery about dark energy

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Using the eROSITA space telescope, scientists have discovered that dark energy is uniformly distributed in space and time.

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The Idea of HYPER

Robert McGehee, Aaron Pierce and Gilly Elor, researchers at the University of Michigan (USA) and University of Mainz (Germany) have a new model for Dark Matter called HYPER

HYPER stands for HighlY interactive ParticlE Relics

The HYPER model believes that in the initial stages of the universe, after dark matter was formed, the interactions of dark matter and normal matter were at a sudden increase in strength

This idea allows for not only the possibility of dark matter to be detectable today but also explains all of the dark matter scattered throughout the universe

Physicists have been looking for heavy dark matter particles (WIMPS), however, they’ve been unsuccessful which led to a pivot onto lighter alternative dark matter particles

The HYPER model has been the closest we’ve been to picturing a phase transition which helps make dark matter easier to detect

The usual problem with other models is dark matter and normal matter interacting with each other too strongly, because then what’s formed is too minuscule, contradicting astrophysical observations

On the other hand, if the amount formed is the exact perfect amount, the interaction would be too weak to detect with the experiments being done

How it works

Interactions with dark matter and normal have been facilitated using a certain particle (mediator)

The interaction depends on the weight of the mass; the dark matter and its detection using this mediator is dependent on it

It is essential the mediator is initially heavy so the right sample of dark matter is made and then turned lighter to continue the detection of the dark matter

This process requires the mediator to have a phase transition right after the dark matter is formed

Next Question:

What have models of dark matter and/or dark energy led to in terms of discoveries, concepts and perspectives?

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This NASA Hubble Space Telescope image shows the distribution of dark matter in the center of the giant galaxy cluster Abell 1689, containing about 1,000 galaxies and trillions of stars. Dark matter is an invisible form of matter that accounts for most of the universe's mass. Hubble cannot see the dark matter directly. Astronomers inferred its location by analyzing the effect of gravitational lensing, where light from galaxies behind Abell 1689 is distorted by intervening matter within the cluster. Researchers used the observed positions of 135 lensed images of 42 background galaxies to calculate the location and amount of dark matter in the cluster. They superimposed a map of these inferred dark matter concentrations, tinted blue, on an image of the cluster taken by Hubble's Advanced Camera for Surveys. If the cluster's gravity came only from the visible galaxies, the lensing distortions would be much weaker. The map reveals that the densest concentration of dark matter is in the cluster's core. Abell 1689 resides 2.2 billion light-years from Earth. The image was taken in June 2002. Credit: NASA, ESA, D. Coe (NASA Jet Propulsion Laboratory/California Institute of Technology, and Space Telescope Science Institute), N. Benitez (Institute of Astrophysics of Andalusia, Spain), T. Broadhurst (University of the Basque Country, Spain), and H. Ford (Johns Hopkins University)

Dark matter remains one of the greatest mysteries of modern physics. It is clear that it must exist, because without dark matter, for example, the motion of galaxies cannot be explained. But it has never been possible to detect dark matter in an experiment.

Currently, there are many proposals for new experiments: They aim to detect dark matter directly via its scattering from the constituents of the atomic nuclei of a detection medium, i.e., protons and neutrons.

A team of researchers-Robert McGehee and Aaron Pierce of the University of Michigan and Gilly Elor of Johannes Gutenberg University of Mainz in Germany-has now proposed a new candidate for dark matter: HYPER, or "HighlY Interactive ParticlE Relics."

In the HYPER model, some time after the formation of dark matter in the early universe, the strength of its interaction with normal matter increases abruptly-which on the one hand, makes it potentially detectable today and at the same time can explain the abundance of dark matter.

The new diversity in the dark matter sector

Since the search for heavy dark matter particles, or so-called WIMPS, has not yet led to success, the research community is looking for alternative dark matter particles, especially lighter ones. At the same time, one generically expects phase transitions in the dark sector-after all, there are several in the visible sector, the researchers say. But previous studies have tended to neglect them.

"There has not been a consistent dark matter model for the mass range that some planned experiments hope to access. However, our HYPER model illustrates that a phase transition can actually help make the dark matter more easily detectable," said Elor, a postdoctoral researcher in theoretical physics at JGU.

The challenge for a suitable model: If dark matter interacts too strongly with normal matter, its (precisely known) amount formed in the early universe would be too small, contradicting astrophysical observations. However, if it is produced in just the right amount, the interaction would conversely be too weak to detect dark matter in present-day experiments.

"Our central idea, which underlies the HYPER model, is that the interaction changes abruptly once-so we can have the best of both worlds: the right amount of dark matter and a large interaction so we might detect it," McGehee said.

Constraints in the mediator mass-nucleon coupling plane from cooling of HB stars [25] and SN 1987A [12], as well as rare kaon decays [26] (gray shading). Credit: Physical Review Letters (2023). DOI: 10.1103/PhysRevLett.130.031803

And this is how the researchers envision it: In particle physics, an interaction is usually mediated by a specific particle, a so-called mediator-and so is the interaction of dark matter with normal matter. Both the formation of dark matter and its detection function via this mediator, with the strength of the interaction depending on its mass: The larger the mass, the weaker the interaction.

The mediator must first be heavy enough so that the correct amount of dark matter is formed and later light enough so that dark matter is detectable at all. The solution: There was a phase transition after the formation of dark matter, during which the mass of the mediator suddenly decreased.

"Thus, on the one hand, the amount of dark matter is kept constant, and on the other hand, the interaction is boosted or strengthened in such a way that dark matter should be directly detectable," Pierce said.

New model covers almost the full parameter range of planned experiments

"The HYPER model of dark matter is able to cover almost the entire range that the new experiments make accessible," Elor said.

Specifically, the research team first considered the maximum cross section of the mediator-mediated interaction with the protons and neutrons of an atomic nucleus to be consistent with astrophysical observations and certain particle-physics decays. The next step was to consider whether there was a model for dark matter that exhibited this interaction.

"And here we came up with the idea of the phase transition," McGehee said. "We then calculated the amount of dark matter that exists in the universe and then simulated the phase transition using our calculations."

There are a great many constraints to consider, such as a constant amount of dark matter.

"Here, we have to systematically consider and include very many scenarios, for example, asking the question whether it is really certain that our mediator does not suddenly lead to the formation of new dark matter, which of course must not be," Elor said. "But in the end, we were convinced that our HYPER model works."

The research is published in the journal Physical Review Letters.

More information: Gilly Elor et al, Maximizing Direct Detection with Highly Interactive Particle Relic Dark Matter, Physical Review Letters (2023). DOI: 10.1103/PhysRevLett.130.031803

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Dark matter remains one of the greatest mysteries of modern physics. It is clear that it must exist, because without dark matter, for example, the motion of galaxies cannot be explained. But it has never been possible to detect dark matter in an experiment.

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The universe is not only expanding, but accelerating in this expansion at this point in time

It was expected that the universe’s expansion would slow down and eventually, stop, but because of the presence of dark energy, the universe continues to rapidly expand

Dark energy fills the empty space in our universe, pushing everything away from each other, creating more space for dark energy to fill

Physicists have been studying to measure the Hubble Constant, which is the rate at which the universe is expanding

The calculations of the Hubble Constant based on the data from the early years of our universe were not matching the recent studies on it, the mismatch of answers being called Hubble Tension

Scientists have not found a certain answer to what will happen to the universe, however there are some theories

If the universe continues as it is, the universe and everything in it will eventually go cold and die out since no new stars will form

The strength of dark energy could possibly change as well and accelerate even more, causing speeds which will cause the universe to tear itself apart

Both these possibilities would mean the universe will continue to expand infinitely

There is a possibility that dark energy works very different than we imagine, resulting in the slowing down of the universe’s expansion and eventual halt, starting to recompress, and then re expanding

Next Question:

How is the Hubble Constant being calculated? What may be the areas of fault in these calculations causing Hubble Tension? How are we continuing to research?

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Scientists debate what the future of the cosmos looks like and whether space will ever stop getting bigger and bigger

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A diagram outlining the information used in calculations of the Hubble Constant.

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The Discovery of the Hubble Constant

In the 1920’s, Edwin Hubble used his experience with telescopes to make measurements of Cepheid variable stars (stars that vary in diameter and temperature)

These measurements were used to calculate their distance from Earth, but also allowed the discovery that distant stars were moving away faster than ones nearby.

Using this new observation, Hubble and Milton Humason calculated the relationship between the distance to a star and the speed that it is moving away from our planet - this is the Hubble constant!

The original estimate was 500km/s/Mpc, much higher than what it is agreed to be today

Through improvements on Hubble’s methods, the more accurate Hubble constant was determined to be about 70/km/s/Mpc

Even still, different methods for measuring this constant disagree with one another, so science still has a long way to go

Measuring the Hubble Constant

To find the Hubble constant, scientists must find the distance to an object in space, as well as how fast it is moving from the observer

Using the Doppler effect, scientists can find the velocity of the object relatively easily

Similar to its effect on sound, the Doppler effect stretches light, increasing its wavelength and “redshifting” it

Astronomers can measure how much the light shifted and use it to find how fast the object is traveling away from the observer

The main source of inaccuracy is the distance between Earth and an object, which is much harder to measure

Many methods have been used to find this distance, each giving different results (This is Hubble Tension!)

Some methods include:

Comparing the brightness of an object to how bright it looks from Earth

Analyzing the strength of gravitational waves

Modeling the cosmic microwave background radiation

As technology continues to advance, astronomers will continue to inch closer to the true value of the Hubble constant. Some are working on a new generation of telescopes, others continue to refine their analysis of gravitational waves - our understanding of this Hubble value will be in constant evolution.

Next Question: What other applications exist for the Doppler effect as it "redshifts" light (or other things)? Does it allow us to learn anything else about our universe?

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Quantum Tunneling: Defying Classical Limits with Q

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Quantum Tunneling: Defying Classical Limits with Q

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Key Information of Quantum Tunneling

Quantum Tunneling (part of quantum mechanics) does not align with classical physics mainly because particles can exist by going through barriers even if they cannot go over barriers. This is because particles can exist in more than one place at the same time

The thinner and shorter the barrier, the higher the probability of the particle going through the barrier

An external electrical field or reducing the temperature helps in the tunneling process. Lowering the temperature means particles need less thermal/activation energy

Tunneling can also help in catalyzing reactions through enzymes or be used to make semiconductor walls thinner, in order to transfer electrons faster and easily

Classical Mechanics Limits of Quantum Tunneling:

Particles will take the path of least resistance through a barrier, but this is not true for quantum particles

Energy conservation does not apply to quantum particles as it does in classical mechanics. This is because particles do not have defined trajectories, rather they exist in multiple states at once

Analogy: (Classical Mechanics) The Ball rolls uphill and comes back down if it does not have enough energy to overcome the hill vs (Quantum Mechanics) The Ball will tunnel through the hill and make it to the other side

The ball exists on both sides of the barrier with a specific probability of being on each side even if it does not have enough energy to overcome

The Quantum Viewpoint:

It may seem counterintuitive to tunnel through a barrier for a particle that already doesn't have enough energy to overcome a barrier. This is why we have to understand the probabilistic nature of the quantum viewpoint (particles have superpositions and don't have to be either on one side or the other)

Heisenberg Uncertainty Principle: We do not know both the position and momentum of a particle accurately at the same time, it is always somewhat uncertain. That is why we can also consider the circumstance that the particle has passed through the barrier

The Wave-Particle duality concept: particles also have wave-like behavior and can tunnel through the barrier by diffracting around it, this is crucial as depending on how we observe and measure a particle, it can be acting like a particle or acting like a wave

Particles will take an infinite amount of time to overcome an infinitely expanding barrier, however, there is a finite amount of time a particle will take to tunnel through a barrier, this allows us to explore the concept of tunneling time

Next Question: How can we use Quantum Tunneling to predict/model the probability of an ultimate fate of the universe such as the Big Crunch or Big Freeze?

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***Post Based on Dark Energy being a Constant 

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Could Dark Energy Wind Up Destroying The Universe?

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Could Dark Energy Wind Up Destroying The Universe?

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Big Bang was incredibly dense and hot at the beginning, expanding at a rate that perfectly balanced out the effects of gravity

The Expansion rate changes over time because the expansion causes the universe to become less dense while also cooling the universe as the radiation loses its energy

3 scenarios:

The universe collapses because gravitational attractive force overcomes the expansion of the universe

Expansion overcomes gravitation and ends in a Big Freeze

Expansion of the universe and gravitation balance out as the universe never collapses but expansions are ceased

Dark Energy as a Constant

Measuring the shift in light that travels from galaxies can help us measure the change in expansion

Galaxies observed in the 1990s were seen as receding rapidly and then eventually slowing down. Based on this information, it was predicted that we could tell if the universe would collapse based on how the slowdown of the recession speed

However, through the collection of data from observing many galaxies, scientists found that galaxies were not receding at a slower rate but were instead receding at a faster rate away from us

^ This is where dark energy, another component of the universe was conceptualized. Based on the current data collected, many believe that dark energy is a constant because there is no change in its energy density.

Based on the theory that dark energy is constant, this will also mean that as the universe expands, and its volume increases, there is more energy in the universe.

Structures in the grand scheme of things will expand further apart, yet the Local Groups and Milky Way will remain the same as they were previously bound together through gravity. Eventually, the Milky Way and Local Groups will look like one giant galaxy as the universe expands. With consideration to this information and experiment, there is a possibility that the universe will end up in a Big Freeze (heat death where the universe has no thermodynamics and can no longer sustain entropy)

Next Questions:

What are the possible ultimate fates of the universe if Dark Energy is not a constant, but rather changes with the expansion of the universe?

Are current experiments being conducted on the signs of Dark Energy and whether it is constant or not?

What leads scientists to believe that dark energy is constant, despite this invalidating the Law of Conservation of Energy?

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Could Dark Energy Wind Up Destroying The Universe?

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Our ultimate fate depends on something we’ve barely begun to measure.

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Harnessing Quantum Tunneling for Quantum Computing

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Harnessing Quantum Tunneling for Quantum Computing

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Quantum computers leverage the phenomenon of quantum tunneling to perform computational tasks in a way that offers several advantages over classical computing methods:

1. Qubits and Superposition: Quantum computers use qubits as their basic units of information, which can exist in superposition, representing both 0 and 1 simultaneously. Quantum tunneling allows qubits to explore multiple computational paths simultaneously, thanks to superposition.

2. Quantum Gates and Algorithms: Quantum gates manipulate qubits, similar to classical logic gates, but exploit quantum properties like tunneling. Quantum algorithms, such as Shor's algorithm and Grover's algorithm, utilize quantum tunneling to solve certain problems much faster than classical algorithms.

3. Exponential Speedup: Quantum computers offer the potential for exponential speedups over classical computers for specific tasks. This is because quantum algorithms can leverage quantum tunneling and other quantum phenomena to process information in parallel, exploring a vast solution space simultaneously.

4. Quantum Annealing: Quantum annealers, a type of quantum computing architecture, utilize quantum tunneling to optimize solutions to optimization problems. By encoding the problem into the configuration of qubits and controlling the system's parameters, quantum annealers exploit quantum tunneling to find optimal solutions efficiently.

Quantum tunneling enables quantum computers to perform computations in a fundamentally different way from classical computers, offering the potential for significant speedups and breakthroughs in various fields of science and technology.

Next question: What are some of the challenges and limitations facing the practical implementation of quantum computers, and how are researchers addressing them?

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Abstract

We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian \(H_A(F)\) which is specifically constructed to solve a K-SAT instance F. The heuristic algorithm aims at iteratively reducing the Hamming distance between an evolving state \({|{\psi _j}\rangle }\) and a state that represents a satisfying assignment for F. Each iteration consists on the evolution of \({|{\psi _j}\rangle }\) (where j is the iteration number) under \(e^{-iH_At}\), a measurement that collapses the superposition, a check to see if the post-measurement state satisfies F and in the case it does not, an update to \(H_A\) for the next iteration. Operator \(H_A\) describes a continuous time quantum walk over a hypercube graph with potential barriers that makes an evolving state to scatter and mostly follow the shortest tunneling paths with the smaller potentials that lead to a state \({|{s}\rangle }\) that represents a satisfying assignment for F. The potential barriers in the Hamiltonian \(H_A\) are constructed through a process that does not require any previous knowledge on the satisfying assignments for the instance F. Due to the topology of \(H_A\) each iteration is expected to reduce the Hamming distance between each post measurement state and a state \({|{s}\rangle }\). If the state \({|{s}\rangle }\) is not measured after n iterations (the number n of logical variables in the instance F being solved), the algorithm is restarted. Periodic measurements and quantum tunneling also give the possibility of getting out of local minima. Our numerical simulations show a success rate of 0.66 on measuring \({|{s}\rangle }\) on the first run of the algorithm (i.e., without restarting after n iterations) on thousands of 3-SAT instances of 4, 6, and 10 variables with unique satisfying assignments.

Rebekah Herrman, Phillip C. Lotshaw, ... George Siopsis

Danylo Lykov, Jonathan Wurtz, ... Yuri Alexeev

Taichi Kosugi, Hirofumi Nishi & Yu-ichiro Matsushita

Introduction

K-SAT is a most important problem in computer science which is known to be \(\mathsf {NP-Complete}\) for all \(\text {K}>2\) 1,2. We focus on the set of satisfiable instances of the K-SAT, i.e. those instances for which there is at least one assignment that returns a logical 1. Satisfying a K-SAT instance may be easy or require very significant efforts, it all depends on its number of variables and clauses. Instances with one or few satisfactory assignments are harder to solve because satisfactory assignments live in a set with cardinality \(2^n\), where n is the number of binary variables upon which the instance is built. Those K-SAT instances for which there is only one satisfying assignment constitute the focus of this paper.

A central activity in quantum computing consists of using quantum mechanical properties as resources in order to build algorithms that may outperform their classical counterparts. Along these lines, we find several quantum algorithms that have been designed to solve K-SAT instances3,4,5. In this paper, we present a quantum algorithm in which continuous quantum walks and quantum tunneling are explicitly used as computational resources.

Originally designed to model quantum phenomena6,7,8,9, quantum walks are an advanced tool for building quantum algorithms (e.g.10,11,12,13,14,15,16,17,18) that has been shown to constitute a universal model of quantum computation19,20. Quantum walks come in two variants: discrete and continuous21. The continuous-time quantum walk was proposed by Childs et al11 and the continuous time quantum walk on a hypercube was first studied by Kendon and Tregenna22 and later extensively analyzed by Drezgich et al23.

Quantum tunneling is a quantum mechanical phenomenon in which a particle passes through a potential barrier. The probability of finding a particle inside the potential barrier decreases exponentially, which in turn increases the probability of finding the particle outside the potential24, as if the particle were expelled from the barrier. This behavior suggests that quantum tunneling can be used as an algorithmic tool in which researchers can assign a meaning to the potential barriers in the context of the problem being solved. So far and to the best of our knowledge, quantum tunneling has only been used as a computational resource as a passive byproduct of quantum phenomena, i.e. as a physical phenomenon that comes out as a result of quantum evolution and that is utilized as a free variable upon which no explicit manipulation or control is performed. This is indeed the case of the most celebrated computational use of quantum tunneling as a component of quantum annealing25,26, and for alternative methods for hybrid search based on quantum annealing27,28,29. The closest to the use of tunneling in our algorithm is the work by Kechedzhi et al.30, where they use tunneling as a part of a quantum subroutine. Furthermore, the effects of potential barriers in discrete and continuous quantum walks in the quantum walk versions of the Grover's algorithm were studied in31 but only as a type of noise that can be avoided, not as an algorithmic resource.

In this paper, we present a proposal for using quantum walks and quantum tunneling together, under explicit manipulation, in a quantum heuristic algorithm aimed at solving hard instances of the K-SAT problem. Our algorithm can deal with instances with multiple satisfying assignments but we have chosen to focus on instances with a unique satisfying assignment as instances with a few or unique satisfying assignments tend to be harder and the most important. For our testing we randomly generated, using a uniform distribution, thousands of 3-SAT instances of 4, 6, and 10 variables with one satisfying assignment. Such instances can be found in our repository32. We do not make any claim regarding computational speed-up. Our only claim is, based on the algorithm, theoretical results and simulations & experiments run on digital computers, that the use of quantum tunneling and quantum walks to solve K-SAT instances with unique solution is a promising approach.

The order in this paper is the following. The second section explains the " K-SAT problem", " Continuous time quantum walk on a hypercube graph" shows the behavior of a quantum walk in a hyper cube graph, " Formal description of our algorithm" presents a formal description of the algorithm, its parameters, and a more complex example, " Implementation details" presents details on the algorithm implementation, the sixth section presents the " Analytical approximation of the algorithm behavior", " Numerical calculation of parameters and additional simulations" presents numerical simulations and the calculation of the parameters of \(H_A\), and " Conclusions " presents conclusions, observations and further work.

K-SAT problem

The K-SAT problem is one of the most important problems in computer science. It is derived from the SAT problem, the first problem proved to be \(\mathsf {NP-complete}\) 1. K-SAT is a constraint satisfaction problem where a Boolean formula has to be satisfied, i.e. its result has to be true. A K-SAT instance is written in conjunctive normal form (CNF), that is, a conjunction of clauses, where a clause is a disjunction of literals, and a literal can be a logical variable or its negation.

Formally speaking, the K-SAT problem can be stated as follows: given a K-SAT instance P, is there an satisfying truth assignment for P? In other words, the K-SAT problem consists of finding assignments that satisfy a given CNF Boolean formula, where K is the number of literals in each clause. For example:

$$\begin{aligned} P({\mathbf {x}})=\left( x_1\vee \overline{x}_3\vee \overline{x}_5\right) \wedge \left( x_2\vee \overline{x}_3\vee \overline{x}_6 \right) \wedge \left( x_1\vee \overline{x}_4\vee x_6 \right) \end{aligned}$$

where \( \vee \) is the logical symbol for OR, \( \wedge \) is the logical symbol for AND, \(\{x_1, \ldots x_6\}\) is the set of logical binary variables, and \(\overline{x}_j\) is the negation of \(x_j\). This is a 3-SAT instance since each clause contains exactly 3 variables, and the binary string \({\mathbf {x}}_s=[1,1,0,0,1,0]\) is one of the assignments that satisfy \(P({\mathbf {x}})\). This can be verified by substituting \({\mathbf {x}}_s\) into \(P({\mathbf {x}})\)

$$\begin{aligned} P({\mathbf {x}}_s)=\left( 1\vee 1\vee 0\right) \wedge \left( 1\vee 1\vee 1 \right) \wedge \left( 1\vee 1\vee 0 \right) =1 \end{aligned}$$

Satisfying a K-SAT instance P can be easy, difficult or impossible, depending on the number of bit strings that make P true. For example, let us suppose we have a K-SAT instance P defined over n binary variables. If we run a brute-force approach on P, we may simply substitute each and every bit string from \({\mathbf {x}}_0 = 000 \ldots 0\) to \({\mathbf {x}}_{2^n-1} = 111 \ldots 1\) on P, only stopping either when finding \(P({\mathbf {x}}_s) = 1\) or when we have substituted all bit strings from \({\mathbf {x}}_0\) to \({\mathbf {x}}_{2^n-1}\) without finding any satisfying assignment. If P has many satisfying assignments, we will eventually find one of those assignments, possibly sooner than later. However, if P has only a few satisfying assignments (along these lines, the worst case scenario would be having only one satisfying assignment), finding them will eventually happen but it may well take a substantial (i.e. exponential) amount of time. If no bit string from \({\mathbf {x}}_0\) to \({\mathbf {x}}_{2^n-1}\) satisfies P, then P is unsatisfiable but it took us an exponential amount of resources to find that out.

As stated above, the K-SAT is an element of the set of \(\mathsf {NP-Complete}\) problems, which in turn is a subset of \(\mathsf {NP}\) (the abbreviation of "Non-deterministic Polynomial Time"), the set of problems in which a solution can be tested in polynomial time. Being an \(\mathsf {NP-Complete}\) problem means that an arbitrary instance of any problem under the \(\mathsf {NP}\) category can be rewritten as an instance of an \(\mathsf {NP-Complete}\) problem in a polynomial number of steps2,33,34. The first problem proved to be \(\mathsf {NP-Complete}\) was the SAT problem, and later the K-SAT, which is a variation of SAT, was proven to be also \(\mathsf {NP-Complete}\) for \(\text {K}>2\) 1,2,34,35,36. If an algorithm that runs on a Deterministic Turing Machine in polynomial time is discovered for an \(\mathsf {NP-Complete}\) problem, it implies every \(\mathsf {NP}\) problem can be solved in polynomial time, hence proving \(\mathsf {P}=\mathsf {NP}\).

Continuous time quantum walk on a hypercube graph

An n-dimensional hypercube graph can be constructed by assigning each vertex to one of the \(2^n\) possible n-variable binary combinations, and connecting the nodes that have a distance of 1 in the Hamming distance (the number of positions/bits at which the corresponding elements between two arrays are different). The Hamiltonian \(H_A\) used in our algorithm describes the dynamics of a quantum walk over a hypercube with potential barriers. The reason why we chose to use the hypercube graph is to reduce the distance between \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) with every iteration of the algorithm by taking advantage of the hypercube topology, where closer states have a shorter Hamming distance. During the evolution of an n-qubit state \({|{\psi _j}\rangle }\) ( j is the iteration number) under \(e^{-iH_At}\), most of the probability flux will tunnel through the paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\), and after a time \(t_f\) the system is measured and the superposition collapses. If the post-measurement state \({|{\psi _{j+1}}\rangle }\) is different from \({|{s}\rangle }\), there is a good chance \({|{\psi _{j+1}}\rangle }\) is a state in a tunneling path from \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) which thanks to the hypercube topology will be at shorter Hamming distance to \({|{s}\rangle }\), \(h_H({|{\psi _{j+1}}\rangle },{|{s}\rangle })<h_H({|{\psi _{j}}\rangle },{|{s}\rangle })\).

The structure of the the adjacency matrix A( n) for an n-dimensional hypercube can be written as

where \({A_j}=\bigotimes ^{n}_{i=1}O_{ij}\) and \( O_{ij}= {\left\{ \begin{array}{ll} \hat{\sigma }_x &{}\text { if } i=j\\ I &{}\text { if } {i\ne j} \end{array}\right. } \)

If a state that is the tensor product of n qubits in the computational basis evolves under the unitary operator \(e^{-iA(n)t}\) (which is equivalent to evolving under \(H_A\) without potential barriers), the probability at some time of measuring a state \({|{z}\rangle }\), that is also an n-qubit tensor product, can be calculated as follows. Firstly, we need to realize that the Hermitian matrices \(A_i\) commute with each other, for example

$$ \begin{aligned} [A_{1} ,A_{3} ] = & (\hat{\sigma }_{x} \otimes I \otimes I \otimes ...)(I \otimes I \otimes \hat{\sigma }_{x} \otimes ...) - (I \otimes I \otimes \hat{\sigma }_{x} \otimes ...)(\hat{\sigma }_{x} \otimes I \otimes I \otimes ...) \\ {\text{ = }} & (\hat{\sigma }_{{\text{x}}} \otimes {\text{I}} \otimes \hat{\sigma }_{{\text{x}}} \otimes ...) - (\hat{\sigma }_{{\text{x}}} \otimes {\text{I}} \otimes \hat{\sigma }_{{\text{x}}} \otimes ...) \\ [{\text{A}}_{{\text{1}}} ,{\text{A}}_{{\text{3}}} ]{\text{ = }} & {\text{0}} \\ \end{aligned} $$

which means we can express \(e^{-iA(n)t}\) as a product of unitary operators \(e^{-iA(n)t}=e^{-iA_1t}e^{-iA_2t}...e^{-iA_nt}\). Then, by expanding \({e^{-iA_jt}}\) into a Taylor series and gathering the terms, we obtain

$$ \begin{aligned} e^{{ - iA_{j} t}} & = I^{{ \otimes n}} \cos (t) - iA_{j} \sin (t) \\ & = I^{{ \otimes n}} \cos (t) - i(I \otimes ... \otimes \hat{\sigma }_{x} \otimes ... \otimes I)\sin (t) \\ & = I \otimes ... \otimes (I\cos (t) - i\hat{\sigma }_{x} \sin (t)) \otimes ... \otimes I \\ e^{{ - iA_{j} t}} & = I \otimes ... \otimes e^{{ - i\hat{\sigma }_{x} t}} \otimes ... \otimes I \\ \end{aligned} $$

which means \({e^{-iA_jt}}\) is equivalent to only evolving the i th qubit under \(e^{-i\hat{\sigma }_xt}\). As a result, we get \(e^{-iA(n)t}=e^{-iA_1t}e^{-iA_2t}...e^{-iA_nt}={\bigotimes _{j=1}^{n}e^{-i\hat{\sigma }_x t}}\).

The probability \(P_z(t)\) of measuring a state \({|{z}\rangle }\) from \({|{x}\rangle }\) evolving under \(e^{-iA(n)t}\) depends only on the Hamming distance between \({|{x}\rangle }\) and \({|{z}\rangle }\). If we evolve \({|{x}\rangle }={|{0}\rangle }^{\otimes n}\), the probability of finding an arbitrary state \({|{z}\rangle }\) is given by

where \(n_0\) and \(n_1\) are the number of 0's and 1s appearing in \({|{z}\rangle }\) respectively, meaning \(n_1\) is also the Hamming distance between \({|{x}\rangle }\) and \({|{z}\rangle }\).

Formal description of our algorithm

Our algorithm is formally introduced in ALGORITHM 1. A key element of our algorithm is the construction of the Hamiltonian \(H_A(F,{|{\psi _j}\rangle },j)\), which determines the dynamics under which a state \({|{\psi _j}\rangle }\) will evolve during the jth iteration of the algorithm. Each iteration of the algorithm consists of the evolution of a state \({|{\psi _j}\rangle }\) during a time \(t_f\) (this is the measurement frequency, for the cases analyzed in this paper \(t_f=\frac{3}{2}\pi \) as discussed in " Numerical calculation of parameters and additional simulations") followed by a measurement that collapses the superposition, and then a check of the post measurement state \({|{\psi _{j+1}}\rangle }\) to see if it represents a satisfying assignment for F (the checking can be done in O( mK)), in case it does not, \(H_A\) gets updated for the next iteration.

The Hamiltonian \(H_A\) for the j th iteration is

where A( n) is the adjacency matrix of an n-dimensional hypercube graph and n is the number of variables in the instance F, which can be constructed using Eq. ( 1), and

is the potential part of \(H_A\). The components of V are:

\({|{\psi _j}\rangle }\) is the tensor product of n qubits in the computational basis. \({|{\psi _j}\rangle }\) represents an assignment \(\psi _j\) for the instance F. \({|{\psi _0}\rangle }\) is an initial random state, while \({|{\psi _j}\rangle }\) for \(j>0\) is the post measurement state for the j th iteration.

\(\alpha = \alpha (F,\psi _j)\) is the number of clauses in F that are not satisfied by the assignment represented by \({|{\psi _j}\rangle }\).

\(M=M(F)\) is a diagonal matrix that has as the element corresponding to \({|{a}\rangle }{\langle {a}|}\) the number of clauses in F not satisfied by the assignment represented by \({|{a}\rangle }\). It is worth noting that the elements of M( F) arise naturally from a construction process (shown later in this section) without any prior knowledge on the satisfying solutions of F.

\(\gamma (n,m,K)\) and \(\delta (n,m,K)\) are both positive real parameters dependent in the number of variables n, the number of clauses m, and the number of literals per clause K in the instance F. The reason of their dependency on n, m, and K is shown later in this section. \(\gamma \) increases an amount \(\delta \) every iteration which, as we explain later in this section, helps with the amplification of the state \({|{s}\rangle }\) that represents a satisfying assignment for F for an evolution during a time \(t_f\) (the measurement frequency). We do not have an analytical expression for \(\gamma \) and \(\delta \), so we have numerically estimated their values for some cases. The values and their effects are discussed in more detail in " Effective Hamiltonian via degenerate perturbation theory" and " Numerical calculation of parameters and additional simulations ".

The rationale behind the elements of \(H_A\) is the following. The potential V of Hamiltonian \(H_A\) is a diagonal matrix with potential values specific to each state. The potential barriers between the states are designed to increase the tunneling between states \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\). The reason most of the tunneling occurs between \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) is because the elements in the potential V corresponding to the outer products \({|{\psi _j}\rangle }{\langle {\psi _j}|}\) and \({|{s}\rangle }{\langle {s}|}\) are the only ones with a value of 0, which is a consequence of the construction process of V.

To construct the potential V, the first step is to build the Non-Satisfied Clauses matrix M( F). As stated before, M( F) is a diagonal matrix whose non-zero entries are the number of clauses in F which an assignment a does not satisfy. Assignment a is represented by state \({|{a}\rangle }\) and its corresponding non-zero entry in M( F) is located on the position that corresponds to the outer product \({|{a}\rangle }{\langle {a}|}\).

To show how M( F) is built, we now introduce an example based on \(F_1({\mathbf {x}})=(x_1 \vee x_2) \wedge (x_1 \vee \overline{x}_2) \wedge (\overline{x}_1 \vee x_2) \wedge (\overline{x}_2 \vee x_3)\). The first clause in \(F_1({\mathbf {x}})\) is \(C_1=( x_1 \vee x_2)\), and the assignments that do not satisfy \(C_1\) are : \(x_1=0\), \(x_2=0\), \(x_3=0\), and \(x_1=0\), \(x_2=0\), \(x_3=1\). The states that represent the assignments that do not satisfy \(C_1\) are \({|{000}\rangle }\) and \({|{001}\rangle }\), and the sum of their outer products is \(NS(C_1)={|{0}\rangle }{\langle {0}|} \otimes {|{0}\rangle }{\langle {0}|} \otimes {|{0}\rangle } {\langle {0}|}+{|{0}\rangle }{\langle {0}|} \otimes {|{0}\rangle }{\langle {0}|}\otimes {|{1}\rangle }{\langle {1}|}\), i.e.

The \(NS(C_1)\) matrix form is given by \( {\text {diag}}({1,1,0,0,0,0,0,0})\). Note that we can use I when doing the outer product summation to represent the variables that do not appear in a clause (\(x_3\) in the case of \(C_1\)), hence reducing the number of elements needed to construct each \(NS(C_i)\). The number of non I elements in every \(NS(C_i)\) is only dependent on K, the number of binary variables on each and every clause of a K-SAT instance, which is fixed for a K-SAT instance. This allows us to construct each \(NS(C_i)\) with a non-exponential number of elements (see " Analytical approximation of the algorithm behavior " for a detailed explanation of how to efficiently construct \(NS(C_i)\) matrices). \(NS(C_1)\) is a diagonal matrix with 1s in the elements corresponding to the outer products of the states that do not satisfy \(C_1\). In this case the 1s are clearly in the positions corresponding to the outer products of \({|{000}\rangle }\) and \({|{001}\rangle }\), while the zeroes correspond to the outer products of the states that represent a satisfying assignment for \(C_1\).

If the matrices \(NS(C_i)\) are calculated for every clause in an instance F, the diagonal elements that are 0 in every \(NS(C_i)\) correspond to the outer products of the states that satisfy every clause, thus satisfying instance F. The sum of every \(NS(C_i)\) results in the diagonal matrix

where an entry in M( F) corresponding to an outer product \({|{a}\rangle }{\langle {a}|}\) is equal to the number of clauses not satisfied in F by the assignment represented by \({|{a}\rangle }\), and m is the number of clauses in F. For our example \(M(F_1)\) is

$$\begin{aligned} M(F_1) = {|{0}\rangle }{\langle {0}|} \otimes {|{0}\rangle } {\langle {0}|}\otimes I+{|{0}\rangle }{\langle {0}|} \otimes {|{1}\rangle } {\langle {1}|}\otimes I + {|{1}\rangle }{\langle {1}|} \otimes {|{0}\rangle } {\langle {0}|}\otimes I+I\otimes {|{1}\rangle }{\langle {1}|}\otimes {|{0}\rangle }{\langle {0}|} \end{aligned}$$

which in matrix notation is given by

From Eq. ( 7) we can see that the assignment corresponding to \({|{000}\rangle }\) does not satisfy 1 clause in \(F_1\), \({|{001}\rangle }\) does not satisfy 1 clause, \({|{010}\rangle }\) does not satisfy 2 clauses, and so on. There is only one zero in the diagonal of \(M(F_1)\), which corresponds to the outer product of \({|{111}\rangle }\), meaning the assignment \(x_1=1\), \(x_2=1\), \(x_3=1\) is the only satisfying assignment for \(F_1\).

Since the probability of tunneling is higher between states at potentials with similar values, we want to start with an initial state in a very small potential, expecting its probability flux to mostly tunnel through the shortest tunneling paths with the smaller potentials leading to \({|{s}\rangle }\) that represent a satisfying assignment for an instance F (the element corresponding to \({|{s}\rangle }{\langle {s}|}\) in M( F) is 0, the global minimum) thus amplifying the probability of measuring \({|{s}\rangle }\). Since we do not know a priori which states correspond to a small potential, we can just set to 0 the potential corresponding to the outer product \({|{\psi _j}\rangle }{\langle {\psi _j}|}\) by classically calculating the number of clauses \(\alpha (\psi _j,F)\) the assignment corresponding to \({|{\psi _j}\rangle }\) does not satisfy (this can be done in O( mK)), and then subtracting the outer product \({|{\psi _j}\rangle }{\langle {\psi _j}|}\) multiplied by \(\alpha (\psi _j,F)\) from the matrix M( F). As a result we get a 0 in the element corresponding to \({|{\psi _j}\rangle }{\langle {\psi _j}|}\) in M( F).

From Eq. ( 3), the Hamiltonian \(H_A\) is

$$\begin{aligned} H_A(F,{|{\psi _j}\rangle },j)=A(n)+(\gamma +\delta j)(M-\alpha {|{\psi _j}\rangle }{\langle {\psi _j}|}) \end{aligned}$$

Suppose \({|{\psi _j}\rangle }={|{000}\rangle }\). We build A( n) using Eq. ( 1). Then, the resulting \(H_A\) is given by

Note that explicitly calculating every element in M(F) consumes an exponential amount of resources in the number of qubits. We do it explicitly in this paper with the purpose of showing the logic and behavior of the algorithm. In a real world implementation, M( F) would be a physical process that acts as a blackbox.

Parameters discussion

As stated before, \(\gamma \) and \(\delta \) are both real positive parameters. First we will explain the effect \(\gamma +\delta j\) has in the probability of measuring \({|{s}\rangle }\), then we proceed to explain why it increases with every iteration, and finally why they are dependent on n, m and K, the number of variables, clauses, and literals per clause in F respectively.

The effect \(\gamma +\delta j\) has on \(H_A\) is clearly a change in the size of the potential barriers. If \((\gamma +\delta j)<1\) the size of the potential barriers get reduced which makes a state \({|{\psi _j}\rangle }\) evolving under \(e^{-iH_At}\) to tunnel and amplify \({|{s}\rangle }\) faster than if \((\gamma +\delta j)=1\). This occurs because the smaller the potential barriers the easier it is for the probability flux to transmit through them, but it has the unwanted side effect of letting more of the probability flux to transmit to states outside the shortest paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\) which translates into a smaller amplification of \({|{s}\rangle }\). On the other hand, if \((\gamma +\delta j)>1\) the time it will take \({|{\psi _j}\rangle }\) to tunnel to \({|{s}\rangle }\) increases but has the benefit of less probability flux getting out of the main tunneling paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\) which in turn increases the amplification and also reduces the small oscillations created by the reflections in the potentials and the probability flux transmitting out of the main tunneling paths. Simulations and analytical approximations showing the previously mentioned effects are shown in " Numerical calculation of parameters and additional simulations ".

To understand why parameter \(\gamma \) gets an increment \(\delta \) on every iteration, it is required to understand how the time it takes to get the maximum amplification of \({|{s}\rangle }\) varies depending on its Hamming distance to \({|{\psi _j}\rangle }\), \(d_H({|{s}\rangle },{|{\psi _j}\rangle })\). Intuitively, a state \({|{\psi _j}\rangle }\) evolving under \(e^{-iH_At}\) will amplify faster \({|{s}\rangle }\) the smaller \(d_H({|{\psi _j}\rangle },{|{s}\rangle })\) is, since it has to tunnel through less potential barriers (see " Numerical calculation of parameters and additional simulations " for simulations and approximations exhibiting this behavior). As explained above, thanks to the hypercube topology we have a good probability of getting a post measurement state \({|{\psi _{j+1}}\rangle }\) from the tunneling paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\) with a smaller Hamming distance to \({|{s}\rangle }\) compared to \({|{\psi _j}\rangle }\), \(d_H({|{\psi _{j+1}}\rangle },{|{s}\rangle })<d_H({|{\psi _{j}}\rangle },{|{s}\rangle })\). So, the purpose of increments \(\delta \) to \(\gamma \) on every iteration is to keep approximately the same value for the measurement frequency \(t_f\), so that the probability \(P_s(t)\) of measuring \({|{s}\rangle }\) is greatly amplified for multiple distances \(d_H({|{\psi _j}\rangle },{|{s}\rangle })\) approximately at the same time.

Parameters \(\gamma \), \(\delta \) depend on n, m and K (the number of variables, clauses and literals per clause of F respectively) because as n increases, the average number of iterations also increases. This is because the average Hamming distance is also increased, and having greater \(\gamma \) and \(\delta \) can make tunneling too difficult from longer Hamming distances in few iterations. As for the dependency on m and K it is because we cannot know the value of every region in the potential V since they are \(2^n\) values, but the average potential value \(apv=\frac{m}{2^K}\) is only dependent on m and K for \((\gamma +\delta j)=1\) (the derivation of apv is presented in " Numerical calculation of parameters and additional simulations", Eq. ( 11)). If an instance \(F_1\) has a greater apv than an instance \(F_2\) , and both instances have the same number of variables, we can expect the optimal values of \(\gamma \) and \(\delta \) for \(F_1\) to be smaller than the optimal values for \(F_2\) since smaller values of \(\gamma \) and \(\delta \) will facilitate the tunneling through the bigger potential barriers found in \(H_A(F_1)\). We have estimated, via digital computer algorithms, optimal values of \(\gamma \) and \(\delta \), by doing a sweep over the values, for 3-SAT instances with 4, 6 and 10 variables with 16, 24, and 40 clauses respectively. Table 1 shows smaller values for larger instances with more variables in accordance to results presented above. The details on the numerical approximations of \(\gamma \) and \(\delta \) are shown in " Conclusions ".

Table 1 Optimal values for \(\gamma \) and \(\delta \) for various cases.

We have also numerically estimated, for different values of \(\gamma \) and \(\delta \), the time during which we evolve the system \(t_f\), the measurement frequency, giving the same value \(t_f=4.72\) in every case (details in " Conclusions"). The analytical approximation of \(t_f\) via perturbation theory (shown in " Numerical calculation of parameters and additional simulations ") gave \(t_f=\frac{3}{2}\pi \), in close accordance with the numerical results.

Implementation details

The algorithm may be efficiently implemented by approximating the evolution operator as a set of quantum gates by using techniques like the Trotter-Suzuki decomposition37 or its variations. We have developed this technique to show that there is at least one efficient method to implement our algorithm. It remains, as future work, to identify optimal implementation methods that may preserve the continuous nature of the algorithm38.

\(H_A\) can be decomposed as a sum of Hermitian matrices acting on small subspaces

$$\begin{aligned} H_A(F,{|{\psi _j}\rangle },j)=(\gamma +\delta j)\left( \sum _{i=1}^m NS(C_i)-\alpha {|{\psi _0}\rangle }{\langle {\psi _0}|}\right) +A \end{aligned}$$

where \(NS(C_i)\) are the Hermitian matrices that compose M( F). M( F) and A( n) (Eq. ( 1)) seem to be expensive to construct, but in reality M( F) can be constructed using O( m) unitary matrices, while A( n) can be constructed in O( n). Each \(NS(C_i)\) can be built from \(2^K\) unitary matrices acting on a small subspace, this is because K is the number of variables contained in each and every clause of a K-SAT instance and, consequently, \(2^K\) is a rather small number, certainly much less than \(2^n\), where n is the number of binary variables over which a K-SAT instance is defined. So, for any instance of the K-SAT problem K is a fixed integer number equal to the number of variables in each and every clause, then \(2^K = O(1)\) for any given K-SAT instance.

For example, the first clause of \(F_1({\mathbf {x}})=(x_1 \vee x_2) \wedge (x_1 \vee \overline{x}_2) \wedge (\overline{x}_1 \vee x_2) \wedge (\overline{x}_2 \vee x_3)\) reads \(C_1=(x_1 \vee x_2)\) and the states that represent the assignments that do not satisfy \(C_1\) are \({|{000}\rangle }\) and \({|{001}\rangle }\). The sum of their outer products is the matrix \(NS(C_1)\)

where \(\hat{\sigma }_z\) is the Pauli operator \(\hat{\sigma }_z = |0\rangle \langle 0 | - |1\rangle \langle 1 |\). So, M( F) can be expressed as the sum of \(2^Km\) Hermitian operators. In general, the number of Is in \(NS(C_i)\) is \(n-K\), in our example (Eq. ( 9)) \(K=2\) and \(n=3\) so it has one I. For instances with more variables, the number of I increases which means that each \(NS(C_i)\) only acts in a small subspace. Expressing \(NS(C_i)\) as the sum of \(2^K\) Hermitian operators, as in Eq. ( 10), requires using \(K2^{K-1}\) Pauli operators \(\hat{\sigma }_z\). The number of \(\hat{\sigma }_z\) operators in \(NS(C_i)\) is independent from the number of variables in the instance, if the instance had more variables it would just add more Is to each \(NS(C_i)\). Consequently, we do not need an exponential number of Pauli matrices to construct M( F).

The unitary \(e^{i\alpha {|{\psi _j}\rangle }{\langle {\psi _j}|}t}\) can be efficiently implemented using ancilla qubits. A controlled gate applies a NOT gate to an ancilla qubit initially in \({|{0}\rangle }\) if the original n qubits are in \({|{\psi _j}\rangle }\) (this can be done with a polynomial number of gates and ancillas39). Then, if the ancilla qubit is in \({|{1}\rangle }\) a controlled gate applies \(e^{i\alpha t}\) to any of the original n qubits. The required projector for lowering the energy of \({|{\psi _j}\rangle }\) can also be built with a gadget proposed by Dodds et al40.

Analytical approximation of the algorithm behavior

To analytically approximate the probability \(P_s(t)\) of finding a satisfying assignment to a K-SAT instance F, which is equivalent to having \({|{s}\rangle }\) as post-measurement quantum state, and to understand how \(\gamma +\delta j\) affects this probability, we use the effective Hamiltonian \(H_{\text {eff}}\) obtained by degenerate perturbation theory. The effective Hamiltonian only acts on a reduced space and only describes part of the energy spectrum of the complete Hamiltonian. It has the advantage of being simpler than the complete Hamiltonian and it is more general than normal perturbation theory since it takes into account the effects of the off-diagonal elements that link the states of interest. For example, it is used in the Born-Oppenheimer approximation to separate the electronic wavefunction from the nuclear wavefunction of a molecule41. In our case we want to separate the subspace spanned by \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) in \(H_A\) to later approximate \(P_s(t)\).

The perturbed Hamiltonian is \(H_p=H_0+\lambda H_1\) where \(\lambda \) is a dimensionless parameter small enough so that the spectrum of \(H_0\) constitutes a good starting point to approximate the spectrum of \(H_p\). In the following examples \(\lambda \) is set to 1 (as it is frequently done in the literature) in the understanding that the elements of \(H_1\) are smaller than those of \(H_0\).

As for our algorithm, to approximate the Hamiltonian already presented in Eq. ( 3)

$$\begin{aligned} H_A=A(n)+(\gamma +\delta j)(M-\alpha {|{\psi _j}\rangle }{\langle {\psi _j}|}) \end{aligned}$$

we use the potential part

$$\begin{aligned} V=(\gamma +\delta j)(M-\alpha {|{\psi _j}\rangle }{\langle {\psi _j}|}) \end{aligned}$$

as the unperturbed Hamiltonian \(H_0=V\), and \(\lambda H_1=A(n)\) is used as the perturbation Hamiltonian. Hence,

$$\begin{aligned} H_A=H_0+\lambda H_1 \end{aligned}$$

We have chosen \(H_0\) and \(\lambda H_1\) to be V and A( n) respectively because the entries of \(\lambda H_1=A\) are much smaller than those of \(H_0=V\). Entries of \(H_1=A\) are composed by 1s and 0s while entries of \(H_0=V\) have average value potential value calculated as follows:

For big instances, the contribution of \(-\alpha {|{\psi _j}\rangle }{\langle {\psi _j}|}\) can be ignored since it only affects one of the \(2^n\) possible states. So,

$$\begin{aligned} V\approx M(F). \end{aligned}$$

Also, from Eq. ( 6), we know that

$$\begin{aligned} M(F)=\sum _{i=1}^{m}NS(C_i). \end{aligned}$$

So, the average potential value is the trace of M( F) divided by \(2^n\).

$$\begin{aligned} apv=\frac{(\gamma +\delta j)\text{tr}(M(F))}{2^n} \end{aligned}$$

To compute \(\text{tr}(M(F))\) we need to know how many elements in M( F) each of the m \(NS(C_i)\) matrices contribute (\(C_i\) are the clauses in F). As it can be seen in " Implementation details" Eq. ( 5), the trace of each \(NS(C_i)\) is \(\text{tr}(NS(C_i))=2^{n-K}\), where K is the number of literals per clause, so

which means that the larger the number of variables and clauses, and the higher value of \(\gamma +\delta j\), the better the approximation.

It is convenient to use \(H_0=V\) since it is a diagonal matrix and calculating its energies and eigenstates is straightforward. Also, there is degeneracy in the energy spectrum of V since the energies of \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) (both eigenstates of \(H_0=V\)) are \(E_s=E_{\psi _j}=0\).

Effective Hamiltonian via degenerate perturbation theory.

The effective Hamiltonian \(H_{\text {eff}}\) acts on a reduced space and only describes part of the energy spectrum of the complete Hamiltonian, which in our case is \(H_A\). The effective Hamiltonian we want to approximate is the one spanned by the states \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) since \(H_A\) has two eigenvectors that can be approximated as a linear combination of \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) by the use of the effective Hamiltonian \(H_{\text {eff}}\). The approximated eigenvectors and energies then will be used to approximate \(P_s(t)\), the probability of getting \({|{s}\rangle }\) as post-measurement state, i.e., finding a satisfying assignment for a K-SAT instance F. As before, our analysis is focused on instances with a unique satisfying solution to make it clearer and because those instances are the hardest to solve.

States \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) may be far from each other in terms of the Hamming distance. If the Hamming distance between \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) is d, i.e. \(d_H({|{\psi _j}\rangle },{|{s}\rangle }=d\), then a d th order approximation to \(H_{\text {eff}}\) is needed for the effects of quantum tunneling between \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) to be present in \(H_{\text {eff}}\). In this analysis we use a 3rd order approximation but the process for higher orders can be easily generalized.

From Gottfried42, the spectrum of \(H_0=V\) has degeneracy from the energies \(E_s=E_{\psi _j}=0\) corresponding to the states \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\). We will call D the degenerate subspace in \(H_0=V\) spanned by the states \(\{{|{\psi _j}\rangle },{|{s}\rangle }\}\in D\).

The perturbed states \({|{a}\rangle }\) that correspond to the degenerate states \({|{\alpha }\rangle }\) (where \({|{\alpha }\rangle }\) represent any of the states in D) that split due to the perturbation can be written as the following linear combination

$$\begin{aligned} {|{a}\rangle }=\sum _\alpha c_\alpha {|{\alpha }\rangle }+\sum _\mu d_\mu {|{\mu }\rangle } \end{aligned}$$

where \({|{\mu }\rangle }\) are the states outside D.

To obtain the effective Hamiltonian we will approximate \(d_\mu \) and do some manipulations so we end up with an eigenvalue problem in the subspace D. As we will see, the order of coefficients \(c_\alpha \) are O(1) while coefficients \(d_\mu \) are \(O(\lambda )\). Based on this, we will approximate the states \({|{a}\rangle }\) as a linear combination of the degenerate states \({|{\alpha }\rangle }\).

Using \(H_0{|{\alpha }\rangle }=E_\alpha {|{\alpha }\rangle }=0\) and \((H_A-E_a){|{a}\rangle }=0\)

Projecting ( 12) onto a state \({|{\beta }\rangle }\) in D, and a state \({|{\nu }\rangle }\) outside D gives the equations

Since \(d_\mu =O(\lambda )\) the last term in ( 14) is \(O(\lambda ^2)\) and can be drooped for the approximation. Since the difference between \(E_\alpha \) and \(E_a\) is negligible compared to \(E_\nu -E_a\), \(E_a\) can be replaced by \(E_\alpha =0\)

Substituting ( 15) in ( 14) we obtain a better approximation of \(d_\nu \) than ( 15). We can iteratively repeat this process to generate higher order approximations of \(d_\nu \) (and in turn, as we will see in Eq. ( 18), higher order approximations of \(H_{\text {eff}}\)).

Noting \(d_\nu \) and \(d_\mu \) both represent the coefficients of the states outside D, we can substitute ( 16) in ( 13)

This is equivalent to the eigenvalue problem ( 13)

$$\begin{aligned} -c_\beta E_a+\lambda \sum _\alpha c_\alpha {\langle {\beta }|}H_1{|{\alpha }\rangle }+\lambda \sum _\mu d_\mu {\langle {\beta }|}H_1{|{\mu }\rangle }=0 \end{aligned}$$

For the third order approximation of the effective Hamiltonian \(H_{\text {eff}}^{(3)}\) in the subspace D ( which is 2 dimensional in this case since we have a 2-fold degeneracy), that is, a Hamiltonian that only acts in this reduced space and only describes part of the spectrum of the true Hamiltonian \(H_A\).

For a dth order approximation, when the distance between the two states in D, \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\), is \(d_H=d\), the off diagonal elements of \(H_{\text {eff}}^{(d)}\) in the subspace D will only have non zero contributions when the states evaluated in the dth order term are those states that form a tunneling path of length d from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\). Without the off-diagonal elements in \(H_{\text {eff}}^{(d)}\) the approximation would not include the effects of tunneling from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\).

Approximation of the probability of finding a satisfying assignment to a K-SAT instance

We now present an approximation of the probability \(P_s(t)\), the probability of finding a satisfying assignment to a K-SAT instance F, which is equivalent to obtain \({|{s}\rangle }\) as post-measurement state of a quantum evolution process starting with the quantum state \({|{\psi _j}\rangle }\). For this purpose, we use the effective Hamiltonian \(H_{\text {eff}}\) from Eq. ( 18) for three different cases: \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=3\), \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=2\), and \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=1\). The approximation for \(P_s(t)\) for distances not presented here can be easily obtained by slightly modifying the procedure.

where \({|{\mu }\rangle }\) and \({|{\nu }\rangle }\) are eigenvectors with eigenvalues \(E_\mu ,E_\nu \) in \(H_0=V\) different from \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\). The resulting effective Hamiltonian will be \(2\times 2\) since it is from the subspace spanned by \(\{{|{\psi _j}\rangle },{|{s}\rangle }\}\) from which we will approximate two eigenvectors of \(H_A\) as a linear superposition of \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\), and later use them to approximate \(P_S(t)\).

The linear term in ( 19) has no contribution to the diagonal of \(H_{\text {eff}}\) since A (as any adjacency matrix) has a diagonal full of 0s. As for the off diagonal elements of the linear term, there is no way for \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) to tunnel from one to the other without going through intermediate states in the case \(d_H=3\), so there are no off diagonal elements contributed by the lineal term.

Evaluating the second order term, we can see it only has diagonal elements, the contributions to the diagonal come from evaluating the term with states at \(d_H({|{\mu }\rangle }, {|{s}\rangle })=1\) or \(d_H({|{\mu }\rangle },{|{\psi _j}\rangle })=1\). Each contribution to this term is divided by its corresponding \(E_\mu \) which is a product of \(\gamma +\delta j\) and the number of clauses the corresponding state \({|{\mu }\rangle }\) does not satisfy.

The third order term is the only contributor to the off-diagonal elements of \(H_{\text {eff}}\) in the case \(d_H=3\). These contributions come from evaluating the third order term with the states that form the shortest tunneling paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\), these states are the ones at a distance \(d_H=2\) from \({|{\psi _j}\rangle }\) or \({|{s}\rangle }\) and \(d_H=1\) from the other. The easier it is for the states \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) to tunnel through these paths (when the potential barriers are smaller) the greater the value of the off-diagonal elements will be. We can check the previous statement from Eq. ( 19) since every contribution from each tunneling path is 1 divided by \(E_\mu E_\nu \).

The resulting effective Hamiltonian \(H_{\text {eff}}\) in the sub space spanned by \(\{{|{\psi _j}\rangle },{|{s}\rangle }\}\) is

Finding the energies of Eq. ( 20) is straightforward

$$\begin{aligned} \begin{vmatrix} \Updelta v_{11}-E_\pm&\Updelta ^2 v_{12}\\ \Updelta ^2 v_{21}&\Updelta v_{22}-E_\pm \end{vmatrix} = \Updelta ^2 v_{11}v_{22}+E_\pm (-\Updelta v_{11}-\Updelta v_{22})+E_\pm ^2-\Updelta ^4v_{12}^2=0 \end{aligned}$$

where \(\Updelta =\frac{1}{\gamma +\delta j}\) for simplicity. To obtain the eigenstates

$$\begin{aligned} \begin{bmatrix} \Updelta v_{11}-E_\pm &{} \Updelta ^2 v_{12}\\ \Updelta ^2 v_{21} &{} \Updelta v_{22}-E_\pm \end{bmatrix} \begin{bmatrix} a\\ b \end{bmatrix} =0 \end{aligned}$$

From Eqs. ( 22) and ( 23), eigenvectors are computed

$$ \begin{aligned} & b(E_{ \pm } - \Delta v_{{22}} ) = a\Delta v_{{12}} \\ & b\Delta ^{2} v_{{12}} = a(E_{ \pm } - \Delta v_{{11}} ) \\ & |\phi _{ \pm } \rangle = k_{ \pm } \left[ {\begin{array}{*{20}c} {E_{ \pm } - \Delta v_{{22}} } \\ {\Delta ^{2} v_{{12}} } \\ \end{array} } \right] = k^{\prime}_{ \pm } \left[ {\begin{array}{*{20}c} {\Delta ^{2} v_{{12}} } \\ {E_{ \pm } - \Delta v_{{11}} } \\ \end{array} } \right] \\ \end{aligned} $$

where \(k_\pm \) and \(k'_\pm \) the are normalization constants

$$ \begin{aligned} k_{ \pm } & = (\Delta ^{4} v_{{12}}^{2} + (E_{ \pm } - \Delta v_{{22}} )^{2} )^{{ - \frac{1}{2}}} , \\ k^{\prime}_{ \pm } & = (\Delta ^{4} v_{{12}}^{2} + (E_{ \pm } - \Delta v_{{11}} )^{2} )^{{ - \frac{1}{2}}} \\ \end{aligned} $$

and can be related as follows:

$$\begin{aligned} {|{\phi _\pm }\rangle }=k_\pm (E_\pm -\Updelta v_{22}){|{\psi _j}\rangle }+k_\pm \Updelta ^2v_{12}{|{s}\rangle }=k'_\pm \Updelta ^2v_{12}{|{\psi _j}\rangle }+k_\pm \Updelta ^2v_{12}{|{s}\rangle } \end{aligned}$$

From Eq. ( 24) and orthogonality, the eigenvectors can be rewritten as

$$\begin{aligned} {|{\phi _+}\rangle }&=k_-\Updelta ^2v_{12}{|{\psi _j}\rangle }+k_+\Updelta ^2v_{12}{|{s}\rangle }\\ {|{\phi _-}\rangle }&=-k_+\Updelta ^2v_{12}{|{\psi _j}\rangle }+k_-\Updelta ^2v_{12}{|{s}\rangle } \end{aligned}$$

The state \({|{\psi _j}\rangle }\) can be written as a linear combination of \({|{\phi _\pm }\rangle }\)

$$\begin{aligned}&{{|{\psi _j}\rangle }=\frac{k_-}{gk_+}{|{\phi _+}\rangle }-\frac{1}{g}{|{\phi _-}\rangle }}\\&{g=\Updelta ^2k_+v_{12}\left( 1+\left( \frac{k_-}{k_+}\right) ^2\right) } \end{aligned}$$

where g is a normalization constant.

We can use \({|{\phi _\pm }\rangle }\) to approximate the evolution of the real eigenstates of the Hamiltonian \(H_A\). The evolution of a true eigenstate of \(H_A\) is

$$\begin{aligned} e^{-iH_At}{|{a_\pm }\rangle }=e^{-iE_{a_\pm }t}{|{a_\pm }\rangle } \end{aligned}$$

by substituting the eigenstates and energies by their approximation we obtain

$$\begin{aligned} e^{-iH_At}{|{\phi _\pm }\rangle }\approx e^{-iE_\pm t} {|{\phi _\pm }\rangle } \end{aligned}$$

Using it, we can get an approximation of probability \(P_s^{(3)}(t)\) of measuring \({|{s}\rangle }\) at a time t evolving from \({|{\psi _j}\rangle }\), when \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=3\).

$$ \begin{aligned} & P_{s}^{{(3)}} (t) = |\langle s|e^{{ - iHt}} |\psi _{j} \rangle |^{2} \\ P_{s}^{{(3)}} (t) & \approx \left| {\frac{{k_{ - } }}{{gk_{ + } }}e^{{ - iE_{ + } t}} \langle s||\phi _{ + } \rangle - \frac{1}{g}e^{{ - iE_{ - } t}} \langle s||\phi _{ - } \rangle } \right|^{2} \\ & \approx \left| {\frac{{\Delta ^{2} k_{ - } v_{{12}} }}{g}e^{{ - iE_{ + } t}} - \frac{{\Delta ^{2} k_{ - } v_{{12}} }}{g}e^{{ - iE_{ - } t}} } \right|^{2} \\ & \approx \left| {\frac{{\Delta ^{2} k_{ - } v_{{12}} }}{g}\left( {e^{{ - iE_{ + } t}} - e^{{ - iE_{ - } t}} } \right)} \right|^{2} \\ & \approx \left( {\frac{{\Delta ^{2} k_{ - } v_{{12}} }}{g}} \right)^{2} \left( {2 - 2\cos ((E_{ + } - E_{ - } )t)} \right) \\ \end{aligned} $$

That is,

The frequency of the probability \(P_s^{(3)}(t)\) is approximately

$$\begin{aligned} E_+-E_-=\Updelta \sqrt{(v_{11}-v_{22})^2+4\Updelta ^2v_{12}^2} \end{aligned}$$

Since \(\Updelta =\frac{1}{\gamma +\delta j}\), for \(\gamma +\delta j>>1\Rightarrow E_+-E_- \approx O(\Updelta )\) since the term with \(\Updelta ^2\) vanishes.

It is easy to follow a similar process to calculate \(P_s^{(2)}(t)\), the case with \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=2\). In this case at least a second order approximation is needed for the effects of tunneling (the off diagonal elements) to appear in \(H_{\text {eff}}^{(2)}\) .

$$\begin{aligned} H_{\text {eff}}^{(2)}=A-\sum _\mu \frac{ A{|{\mu }\rangle } {\langle {\mu }|}A}{E_\mu } \end{aligned}$$

The corresponding \(H_{\text {eff}}\) is slightly different to the previous case which has an impact in the frequency of \(P_s^{(2)}(t)\). The first order contribution to \(H_{\text {eff}}\), as in the previous case, is also null for the same reasons. In this case the second order term gives all the elements for \(H_{\text {eff}}\). The contributions from the second order come from evaluating the states at \(d_H({|{\mu }\rangle },{|{\psi _j}\rangle })=1\) or \(d_H({|{\mu }\rangle },{|{s}\rangle })=1\). Some of these states will be part of the shortest tunneling paths from \({|{\psi _0}\rangle }\) to \({|{s}\rangle }\), meaning \(d_H({|{\mu }\rangle },{|{\psi _j}\rangle })=d_H({|{\mu }\rangle },{|{s}\rangle })=1\), and are the ones that contribute to the off-diagonal elements. As in the previous case (\(d_H=3)\), the easier it is for \({|{\psi _j}\rangle }\) to tunnel through a barrier to get to \({|{s}\rangle }\) the greater the contribution from that path. Each contribution is divided by the corresponding \(E_\mu \), that is a product of \(\gamma +\delta j\) and the number of clauses the corresponding state \({|{\mu }\rangle }\) does not satisfy. This gives the following effective Hamiltonian

Following a process similar to the one presented for \(d_H=3\), we obtain

$$\begin{aligned} P_s^{(2)}(t)\approx \left( \frac{k_+k_-}{k_-^2+k_+^2}\right) ^2\left( 2-2\cos ((E_+-E_-)t)\right) \end{aligned}$$

where

$$\begin{aligned}&E_\pm =\frac{\Updelta }{2}\left( v_{11}+v_{22}\pm \sqrt{(v_{11}-v_{22})^2+4v_{12}^2}\right) \\&k_\pm =(\Updelta ^2v_{12}^2+(E_\pm -\Updelta v_{22})^2)^{-\frac{1}{2}} \end{aligned}$$

\(P_s^{(2)}(t)\) has a frequency \(E_+-E_-=\Updelta \sqrt{(v_{11}-v_{22})^2+4v_{12}^2}\) which is \(O(\Updelta )\), Comparing it to the \(P_s^{(3)}(t)\) frequency, when \(\gamma +\delta j>>1\), remembering \(\Updelta =\frac{1}{\gamma +\delta j}\), \(P_{s}^{(2)}(t)\) has the term \(4v_{12}^2\) inside the square root compared to \(P_{s}^{(3)}(t)\) where \(\Updelta ^24v_{12}^2\) vanishes. Also, the off-diagonal elements \(v_{12}\) from \(H_{\text {eff}}\) when \(d_H=2\) tend to be greater compared to the case \(d_H=3\) since there are less potential barriers in the tunneling path from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\). Consequently, in most cases the frequency of \(P_{s}^{(2)}(t)\) will be higher than he frequency of \(P_{s}^{(3)}(t)\).

For \(d_H=1\), the first order approximation can be used. The \(H_{\text {eff}}\) is simply

$$\begin{aligned} H_{\text {eff}}^{(1)}= \begin{bmatrix} 0 &{} 1\\ 1 &{} 0 \end{bmatrix} \end{aligned}$$

The probability \(P_s^{(1)}(t)\) can be obtained as follows

Equation ( 27) does not have dependency on \(\Updelta \) and has a period of \(\pi \). Hence, for \(t=\frac{\pi }{2}+\pi m\), where \(m \in {\mathbb {Z}}\), \(P_s^{(1)}(t)\) is at its highest value. As \(\gamma +\delta j\) increases, the potential barriers around the states \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) get larger, reducing the tunneling to the states outside of D (for \(d_H=1\) the states in D are neighbors) which makes the real behavior of \(P_s^{(1)}(t)\) closer to the approximation of Eq. ( 27). Due to this lack of dependency on \(\Updelta \) from the approximation of \(P_s^{(1)}(t)\), the actual period of \(P_s^{(1)}(t)\) (without approximations) is practically always the same. With this in mind, the values of \(\gamma \) and \(\delta \) have to keep the evolution time \(t_f=\frac{\pi }{2}+\pi m\), where \(m \in {\mathbb {Z}}\), as a time that greatly amplifies \(P_s(t)\) for multiple values of \(d_H({|{\psi _j}\rangle },{|{s}\rangle })\). In fact, our numerical simulations using the optimal parameters, found by doing a sweep, show this property of preserving \(t_f\) as an evolution time that greatly amplifies \(P_s(t)\). Our simulations give us an optimal measurement frequency of \(t_f=4.72\), that is approximately \(\frac{3}{2}\pi \), in agreement with analytical results. Keep in mind that for larger numbers of variables and clauses this value may change to \(t_f=\frac{\pi }{2}+\pi m\), with a different value of m.

Comparisons between the analytical approximation and simulations

In this section, we compare the analytical approximations of \(P_s(t)\) from the previous section

$$ \begin{aligned} P_{s}^{{(3)}} (t) & \approx \left( {\frac{{k_{ + } k_{ - } }}{{k_{ - }^{2} + k_{ + }^{2} }}} \right)^{2} \left( {2 - 2\cos (t\Delta \sqrt {(v_{{11}} - v_{{22}} )^{2} + 4\Delta ^{2} v_{{12}}^{2} } )} \right) \\ P_{s}^{{(2)}} (t) & \approx \left( {\frac{{k_{ + } k_{ - } }}{{k_{ - }^{2} + k_{ + }^{2} }}} \right)^{2} \left( {2 - 2\cos (t\Delta \sqrt {(v_{{11}} - v_{{22}} )^{2} + 4v_{{12}}^{2} } )} \right) \\ P_{s}^{{(1)}} (t) & \approx \sin ^{2} (t) \\ \end{aligned} $$

to our numerical simulations of \(P_s(t)\) for the distances \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=1,2,3\). The instance used for these comparisons is the following 6 variable instance with 24 clauses and the unique satisfying assignment \(s=[110111]\).

To exemplify the case \(d_H=3\) we used the state \({|{\psi _j}\rangle }={|{010001}\rangle }\). The effective Hamiltonian using degenerate perturbation theory up to the third order, as in Eq. ( 20), is

$$\begin{aligned} H_{\text {eff}}^{(3)}= \begin{bmatrix} \frac{-2.455}{\gamma +\delta j} &{} \frac{2.555}{(\gamma +\delta j)^2}\\ \frac{2.555}{(\gamma +\delta j)^2} &{} \frac{-3.666}{\gamma +\delta j} \end{bmatrix} \end{aligned}$$

Figure 1 compares analytical approximation and numerical simulations of \(P_s(t)\) evolving from \({|{\psi _j}\rangle }={|{010001}\rangle }\) for various values of \(\gamma +\delta j\). As the value of \(\gamma +\delta j\) increases the approximation gets better since the tunneling to states outside the main tunneling paths from \({|{\psi _j}\rangle }\) to \({|{s}\rangle }\) decreases, and the reflections at the potential barriers also decrease. This contributes to reduce the small oscillations seen in the numerical simulations of \(P_s(t)\). We can also see that even for small values of \(\gamma +\delta j\), the approximation of the period is accurate.

Comparison between simulations (blue line) and the analytical approximation (orange line) of \(P_s(t)\) for initial state \({|{010001}\rangle }\), \(d_H=3\).

For \(d_H=2\), we use the initial state \({|{\psi _j}\rangle }={|{110001}\rangle }\). By using perturbation theory up to the second order, as in Eq. ( 26), we obtain the following effective Hamiltonian

$$\begin{aligned} H_{\text {eff}}^{(2)}= \begin{bmatrix} \frac{-0.446}{\gamma +\delta j} &{} \frac{-0.266}{\gamma +\delta j}\\ \frac{-0.266}{\gamma +\delta j} &{} \frac{-0.733}{\gamma +\delta j} \end{bmatrix} \end{aligned}$$

Figure 2 shows comparisons between the analytical approximation and the numerical simulations of \(P_s(t)\) evolving from \({|{110001}\rangle }\) for various values of \(\gamma +\delta j\).

Comparison between simulations (blue line) and the analytical approximation (orange) of \(P_s(t)\) for initial state \({|{110001}\rangle }\), \(d_H=2\).

Lastly, for \(d_H=1\) and using the initial state \({|{110011}\rangle }\) the effective matrix is simply

$$\begin{aligned}H_{\text {eff}}^{(1)}= \begin{bmatrix} 0 &{} 1\\ 1 &{} 0 \end{bmatrix} \end{aligned}$$

Figure 3 shows comparisons between the analytical approximation and the numerical simulations of \(P_s(t)\) evolving from \({|{110011}\rangle }\) for various values of \(\gamma +\delta j\)

Comparison between simulations (blue line) and the analytical approximation (orange) of \(P_s(t)\) for a Hamming distance of 1.

Graphs from Figs. ( 1, 2, 3) show the advantage of including potential barriers compared to the exponentially small overlap \(P_s(t)=\sin ^6(t) \cos ^6(t) = \sin ^6(2t)/2^6\), obtained from Eq. ( 2), corresponding to solely using the hypercube adjacency matrix as the quantum walk Hamiltonian.

For distances \(d_H>3\), the patterns exhibited in Figs. 1, 2 and 3 hold. In general, the period of \(P_s(t)\) grows longer as \(d_H\) increases as a consequence of having to tunnel through longer paths, which is also reflected in the analytical approximation for the period by becoming more dependent on \(\gamma +\delta j\) as a consequence of \(H_{\text {eff}}\) having the mathematical structure

where d is the distance \(d_H({|{\psi _j}\rangle },{|{s}\rangle })=d\) and also the minimum approximation order necessary for the tunneling effects to be present in the approximation. As \(d_H=d\) grows the off-diagonal elements in \(H_{{\text {eff}}}\) that serve as couplings between \({|{\psi _j}\rangle }\) and \({|{s}\rangle }\) get smaller which implies longer periods for transitioning between states.

Numerical calculation of parameters and additional simulations

Our database, fully available in32, consists of thousands of 3-SAT instances from which 5799 have 4 variables and 16 clauses, 1949 have 6 variables and 24 clauses, and 606 have 10 variables and 40 clauses. Every instance in the database has a unique satisfying solution. All instances were randomly created by a uniform distribution, and then solved by exhaustive substitution as our instances are small enough for that. Instances with unique satisfying assignment were saved for the database.

The sets of parameters were computed by doing thousands of simulations on 3-SAT instances for the three different number of variables in our data set. The parameters were computed by slightly varying the value of each parameter and running 500 simulations for each combination of parameters. Half the dataset was used to find the parameters while the other half was used to test their performance. Each simulation was run using a randomly selected instance from the dataset, starting with a randomly selected initial state, and stopped after n measurements, where n is the number of variables of the instance, meaning only the first run of the algorithm (without resetting after n iterations) was considered for these calculations.

In every case the measurement frequency \(t_f\) was around 4.72 which is approximately \(\frac{3}{2}\pi \) in accordance to our analytical approximation. Future versions of the algorithm may make use of variable measurement times, possibly by the use of techniques used in the literature of first detection time for quantum walks43,44.

The optimal sets of parameters we estimated for each number of variables and clauses are shown in Table 2. Alongside them we have included statistical information from 1000 simulations for each of the three different number of variables: The success rate of having \({|{s}\rangle }\) as post-measurement quantum state (that is, the success rate of finding the unique solution of each instance used in this study) for the first run of the algorithm ( n iterations at most), average number of necessary iterations to get \({|{s}\rangle }\) as post-measurement quantum state, and the standard deviation in the number of iterations to get \({|{s}\rangle }\) as post-measurement quantum state.

Table 2 Optimal parameters and statistical information from 1000 simulations.

Taking in account all the simulations we ran for Table 2, the success rate was 0.66 for the first run of the algorithm. The success rate for the 10 variable instances may seem low, but remember it is only for the first run, for the second run its success rate increases to 0.63 and the fifth run is already 0.93. Also, we are using linear increments for \(\gamma +\delta j\), the use of a different function may translate into a better performance. Figure 4 presents simulations of \(P_s(t)\) for two randomly selected 10 variables 3-SAT instances from the database32 showing the expected behavior for runs where the satisfactory state \({|{s}\rangle }\) is measured.

Probabilities of measuring \({|{s}\rangle }\) depicting the behavior of the complete algorithm for 2 randomly selected 10 variables 3-SAT instances with a unique solution from our database. Vertical green lines show when a measurement took place.

Conclusions

We have presented a quantum walk-based algorithm that explicitly utilizes potential barriers and quantum tunneling to find satisfying assignments for hard K-SAT instances.

One of the novelties of our paper is to harness by design quantum tunneling as a computational resource and to use it as a controllable feature that can be engineered to provide additional processing power to a quantum algorithm. Furthermore, we have derived a detailed mathematical procedure to build a family of Hamiltonians to solve K-SAT instances via continuous quantum walks with potential barriers and quantum tunneling. Based on the structure of these Hamiltonians, we have used degenerate perturbation theory to derive an analytical approximation of the probability of finding satisfying assignments to K-SAT instances under our algorithmic approach.

As for the set of parameters required to run our algorithm, we have made an extensive numerical study that has allowed us to present good estimates of those parameters. It remains as future work to derive analytical formulae for this set of parameters, a future work which is encouraged by our numerical results.

Acknowledgements

EC thanks his family for their support. SEVA warmly thanks his family for their unconditional support.

Funding

EC acknowledges the financial support received by Tecnologico de Monterrey and CONACYT (CVU 834995). SEVA gratefully acknowledges the financial support of Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias and CONACyT [SNI number 41594 as well as Fronteras de la Ciencia project number 1007]. ML acknowledges the support received by the U.S. Naval Research Laboratory Base Program on Quantum Computation.

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Authors and Affiliations

1. Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias, Av Eugenio Garza Sada 2501, Mty, NL, Mexico

Ernesto Campos & Salvador E. Venegas-Andraca

2. Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, 121205, Moscow, Russia

Ernesto Campos

3. US Naval Research Laboratory, 4555 Overlook Ave., SW Washington, DC, 20375, USA

Marco Lanzagorta

Contributions

E.C. and S.E.V.A. proposed the idea of writing a quantum algorithm using quantum walks and quantum tunneling to solve hard K-SAT instances. E.C., S.E.V.A. and M.L. made contributions towards the mathematical analysis and results or our algorithm. E.C., S.E.V.A. and M.L. reviewed the manuscript.

Corresponding author

Correspondence to Salvador E. Venegas-Andraca.

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Campos, E., Venegas-Andraca, S.E. & Lanzagorta, M. Quantum tunneling and quantum walks as algorithmic resources to solve hard K-SAT instances. Sci Rep 11, 16845 (2021). https://doi.org/10.1038/s41598-021-95801-1
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DOI: https://doi.org/10.1038/s41598-021-95801-1

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Quantum tunneling and quantum walks as algorithmic resources to solve hard K-SAT instances - Scientific Reports

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We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian $$H_A(F)$$ which is specifically constructed to solve a K-SAT instance F.