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I find this algorithm to be very important because he can Hack the RSA , a well known Encryption all over the world, and with Quantum computer we can hack the RSA in a few seconds.

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Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization.[1] Informally, it solves the following problem: Given an integer , find its prime factors. It was invented in 1994 by the American mathematician Peter Shor.

On a quantum computer, to factor an integer , Shor's algorithm runs in polynomial time (the time taken is polynomial in , the size of the integer given as input).[2] Specifically, it takes quantum gates of order using fast multiplication,[3] thus demonstrating that the integer-factorization problem can be efficiently solved on a quantum computer and is consequently in the complexity class BQP. This is almost exponentially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time - .[4] The efficiency of Shor's algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by repeated squarings.

If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as the widely used RSA scheme. RSA is based on the assumption that factoring large integers is computationally intractable. As far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.

In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored into , using an NMR implementation of a quantum computer with qubits.[5] After IBM's implementation, two independent groups implemented Shor's algorithm using photonic qubits, emphasizing that multi-qubit entanglement was observed when running the Shor's algorithm circuits.[6][7] In 2012, the factorization of was performed with solid-state qubits.[8] Also, in 2012, the factorization of was achieved, setting the record for the largest integer factored with Shor's algorithm.[9]

The problem that we are trying to solve is, given a composite number , to find a non-trivial divisor of (a divisor strictly between and ). Before attempting to find such a divisor, one can use relatively quick primality-testing algorithms to verify that is indeed composite.

We need to be odd (otherwise is a divisor) and not to be any power of a prime (otherwise that prime is a divisor), so we need to check that there are no integer roots for .

Hence we may assume that is the product of two coprime integers greater than . It follows from the Chinese remainder theorem that there are at least four distinct square roots of modulo (since there are two roots for each modular equation). The aim of the algorithm is to find a square root of modulo that is different from and , because then

for a non-zero integer which gives us the non-trivial divisors and of . This idea is similar to other factoring algorithms, like the quadratic sieve.

In turn, finding such a is reduced to finding an element of even period with a certain additional property (as explained below, it is required that the condition of Step 6 of the classical part does not hold). The quantum algorithm is used for finding the period of randomly chosen elements , as this is a hard problem on a classical computer.

Shor's algorithm consists of two parts:

1. A reduction, which can be done on a classical computer, of the factoring problem to the problem of order-finding.

2. A quantum algorithm to solve the order-finding problem.

1. Pick a random number .

2. Compute , the greatest common divisor of and . This may be done using the Euclidean algorithm.

3. If , then this number is a nontrivial factor of , so we are done.

4. Otherwise, use the quantum period-finding subroutine (below) to find , which denotes the period of the following function: This is the order of in the group , which is the smallest positive integer for which , or . By Euler's Theorem, divides , where denotes Euler's totient function.

5. If is odd, then go back to step 1.

6. If , then go back to step 1.

7. Otherwise, at least one of and is a nontrivial factor of , so we are done.

Proceed as follows:

1. Initialize the registers to
where denotes the tensor product.
This initial state is a superposition of states, and is easily obtained by generating independent qubits, each a superposition of and states. We can accomplish this by initializing the qubits to the zero-position, and then applying the hadamard gate in parallel to each of the qubits, where .

2. Construct as a quantum function and apply it to the above state, to obtain This is still a superposition of states. However, the qubits, i.e, the input qubits and ( ) output qubits, are now entangled or not separable, as the state cannot be written as a tensor product of states of individual qubits.

3. Apply the inverse quantum Fourier transform to the input register. This transform (operating on a superposition of states) uses a -th root of unity such as to distribute the amplitude of any given state equally among all of the states, and to do so in a different way for each different . We thus obtain This leads to the final state Now, we reorder this sum as This is a superposition of many more than states, but many fewer than states, as there are fewer than distinct values of . Let

be a -th root of unity,

be the period of ,

be the smallest of the for which (we actually have ), and

write

indexes these , running from to , so that .

Then is a unit vector in the complex plane ( is a root of unity, and and are integers), and the coefficient of in the final state is Each term in this sum represents adifferent path to the same result, and quantum interference occurs - constructive when the unit vectors point in nearly the same direction in the complex plane, which requires that point along the positive real axis.

4. Perform a measurement. We obtain some outcome in the input register and some outcome in the output register. As is periodic, the probability of measuring some state is given by Analysis now shows that this probability is higher the closer the unit vector is to the positive real axis, or the closer is to an integer. Unless is a power of , it will not be a factor of .

5. Since is close to some integer , the known value is close to the unknown value . Performing [classical] continued fraction expansion on allows us to find approximations of it that satisfy two conditions: Given these multiple conditions (and assuming is irreducible), is very likely to be the appropriate period , or at least a factor of it.

6. Check (classically) if . If so, then we are done.

7. Otherwise, (classically) obtain more candidates for by using multiples of , or by using other with near . If any candidate works, then we are done.

8. Otherwise, try again starting from step 1 of this subroutine.

The algorithm is composed of two parts. The first part of the algorithm turns the factoring problem into the problem of finding the period of a function and may be implemented classically. The second part finds the period using the quantum Fourier transform and is responsible for the quantum speedup.

The integers less than and coprime with form the multiplicative group of integers modulo , which is a finite abelian group . The size of this group is given by . By the end of step 3, we have an integer in this group. As the group is finite, must have a finite order , which is the smallest positive integer such that

Therefore, divides (also written ). Suppose that we are able to obtain and that it is even. (If is odd, then see step 5.) Now is a square root of modulo that is different from . This is because is the order of modulo , so , or else the order of in this group would be . If , then by step 6, we have to restart the algorithm with a different random number .

Multiplying both sides by , we obtain

As divides , we find that divides , so that , again contradicting the construction of .

Therefore, is the required proper factor of .

Shor's period-finding algorithm relies heavily on the ability of a quantum computer to be in many states simultaneously. Physicists call this behavior a " superposition" of states. To compute the period of a function , we evaluate the function at all points simultaneously.

Quantum physics does not allow us to access all this information directly, though. A measurement will yield only one of all possible values, destroying all others. If not for the no cloning theorem, we could first measure without measuring , and then make a few copies of the resulting state (which is a superposition of states all having the same ). Measuring on these states would provide different values which give the same , leading to the period. Because we cannot make exact copies of a quantum state, this method does not work. Therefore, we have to carefully transform the superposition to another state that will return the correct answer with high probability. This is achieved by the quantum Fourier transform.

Shor thus had to solve three "implementation" problems. All of them had to be implemented "fast", which means that they can be implemented with a number of quantum gates that is polynomial in .

1. Create a superposition of states. This can be done by applying Hadamard gates to all qubits in the input register. Another approach would be to use the quantum Fourier transform (see below).

2. Implement the function as a quantum transform. To achieve this, Shor used repeated squaring for his modular exponentiation transformation. It is important to note that this step is more difficult to implement than the quantum Fourier transform, in that it requires ancillary qubits and substantially more gates to accomplish.

3. Perform a quantum Fourier transform. By using controlled rotation gates and Hadamard gates, Shor designed a circuit for the quantum Fourier transform (with ) that uses just gates.[10]

After all these transformations a measurement will yield an approximation to the period . For simplicity assume that there is a such that is an integer. Then the probability to measure is . To see this, we notice that then

Note: Another way to explain Shor's algorithm is by noting that it is just the quantum phase estimation algorithm in disguise.

The runtime bottleneck of Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical pre-/post-processing. There are several approaches to constructing and optimizing circuits for modular exponentiation. The simplest and (currently) most practical approach is to mimic conventional arithmetic circuits with reversible gates, starting with ripple-carry adders. Knowing the base and the modulus of exponentiation facilitates further optimizations.[11][12] Reversible circuits typically use on the order of gates for qubits. Alternative techniques asymptotically improve gate counts by using quantum Fourier transforms, but are not competitive with fewer than 600 qubits due to high constants.

Given a group with order and generator , suppose we know that , for some , and we wish to compute , which is the discrete logarithm: . Consider the abelian group , where each factor corresponds to modular addition of values. Now, consider the function

This gives us an abelian hidden subgroup problem, as corresponds to a group homomorphism. The kernel corresponds to the multiples of . So, if we can find the kernel, we can find .

Nielsen, Michael A. & Chuang, Isaac L. (2010), Quantum Computation and Quantum Information, 10th Anniversary Edition, Cambridge University Press, ISBN 9781107002173 .

Phillip Kaye, Raymond Laflamme, Michele Mosca, An introduction to quantum computing, Oxford University Press, 2007, ISBN 0-19-857049-X

"Explanation for the man in the street" by Scott Aaronson, "approved" by Peter Shor. (Shor wrote "Great article, Scott! That's the best job of explaining quantum computing to the man on the street that I've seen."). An alternate metaphor for the QFT was presented in one of the comments. Scott Aaronson suggests the following 12 references as further reading (out of "the 10 quantum algorithm tutorials that are already on the web."):

Shor, Peter W. (1997), "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer", SIAM J. Comput., 26 (5): 1484-1509, arXiv:quant-ph/9508027v2, Bibcode:1999SIAMR..41..303S, doi:10.1137/S0036144598347011. Revised version of the original paper by Peter Shor ("28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 1996").

Quantum Computing and Shor's Algorithm, Matthew Hayward's Quantum Algorithms Page, 2005-02-17, imsa.edu, LaTeX2HTML version of the original LaTeX document, also available as PDF or postscript document.

Quantum Computation and Shor's Factoring Algorithm, Ronald de Wolf, CWI and University of Amsterdam, January 12, 1999, 9 page postscript document.

Shor's Factoring Algorithm, Notes from Lecture 9 of Berkeley CS 294-2, dated 4 Oct 2004, 7 page postscript document.

Chapter 6 Quantum Computation, 91 page postscript document, Caltech, Preskill, PH229.

Quantum computation: a tutorial by Samuel L. Braunstein.

The Quantum States of Shor's Algorithm, by Neal Young, Last modified: Tue May 21 11:47:38 1996.

III. Breaking RSA Encryption with a Quantum Computer: Shor's Factoring Algorithm, Lecture notes on Quantum computation, Cornell University, Physics 481-681, CS 483; Spring, 2006 by N. David Mermin. Last revised 2006-03-28, 30 page PDF document.

Lavor, C.; Manssur, L. R. U.; Portugal, R. (2003). "Shor's Algorithm for Factoring Large Integers". arXiv:quant-ph/0303175.

Lomonaco, Jr (2000). "Shor's Quantum Factoring Algorithm". arXiv:quant-ph/0010034. This paper is a written version of a one-hour lecture given on Peter Shor's quantum factoring algorithm. 22 pages.

Chapter 20 Quantum Computation, from Computational Complexity: A Modern Approach, Draft of a book: Dated January 2007, Sanjeev Arora and Boaz Barak, Princeton University. Published as Chapter 10 Quantum Computation of Sanjeev Arora, Boaz Barak, "Computational Complexity: A Modern Approach", Cambridge University Press, 2009, ISBN 978-0-521-42426-4

A Step Toward Quantum Computing: Entangling 10 Billion Particles, from "Discover Magazine", Dated January 19, 2011.

Josef Gruska - Quantum Computing Challenges also in Mathematics unlimited: 2001 and beyond, Editors Björn Engquist, Wilfried Schmid, Springer, 2001, ISBN 978-3-540-66913-5

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Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization.[1] Informally, it solves the following problem: Given an integer , find its prime factors. It was invented in 1994 by the American mathematician Peter Shor.

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This is a Perspective. By Vedran Dunjko (LIACS, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands) and Peter Wittek (Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada, Creative Destruction Lab, Toronto, Ontario M5S 3E6, Canada, Vector Institute for Artificial Intelligence, Toronto, Ontario M5S 1M1, Canada, and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada).

About this non-review

By mid-2019, both Peter and myself had found ourselves numerous times in situations where we were asked to define what Quantum Machine Learning is (and what it isn't), or, worse, where we were prompted to divine what the ultimate approach to it should be. We discussed the topic in passing, to find that we generally share identical feelings on the topic: there is nothing to be gained by constraining what QML is or isn't, and, we for sure have no clue what the future of QML is guaranteed to bring (albeit, we were both sure it will bring something cool!), as essentially all existent research lines have genuine potential. This discussion became a blog-type text, with the objective to elaborate on this perspective of ours, and to provide an entry point to the rich and diverse spectrum of topics QML could be - a listing of reviews, mostly, with a sprinkle of other promising yet non-mainstream topics - without much fuss, and certainly without ungrateful "expert predictions".

On advice from colleagues, we have sent this, as Peter dubbed it, "non-review" (a somewhat extended blog-post, really) to Quantum reviewers, to see if it would be suitable as a Perspectives article, sometime in September 2019.

At this point Peter had already left for his final expedition to the Himalayas. The article has been on hold while we all were hoping Peter would still be found, and also later while we were slowly accepting his loss. Peter's colleagues and myself believe Peter would prefer this note published.

To honour Peter's original ideas and text, the note is published with no alterations, which could have been made based on useful suggestions from the reviewers and colleagues, or other types of updates (including new works which would have influenced parts of the text - the field is very fluid), from the original minimal version we submitted together.

- Vedran Dunjko
See also Quantum's blog post and testimonials on Peter.

What is quantum machine learning? What are the key issues? As time progresses, these questions are becoming more difficult to answer. As quantum machine learning rapidly grows, even the much simpler question of what the reference on the topic is, seems overly ambitious. While the long-term perspectives of quantum machine learning may still be quite opaque, the field is bustling, growing, and changing, making it difficult to tame with a single review. This perspective is a tribute to review articles and books, while also drawing attention to recent trends and exploratory works that are often overshadowed by the volume of mainstream contributions.

Contents

1. Supervised and unsupervised learning

2. Reinforcement learning and AI aspects

3. Machine learning in (experimental) physics

4. Quantum-inspired machine learning

QML and beyond

Introduction

As time progresses, any attempts to pin down quantum machine learning into a well-behaved young discipline are becoming increasingly more difficult. Quantum machine learning (QML) is not one settled and homogeneous field; partly, this is because machine learning itself is quite diverse. But the situation is more complicated, due to the respective roles that quantum and machine learning may play in "QML". For some, QML is all about using quantum effects to perform machine learning somehow better. For others, it is clear that it is about utilizing machine learning as the key tool for certain quantum problems. A more comprehensive view is that quantum machine learning is simply the field exploring the connections between quantum computing and quantum physics on one hand, and machine learning, and related fields, on the other. Quantum-applied machine learning, and quantum-enhanced machine learning are then the two dominating, but not only aspects of quantum machine learning. For instance, quantum-inspired machine learning and quantum-generalized learning ideas stand out as two very promising research lines, in line with the QML philosophy, but are ultimately not about speed-ups or applications of ML in quantum experiments.

Quantum-inspired machine learning, for instance, draws inspiration from quantum processing to come up with novel classical learning models and new ways to train and evaluate them: examples here are the approaches of Tang for fast stochastic linear-algebraic manipulations [ 1], ideas involving tensor networks as learning models (see works of Stoudenmire [ 2]), and inspirations behind the projective simulation of Briegel [ 3].

Quantum-generalized machine learning generalizes even the basic concepts: just like density matrices generalize classical notions of information, quantum-generalized machine learning asks what machine learning can look like when data, or environments are genuinely quantum objects.

It is clear that it makes little sense to try to write a comprehensive review of all of quantum machine learning (we state this as two authors who have gone down this path only a few years back). So how does one begin exploring quantum machine learning? Well, there are the old-school methods: books, review papers, and papers; but nowadays, we also have Youtube videos, "awesome" pages on GitHub, blogs, online tutorials and similar. In this non-review, we remain old-school, and reflect on recent literature reporting on the developments in QML from our biased perspective.

The books

The available QML books focus mostly on quantum-enhanced supervised learning, and the broad QML field is by now significantly larger. This brings us to the next level.

Surveys and review articles

To learn more, we have to turn our attention to chunks of QML covered in a number of review articles. Here's a few. The first one came out at the same time as the first book: the title is An introduction to Quantum Machine Learning, by Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione [ 8]. It is a short-and-sweet survey of some of the quantum-enhanced algorithms known at the time. The same set of authors undertook a review of long-term research in the perspectives in quantum neural networks in The quest for a Quantum Neural Network [ 9]. At the time, the jury was out on the perspectives. Modern results in shallow architectures (see, e.g., the book [ 5] above) could be seen as a sign of rejuvenation of the field.

If we can label the early excitement in QML as the first generation quantum-enhanced machine learning algorithms, then the review paper in Nature [ 10] closed that era. This review focused on quantum linear algebraic enhancements for machine learning, but it also signalled the expansion of the field to include machine learning applications in design and control of quantum systems: an aspect of quantum-applied machine learning.

One of the authors of this perspective co-authored Machine learning & artificial intelligence in the quantum domain: a review of recent progress [ 13], with the goal of comprehensiveness, providing examples of quantum-enhanced, quantum-applied, and quantum generalized machine learning and AI. A fresh out of the oven review covers the hottest QML topic over the last year, with a self-explanatory title: Parameterized quantum circuits as machine learning models [ 14]. Quantum neural networks finally also achieved a level of maturity, as summarized in Quantum Deep Learning Neural Networks [ 15].

Learning in quantum control: High-dimensional global optimization for noisy quantum dynamics [ 18] reviews machine learning ideas applied in quantum control, but with new results; so a research paper/review paper hybrid.

Not perspectives

Having listed a number of books and reviews on the broad spectrum of topics of quantum machine learning, one is now tempted to put the field in perspective, and there is a number of possible ways to do this. One is to "follow the trends" in order to classify what the consensus on the "hot topics" of the field had been over time.

As mentioned, quantum machine learning actually has quite a long history, ranging back to the early days of quantum computing. In our "broad viewpoint" on what the most explored topics were over time, it is easy to argue that in the period from early 1990s, until mid-2000s the research in what is now recognized as "quantum machine learning" had two themes: quantum computational learning theory, and quantum generalizations of neural networks. The main motivations behind these approaches can be understood as fundamental in flavour: how does access to pure quantum states encoding classical distributions enhance learnability -separating classical distributions from quantum states; in the context of quantum neural networks, the questions ranged from the explorations in the possible quantum nature of the human brain, to finding ways to reconcile the necessary non-linearities in neuronal processing with the fundamental linearity of quantum mechanics.

The early 2000s mark the first quantum-generalized machine learning ideas, but from 2006 onwards, there has been a steady influx of quantum algorithms with a pragmatic objective: using quantum computers to perform some ML computations faster - this is what we labelled above as the first generation of quantum-enhanced machine learning algorithms.

The period from early 2010s brought an explosive rise in this domain, riding on the potential of quantum linear algebra algorithms and quantum databases, which, so we hoped, open up the possibility for a steady supply of exponential speed-ups.

From 2015, we had new sparks of ideas proposing more expressive quantum models (so improving the learning performance, rather than just straightforward computational complexity), but also first significant signs of life in what is now called quantum-applied machine learning - the latter domain had in fact always been present, but almost certainly now due to the successes of machine learning, the term "machine learning" became pervasive in such works.

As indicated above, in 2018, a number of shifts occurred. The progress in experimental quantum computing, and quantum algorithms designed with restricted circuit-based architectures in mind (VQE, QAOA), inspired the idea that limited quantum machines just may be the best models (in the sense of parametrized distributions, or as sets of hypothesis functions for classification) for machine learning. Well, maybe not the best, but certainly models for which the massive body of quantum supremacy research may actually start to provide hard evidence that these models cannot be simulated by a classical machine - barring complexity theoretical consequences which would make any honest theoretical computer scientist blush.

This research line is only more emphasized by the perceived blow that Tang and a number of follow-up works have provided: quantum database-based approaches can (in a precise way) be dequantized, which has removed some of the glitter from the perceived importance of quantum algorithms in this line. However, dequantization does not really mean "classical algorithms are as good"; it means "we now know the separation is not exponential." But a high degree polynomial separation can be just as good in practice (or even better, depending on the fine-grained parameters swallowed by the big-O notation, and actually relevant instance sizes). More on this later.

Currently, perhaps the hottest trend in quantum-enhanced learning algorithms is playing around with parametrized quantum circuits, whose parameters are tuned much like the weights of a neural network: it is actually bringing us back to 1990s, when the term "quantum neural networks" (in a way, even to the very first work on the topic of Lewenstein in '94 [ 19]) was used to mean precisely that: a circuit whose parameters are tuned to realize a desired mapping. Once fringe, these works are now front and center, as one of the most promising applications of NISQ architectures. Why? Do you have no idea how to come up with an algorithm for your restricted quantum machine? No worries, put parameters in, and treat it as a model, it will do the "best it can", and who knows, just maybe it is the best model ever! These days we hear much less about the big-data quantum-database quantum machine learning ideas which were almost synonymous with QML.

Each one of the phases of QML had review works which identified "the key" for QML, the biggest questions, and what should be done next. And it is right that they should do so, as research needs direction. Most of such directions got, very rapidly (especially having in mind that while QML is "hot", objectively few researchers work in this area), substituted with a new series of questions. This, we believe, will be unavoidable for young fields, which are growing faster than maturing.

Right now, QML has a few "obvious" objectives. First and foremost, there is the promise that QML may be the "best application for quantum computers." For this we would need to present classes of algorithms with end-to-end speed up, with all the fine-print accounted for [ 20], and which have strong evidence of a separation relative to classical algorithms. Furthermore, these algorithms should be good learning algorithms; a feature that is often put in the second place in the race for quantum-classical separations.

Second there is the hope that they are also the "best applications for near-term devices". Combine variational circuits with the inherent (well, potentially one could hope such a thing may be true) robustness of ML algorithms to data robustness, and show that this robustness percolates to the learning algorithm itself. Then you have the golden-egg laying goose: robust, near-term application of quantum computers that matters. Now what is missing in this story is any convincing evidence that these algorithms are actually any good, but we shall have to suffer through this for a while until some new theory gets developed. This does not sound particularly appealing, but nobody knows why deep learning neural networks work great either.

The plan for the future of QML seems solid, but then, as Mike Tyson and Joe Louis say: "Everybody's got plans... until they get hit." As we learn more, these obvious objectives will certainly change, and, e.g., with the possibility of dequantizations (what if additive-error-noisy systems can be classically simulated in many cases?), quite abruptly at that.
However, we are now focusing on QML in a too narrow a sense. The entire topics of QML change as new ideas emerge.

So here is what would be ideal. We should remove the pressure from having a clear perspective of "where the field is going", especially since the contours of the field are not set yet. We should remind ourselves that some of the biggest breakthroughs where achieved by pure curiosity-driven research, and not by ticking off this weeks "target achievements". That would be ideal. In reality, modern science, to live, needs publications and goals. This must be acknowledged as well.

We are happy that the vast majority of works follow this trajectory, as long as time and again, some light is shed on less acknowledged, curiosity-driven, fringe topic research. The remainder of this work aims to do that. The selection of these works is not based on them being milestones on a well-planned out roadmap. Rather, they are works we like, we found inspiring, fun, or visionary.

Below is a non-exhaustive list of some of the more recent papers highlighting QML aspects not covered in detail in the above reviews. Here, we can start distinguishing the flavours of QML to more precise detail.

1. Supervised and unsupervised learning

2. Reinforcement learning and AI aspects

Both authors of this perspective are rather fond of old-school AI, also witnessed in the paper Quantum Enhanced Inference in Markov Logic Networks [ 29]. This shows quantum advantages for Gibbs sampling in networks that combine causal networks and formal deduction, but there is plenty of more interesting questions to answer in this domain.

3. Machine learning in (experimental) physics

Next we have some newer lines of thought on machine learning applied to (experimental) physics. Machine learning can of course be used to help us speed up various types of information processing tasks, but in Detecting quantum speedup by quantum walk with convolutional neural networks [ 30], the authors show that neural networks can detect whether a quantum algorithm can produce a speed-up in quantum walk scenarios where theoretical bounds are not known. This result is exciting especially in the context of real-world practical computing, where theoretical worst-case bounds are often less important than heuristic domain-specific performance.

In a different direction, in Machine learning for long-distance quantum communication [ 31] it is shown that AI systems based on reinforcement learning can also be challenged to actually design new quantum communication protocols. Together with works like [ 32], where machine learning is tasked to invent new error correcting codes, such works push the envelopes of what we may come to expect machines to be capable of.

Switching gears from discovering protocols to unveiling nature itself, in Discovering physical concepts with neural networks [ 33], the authors investigate machine-assisted discovery in the physics realm, including inferring the bounds on the dimensionality of quantum systems.

In a similar, but more quantitative sense of machine-assisted research, in Automated discovery of characteristic features of phase transitions in many-body localization [ 34], the authors further illustrate that the true breakthroughs will come when machines discover truly new properties, like new order parameters. This is possible in the unsupervised and weakly supervised regime, as this paper shows.

We finalize this section with a paper which is on the border of genuine ML applications, but it is certainly related; Convex optimization of programmable quantum computers [ 35] provides an interesting observation that finding optimal program states for finite gate arrays to realize a target quantum evolution constitutes a (perhaps unexpectedly) convex optimization problem. This opens the doors to plethora of classical (in the sense of "being a classic") optimization methods for "optimal programming" of programmable quantum circuits.

4. Quantum-inspired machine learning

There has been much movement in quantum-inspired machine learning; although this is a borderline topic for QML, it is easy to imagine that many results here may inspire new quantum algorithms right back.

A prominent new research line considers using tensor networks in place of neural networks for learning, as illustrated in, e.g., Supervised Learning with Quantum-Inspired Tensor Networks [ 2]. This research line is new but also deeply rooted, due to the intricate mathematical connections between neural nets, tensor networks, and learning problems and significant bodies of research that studied some of the aspects. Although this research line is briefly mentioned in review [ 16] (we focus on research not previously covered here), this is a rapidly growing field of research, which will likely deserve its own review papers.

This brings us to the breakthrough results of Ewin Tang, who showed that classical randomized algorithms can achieve exponential improvements over standard classical approaches for many settings previously reserved for quantum linear algebra. That is, the gap between classical and quantum algorithms is no longer exponential, but it is critical to note it is still a high-polynomial separation. Current exact polynomial degrees render the classical algorithms in general insufficiently efficient for real-world use, whereas quantum algorithms would work. The first study in the question of the actual real-world advantages of quantum processing is given in Quantum-inspired algorithms in practice [ 36].

Regarding the dequantization results themselves, Ewin has a small online review of her own on the topic, so best hear it from the expert herself.

QML and beyond

This brings us to numerous topics that are still not generally directly included in QML, but we would not be surprised if an explicit (applied) link emerges presently. Here we list a few interesting examples (and not necessarily the very first papers on the topic).

In Learning Hidden Quantum Markov Models [ 37], the authors explore the learning of (quantum) hidden Markov models, and they explicitly associate their work with the QML domain. Learning hidden Markov models is intimately related to unsupervised learning, can be used in classification, and their connection to machine learning is quite obvious as they are special cases of Bayesian networks. Will hidden quantum Markov models be as relevant for quantum machine learning? Time will tell.

Next we move from learning to elements of meaning, and natural language processing (which is certainly one of the most prominent long-term objectives of AI). Here, quantum logicians have been investigating the suitability of the non-commutative structure of quantum theory to model aspects of natural languages, e.g., Word Vectors and Quantum Logic: Experiments with negation and disjunction [ 38] (the field is much larger and older than this single example). Once again, moving from more fundamentally flavoured research, to research with a pragmatic "what can we enhance"-hue, recently we have seen papers providing algorithmic ideas where quantum computing offers an edge in language processing. Examples include. Quantum Algorithms for Compositional Natural Language Processing [ 39] and Quantum Language Processing [ 40]. These developments will no doubt influence ideas in quantum AI.

As a final comment, the entire field of "genuinely quantum" machine learning (where the data itself is quantum) is still finding its right place and full recognition. Perhaps as quantum technologies mature, and problems of quantum learning become genuinely practical, the field will crystallize and grow. Although this field is acknowledged in a few reviews we mentioned, the interested reader can see new ideas where the field may be heading in, e.g., Unsupervised classification of quantum data [ 41] (where we move from supervised to unsupervised generalizations). In a related vein, we have already mentioned the work of Cong [ 24] where algorithms for the deep neural-network-like analysis of actual quantum states are suggested.

In summary, QML is diverse, growing, inclusive, and it is rich in open questions. We (the authors of this non-review) are biased towards topics we are interested in and we are certain we are missing many new exciting ideas that have been popping up in recent times. Capturing all the QML trends, which will in the end be central is, for the time being, an impossible task - and, in a way, this is the key message of this note.

Acknowledgements
We would like to thank Sofiene Jerbi, Charles Moussa and Casper Gyurik in helping us compile the books, reviews, articles and for the proofreading of the text.

[2] Edwin Stoudenmire and David J Schwab. Supervised learning with tensor networks. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29, pages 4799-4807. Curran Associates, Inc., 2016.

[4] Peter Wittek. Quantum Machine Learning: What Quantum Computing Means to Data Mining. Elsevier Science, 2016.

[5] Maria Schuld and Francesco Petruccione. Supervised Learning with Quantum Computers. Springer International Publishing, 2018.

[12] Srinivasan Arunachalam and Ronald de Wolf. A survey of quantum learning theory, 2017.

[15] Abu Kamruzzaman, Yousef Alhwaiti, Avery Leider, and Charles C. Tappert. Quantum deep learning neural networks. In Lecture Notes in Networks and Systems, pages 299-311. Springer International Publishing, February 2019.

[23] Jonathan Allcock, Chang-Yu Hsieh, Iordanis Kerenidis, and Shengyu Zhang. Quantum algorithms for feedforward neural networks, 2018.

[25] Tongyang Li, Shouvanik Chakrabarti, and Xiaodi Wu. Sublinear quantum algorithms for training linear and kernel-based classifiers. 2019.

[26] Iordanis Kerenidis and Alessandro Luongo. Quantum classification of the mnist dataset via slow feature analysis, 2018.

[27] Pooya Ronagh. Quantum algorithms for solving dynamic programming problems, 2019.

[28] Arjan Cornelissen. Quantum gradient estimation and its application to quantum reinforcement learning, 2019. MSc Thesis.

[31] Julius Wallnöfer, Alexey A. Melnikov, Wolfgang Dür, and Hans J. Briegel. Machine learning for long-distance quantum communication, 2019.

[35] Leonardo Banchi, Jason Pereira, Seth Lloyd, and Stefano Pirandola. Convex optimization of programmable quantum computers, 2019.

[36] Juan Miguel Arrazola, Alain Delgado, Bhaskar Roy Bardhan, and Seth Lloyd. Quantum-inspired algorithms in practice, 2019.

[37] Siddarth Srinivasan, Geoff Gordon, and Byron Boots. Learning hidden quantum Markov models. In Amos Storkey and Fernando Perez-Cruz, editors, Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, volume 84 of Proceedings of Machine Learning Research, pages 1979-1987, Playa Blanca, Lanzarote, Canary Islands, 09-11 Apr 2018. PMLR.

[38] Dominic Widdows and Stanley Peters. Word vectors and quantum logic: Experiments with negation and disjunction. In In Proceedings of the 8th Mathematics of Language Conference, pages 141-154, 2003.

[40] Nathan Wiebe, Alex Bocharov, Paul Smolensky, Matthias Troyer, and Krysta M Svore. Quantum language processing, 2019.

Cited by

[1] Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing 387 (2020) ISBN:9781450369794.

[2] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "Machine learning meets quantum foundations: A brief survey", AVS Quantum Science 2 3, 034101 (2020).

[3] Lucas Lamata, "Quantum machine learning and quantum biomimetics: A perspective", Machine Learning: Science and Technology 1 3, 033002 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2020-07-28 10:39:11). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2020-07-28 10:39:11).

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A non-review of Quantum Machine Learning: trends and explorations

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Vedran Dunjko and Peter Wittek, Quantum Views 4, 32 (2020). About this non-review By mid-2019, both Peter and myself had found ourselves numerous times in situations where we were asked to define what Quantum Machine Learning is (and what it i...

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חוקרים מאוניברסיטת תל אביב בנו בריכת גלים מיוחדת, באורך חמישה מטרים, שמדמה התפשטות גלים קוונטיים. באמצעות הבריכה, צוות החוקרים בהובלת הדוקטורנט גרי גאורגי רוזנמן, פרופ' עדי אריה, פרופ' לב שמר ופרופ' וולפגנג שלייך, צפו לראשונה בתופעה קוונטית בשם פאזת קנארד - נדבך מרכזי בתורת הקוונטים שנחזה בשנת 1927, אבל מעולם לא נמדד במעבדה. מאמרים על התגלית התפרסמו ב-Physical Review Letters וב-Physical Review E- Rapid Communications.

"המטרה של המחקר היא לייצר פלטפורמה שמקיימת אנלוגיה מתמטית שלמה עם משוואת שרדינגר, המשוואה העיקרית של מכניקת הקוונטים, אבל בהתקן ענק שניתן לראותו בעין", מסביר רוזנמן. "הבריכה מייצרת גלי כבידה משטחיים, שמתנהגים באופן דומה לגלי חומר זעירים בעולם הקוונטי, וכך מאפשרת לנו למדוד תופעות קוונטיות במערכת מקרוסקופית. מבחינה תיאורטית יש תופעות במכניקת הקוונטים שקשה מאוד למדוד, ואנחנו מצאנו דרך לעקוף את המגבלה הזאת".

מכניקת הקוונטים היא שמה של תיאוריה מדעית המתארת את עולם החלקיקים ברמה התת-אטומית. אחת מאבני היסוד בתורת הקוונטים היא התיאור הכפול של גופים פיזיקליים - הן כחלקיקים והן כגלים. בדומה לגלים בים, גלים קוונטיים מתארים תנודות מחזוריות במרחב ובזמן, ומתאפיינים בפאזה, או מופע. הפאזה של הגל קובעת האם בנקודה מסוימת במרחב ובזמן יהיה הגל בשיא המשרעת שלו, בשפל המשרעת או בכל מצב ביניים אחר.

בשנת 1927 חזה הפיזיקאי התיאורטי ארל הסה קנארד כי חלקיק קוונטי שמופעל עליו כוח קבוע, כמו כוח כבידה על גוף בעל מסה, או שדה חשמלי קבוע על חלקיק טעון חשמלית, יצבור פאזה הקרויה כיום על שמו - פאזת קנארד. ב-93 השנים שחלפו מאז, פיזיקאים ניסו לחשוב על ניסויים יצירתיים שיאפשרו להם למדוד את הפאזה של קנארד - אך עד כה ללא הועיל.

"אנחנו זיהינו את השקילות בין מערכת קוונטית שמקיימת את משוואת שרדינגר לבין מערכת הידרודינמית של גלים שמתקדמים על פני מים", אומר רוזנמן. "בשלב השני, לקחנו את הקשר הזה שלב אחד קדימה ובנינו מערכת שמקיימת את משוואת שרדינגר וגם מפעילה כוח קבוע על הגל. על ידי שינוי הכוחות, אנחנו יכולים לראות במו עינינו כיצד חלקיקים צוברים את הפאזות שנחזו על ידי קנארד באופן תאורטי. צריך להבין שלפאזת קנארד עשויים להיות גם שימושים מעשיים, למשל במדידה רגישה של כוחות. בימים אלה אנחנו משתמשים בבריכת הגלים כדי לצפות לראשונה באפקטים פיזיקליים אחרים, כמו צבירת הפאזה של גלים הנעים בסמוך לחורים שחורים".

המחקר נערך בין היתר על ידי חוקרים מבתי הספר לפיזיקה, הנדסה מכנית והנדסת חשמל באוניברסיטת תל אביב, בשיתוף פעולה עם חוקרים מהמכון לפיזיקה קוונטית באולם שבגרמניה.

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חוקרים ישראלים פתרו תעלומה מדעית בת 93 שנה

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חוקרים מאוניברסיטת תל אביב בנו בריכת גלים מיוחדת, באורך חמישה מטרים, שמדמה התפשטות גלים קוונטיים. באמצעות הבריכה, צוות החוקרים בהובלת הדוקטורנט גרי גאורגי רוזנמן, פרופ' עדי אריה, פרופ' לב שמר ופרופ' וולפגנג שלייך, צפו לראשונה בתופעה קוונטית בשם פאזת קנארד - נדבך מרכזי בתורת הקוונטים שנחזה בשנת 1927, אבל מעולם לא נמדד במעבדה.

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A joke that I saw on the internet:

Heisenberg is driving home with Schrodinger after a long day of physics when a policeman pulls them over. He knocks on the drivers window and asks: "Excuse me sir, do you know how fast you were going?" To which Heisenberg laughs and answers: "No, but I can tell you exactly where I am." and both of the physicists start to laugh. The policeman does not understand and thinks that there is something fishy with these two, so he asks if he can look in the boot. The physicists open the boot for him and to his surprise there is a dead cat. He is really confused and asks Schrodinger if he knew that the cat in the boot was dead. Schrodinger answers: "Well, I do now."

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Officials say quantum technology could become the basis for a more secure internet architecture

US officials and scientists have begun laying the groundwork for a more secure "virtually unhackable" internet based on quantum computing technology.

At a presentation Thursday, Department of Energy (DOE) officials issued a report that lays out a blueprint strategy for the development of a national quantum internet, using laws of quantum mechanics to transmit information more securely than on existing networks.

The agency is working with universities and industry researchers on the engineering for the initiative with the aim of creating a prototype within a decade.

In February, scientists from DOE's Argonne National Laboratory and the University of Chicago created a 52-mile (83-kilometer) "quantum loop" in the Chicago suburbs, establishing one of the longest land-based quantum networks in the nation.

The aim is to create a parallel, more secure network based on quantum "entanglement," or the transmission of sub-atomic particles.

"One of the hallmarks of quantum transmissions is that they are exceedingly difficult to eavesdrop on as information passes between locations," according to the Energy Department statement.

"Scientists plan to use that trait to make virtually unhackable networks."

The department said early adopters could include the banking and health services sectors, adding that there would be applications for national security and aircraft communications.

"Eventually, the use of quantum networking technology in mobile phones could have broad impacts on the lives of individuals around the world," the statement added.

The agency's 17 national laboratories will serve as the backbone of the coming quantum internet, which has initial government funding.

"The foundation of quantum networks rests on our ability to precisely synthesize and manipulate matter at the atomic scale, including the control of single photons," said David Awschalom, a professor at the University of Chicago and senior scientist at Argonne National Laboratory.

© 2020 AFP

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Quantum loop: US unveils blueprint for 'virtually unhackable' internet

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US officials and scientists have begun laying the groundwork for a more secure "virtually unhackable" internet based on quantum computing technology. At a presentation Thursday, Department of Energy (DOE) officials issued a report that lays out a blueprint strategy for the development of a national quantum internet, using laws of quantum mechanics to transmit information more securely than on existing networks.

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Richard Feynman - Nobody understands Quantum Mechanics

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Richard Feynman - Nobody understands Quantum Mechanics

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Richard Feynman - Nobody understands Quantum Mechanics

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"the first commercial application that depends on a quantum processor to perform a critical live task "

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"the first commercial application that depends on a quantum processor to perform a critical live task "

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Quantum Shuttle: Traffic Navigation with Quantum Computing

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The Web Summit conference in Lisbon, Portugal, is one of the biggest technology conferences in Europe, attended by tens of thousands of people every year. The high influx of people into Lisbon causes significant stress on the city's transit services for the duration of the conference.

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"The Sun Only Shines Because Of Quantum Physics"

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"The Sun Only Shines Because Of Quantum Physics"

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Earth, as we know it, is only teeming with life because of the influence of our Sun. Its light and heat provides every square meter of Earth - when it's in direct sunlight - with a constant ~1500 W of power, enough to keep our planet at a comfortable temperature for liquid water to continuously exist on its surface. Just like the hundreds of billions of stars in our galaxy amidst the trillions of galaxies in the Universe, our Sun shines continuously, varying only slightly over time.

But without quantum physics, the Sun wouldn't shine at all. Even in the extreme conditions found in the core of a massive star like our Sun, the nuclear reactions that power it could not occur without the bizarre properties that our quantum Universe demands. Thankfully, our Universe is quantum in nature, enabling the Sun and all the other stars to shine as they do. Here's the science of how it works.

Starlight is the single greatest source of energy in the Universe throughout its entire 13.8 billion year history, subsequent to the hot Big Bang. These large, massive concentrations of hydrogen and helium contract under their own gravity when they first form, causing their cores to become denser and denser all while heating up. Eventually, a critical threshold is reached - at temperatures of ~4 million kelvin and densities exceeding that of solid lead - where nuclear fusion begins in the star's core.

But here's the puzzle: you can determine exactly how much energy the particles in the Sun must have, and calculate how those energies are distributed. You can calculate what types of collisions occur between protons in the Sun's core, and compare that with how much energy is required to actually bring two protons into physical contact with one another: overcoming the electric repulsion between them.

And when you do your calculations, you find a shocking conclusion: there are zero collisions happening there with enough energy to lead to nuclear fusion. Zero. None at all.

At first glance, this would appear to make nuclear fusion - and hence, the ability of the Sun to shine - completely impossible. And yet, based on the energy we observe coming from the Sun, we know that it does, in fact, shine.

Deep inside the Sun, in the innermost regions where the temperature ranges between 4 million all the way up to 15 million kelvin, the nucleus of four initial hydrogen atoms (i.e., individual protons) will fuse together in a chain reaction, with the end result producing a helium nucleus (made of two protons and two neutrons), along with the release of a significant amount of energy.

That energy is carried away in the form of both neutrinos and photons, and while the photons might spend over 100,000 years before they make it to the Sun's photosphere and radiate into space, the neutrinos exit the Sun in mere seconds, where we've been detecting them on Earth since the 1960s.

You might think about this scenario and be a bit puzzled, since it isn't obvious how energy is released from these reactions. Neutrons, you see, are ever so slightly more massive than protons are: by about 0.1%. When you fuse four protons into a nucleus containing two protons and two neutrons, you might think that reaction would require energy instead of emitting it.

If all of those particles were free and unbound, that would be true. But when neutrons and protons are bound together into a nucleus such as helium, they wound up being bound together so tightly that they're actually significantly less massive than their individual, unbound constituents. While two neutrons have about 2 MeV (where an MeV is one million electron-volts, a measure of energy) more energy than two protons are - via Einstein's E = mc² - a helium nucleus is the equivalent of 28 MeV lighter than four unbound protons.

In other words, the process of nuclear fusion releases energy: about 0.7% of whatever protons fuse together gets converted into energy, carried by both neutrinos and photons.

We observe the Sun emitting, over its entire surface, a continuous power output of 4 × 10 ²⁶ Watts. That amount of energy translates into an enormous number of protons - somewhere upwards of 10 ³⁸ of them - fusing together in this chain reaction every second. This is spread out over an enormous volume of space, of course, since the interior of the Sun is enormous; the average human being metabolizing their daily food produces more energy than an equivalent human-sized volume of the Sun.

But with all of those reactions occurring in the interior of the Sun, you might start to wonder how efficient these reactions are. Do we really get enough of them to generate all the power that the Sun creates? Can this really lead to such an enormous energy output, and explain how the Sun shines?

It's a complex question, and if you start to think about it quantitatively, here are the numbers you arrive at.

The Sun is far larger and more massive than anything we've experienced in our lives. If you were to take the entire planet Earth and line up a series of them across the Sun's diameter, it would take 109 Earths to make it all the way across. If you were to take all the mass contained within planet Earth, you'd have to accumulate more than 300,000 of them to equal the mass of our Sun.

All told, there are some 10 ⁵⁷ particles making up the Sun, with right around 10% of those particles present in the fusion region that defines the Sun's core. Inside the core, here's what's happening:

Individual protons reach tremendous speeds, up to ~500 km/s in the Sun's central core, where temperatures reach up to 15 million K.

These fast-moving particles are so numerous that each proton experiences billions of collisions each second.

And only a tiny fraction of these collisions need to create deuterium - just 1 in 10²⁸ - in a fusion reaction to produce the needed energy.

This sounds reasonable, right? Surely, given the enormous number of proton collisions that occur, how fast they're moving, and the fact that only a tiny, almost imperceptible fraction of them would need to actually fuse, this could be achievable.

So we do the math. We calculate, based on how particles behave and move when you have a whole lot of them under a given set of energies and velocities, how many proton-proton collisions have enough energy to initiate nuclear fusion in those reactions.

To get there, all two protons have to do is get close enough to physically touch, overcoming the fact that they both have positive electric charges, and that like charges repel.

So how many, of the ~10 ⁵⁶ protons in the Sun's core, colliding billions of times per second, actually have enough energy to cause a fusion reaction to occur?

Exactly zero.

And yet, somehow, it happens. Not only does nuclear fusion successfully power the Sun, but stars far less massive - and with far lower core temperatures - than our own. Hydrogen gets converted to helium; fusion occurs; starlight gets created; planets become potentially habitable.

So what's the secret?

This is the key place where quantum physics comes into play. Down at a subatomic level, atomic nuclei don't actually behave as particles alone, but rather as waves. Sure, you can measure the physical size of a proton, but doing so makes its momentum inherently uncertain. You can also measure the momentum of a proton - essentially what we did when we calculated what its velocity is - but doing so makes its position more inherently uncertain.

Each proton, instead, is a quantum particle, where its physical location is better described by a probability function than a pinned-down position.

Because of the quantum nature of these protons, the wavefunctions of two protons can overlap. Even protons which don't have enough energy to overcome the repulsive electric force between them can see their wavefunctions overlap, and that overlapping means they have a finite probability of experiencing quantum tunneling: where they can end up in a more stable bound state than their initial, free state.

Once you form deuterium from two protons - the hard part - the rest of the chain reaction can proceed quite quickly, leading to the formation of helium-4 in short order.

But the probability of forming deuterium is very small. In fact, for any particular proton-proton interaction that occurs in the Sun's core, practically all of them will have the simplest result imaginable: their wavefunctions temporarily overlap, then they stop overlapping, and all you wind up with is two protons, the same as what you started with. But a very small fraction of the time, about 1 in every 10 ²⁸ collisions (remember that number from earlier?), two protons wind up fusing together, creating a deuteron, as well as a positron and a neutrino, and possibly also a photon.

When the wavefunction of two protons in the Sun's core overlap, there's only a minuscule chance that they'll do anything other than return to being two protons. The odds of them fusing together to make a deuterium nucleus are about the same as winning the Powerball lottery three times in a row: astronomically small. And yet, there are so many protons inside the Sun that this successfully occurs so often that it powers not only our Sun, but practically all of the stars in the Universe.

Over the past 4.5 billion years, this has happened enough time in our Sun that it has lost approximately the mass of Saturn due to nuclear fusion and Einstein's most famous equation: E = mc². If it weren't for the quantum nature of the Universe, however, nuclear fusion wouldn't occur in the Sun at all, and Earth would simply be a cold, lifeless rock floating in the abyss of space. It's only because of the uncertainty inherent to position, momentum, energy, and time, that our existence is possible at all. Without quantum physics, the Sun wouldn't be able to shine. In a very real sense, we really did win the cosmic lottery.

... Read More

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The Sun Only Shines Because Of Quantum Physics

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Earth, as we know it, is only teeming with life because of the influence of our Sun. Its light and heat provides every square meter of Earth - when it's in direct sunlight - with a constant ~1500 W of power, enough to keep our planet at a comfortable temperature for liquid water to continuously exist on its surface.