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
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.
- Supervised and unsupervised learning
- Reinforcement learning and AI aspects
- Machine learning in (experimental) physics
- Quantum-inspired machine learning
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 , ideas involving tensor networks as learning models (see works of Stoudenmire ), and inspirations behind the projective simulation of Briegel .
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.
By now, a few books have emerged which align with quantum machine learning, and which emphasize different aspects of the prospective field. The first book which carried the title “quantum machine learning” was Quantum Machine Learning: What Quantum Computing Means to Data Mining, by one of the authors . It was the entry point to quantum-enhanced machine learning, suitable for persons with a machine learning background. Supervised Learning with Quantum Computers, by Maria Schuld and Francesco Petruccione  is a more recent book also focusing on quantum-enhanced machine learning, and places more focus on the potential of near-term gate-based architectures.
Aside from the books above, which directly address the core of modern QML, it is worth while to keep an eye out on books that emphasize less mainstream ideas. Principles of Quantum Artificial Intelligence, by Andraes Wichert , is technically not on machine learning, but rather on “problem solving” aspects important for symbolic AI and reasoning. Given the close relationship between AI and ML, it makes sense to keep this book in mind as well. Quantum Robotics: A Primer on Current Science and Future Perspectives by Prateek Tandon, Stanley Lam and Ben Shih , places a bit more emphasis on QML aspects which could go in the way of robotics, although much of the book is dedicated to more standard quantum computing and QML ideas.
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 . 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 . At the time, the jury was out on the perspectives. Modern results in shallow architectures (see, e.g., the book  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  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.
The number of original results kept growing rapidly, with more interest in near-term feasibility and various cross-overs between quantum physics and machine learning, as well as in theoretical foundations. On the theory side, Quantum machine learning: a classical perspective  covered quantum-enhanced machine learning with an algorithmic and complexity theoretical emphasis. Srinivasan Arunachalam and Ronald de Wolf wrote A Survey of Quantum Learning Theory , dedicated to quantum probably approximately correct learning, and related topics. This formal aspect of quantum machine learning is often underrepresented in literature, yet it is one of the oldest applications of quantum computing.
One of the authors of this perspective co-authored Machine learning & artificial intelligence in the quantum domain: a review of recent progress , 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 . Quantum neural networks finally also achieved a level of maturity, as summarized in Quantum Deep Learning Neural Networks .
Looking at other cross-overs, Machine learning and the physical sciences , is an excellent review of quantum-applied machine learning. It is also one of the first reviews dedicated to the thriving topic. Applications of ML in chemistry and quantum chemistry is perhaps tangential to QML in a narrower sense, but since quantum computing loves quantum chemistry, we can expect many ideas to cross over. This idea is immortalized in Guest Editorial: Special Topic on Data-Enabled Theoretical Chemistry .
Learning in quantum control: High-dimensional global optimization for noisy quantum dynamics  reviews machine learning ideas applied in quantum control, but with new results; so a research paper/review paper hybrid.
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 ) 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 , 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
Let us start with some newer lines of thought on quantum-enhanced machine learning. For a starter, hardware for continuous-variable quantum computing is emerging, although the paradigm is tricky due to the difficulty of error correction. Continuous-variable quantum neural networks  offers a model in the NISQ, uncorrected era, and it may give the edge we are after. Bayesian Deep Learning on a Quantum Computer  benefits from recent classical results that connect Bayesian learning to deep learning, and both can benefit from quantum computers, although the paper is tongue-in-cheek, since it has matrix exponentiation in its core and it shows experimental results on NISQ-era hardware, which are not exactly compelling.
Nevertheless, quantum computers have an irresistible appeal to train feedforward neural networks; see, for instance, Quantum algorithms for feedforward neural networks . Quantum Convolutional Neural Networks  introduces a novel model quantum-generalizing convolutional neural networks, which may be suitable for the problems of learning of quantum states. This paper also fits in the domain of quantum-generalized machine learning.
Sublinear quantum algorithms for training linear and kernel-based classifiers  constitutes a long awaited application of quantum multiplicative weight primal-dual ideas in supervised machine learning. Quantum classification of the MNIST dataset via Slow Feature Analysis  proposed classification based on the Quantum Frobenius Distance, which we can also think of as a kernel function.
2. Reinforcement learning and AI aspects
There have been new developments in theoretical and applied aspects of quantum-enhanced reinforcement learning, as well. Quantum Algorithms for Solving Dynamic Programming Problems  proves separations and lower bounds for the learning of exact optimal policies given quantum access to transition functions in Markov decision processes. Quantum gradient estimation and its application to quantum reinforcement learning  is a truly excellent master thesis in quantum computing, showing the potential of quantum computing for policy gradient methods.
Both authors of this perspective are rather fond of old-school AI, also witnessed in the paper Quantum Enhanced Inference in Markov Logic Networks . 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 , 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  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 , 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 , 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 , 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  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 . 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  (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 .
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 , 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  (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  and Quantum Language Processing . 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  (where we move from supervised to unsupervised generalizations). In a related vein, we have already mentioned the work of Cong  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.
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.
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 Maciej Lewenstein. Quantum perceptrons. Journal of Modern Optics, 41(12):2491–2501, December 1994.
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 Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolás Quesada, and Seth Lloyd. Continuous-variable quantum neural networks, 2018.
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 Patrick Huembeli, Alexandre Dauphin, Peter Wittek, and Christian Gogolin. Automated discovery of characteristic features of phase transitions in many-body localization. Phys. Rev. B, 99:104106, Mar 2019.
 Leonardo Banchi, Jason Pereira, Seth Lloyd, and Stefano Pirandola. Convex optimization of programmable quantum computers, 2019.
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