# Quantum machine learning

**Quantum machine learning** is an emerging interdisciplinary research area at the intersection of quantum physics and machine learning.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]} The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer.^{[6]}^{[7]}^{[8]} This includes hybrid methods that involve both classical and quantum processing, where computationally expensive subroutines are outsourced to a quantum device.^{[9]}^{[10]}^{[11]} Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data.^{[12]} Beyond quantum computing, the term "quantum machine learning" is often associated with machine learning methods applied to data generated from quantum experiments, such as learning quantum phase transitions^{[13]}^{[14]} or creating new quantum experiments.^{[15]} Quantum machine learning also extends to a branch of research that explores methodological and structural similarities between certain physical systems and learning systems, in particular neural networks. For example, some mathematical and numerical techniques from quantum physics carry over to classical deep learning and vice versa.^{[16]}^{[17]}^{[18]} Finally, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as "quantum learning theory".^{[19]}

## Machine learning with quantum computers

Quantum-enhanced machine learning refers to quantum algorithms that solve tasks in machine learning, thereby improving a classical machine learning method. Such algorithms typically require one to encode the given classical dataset into a quantum computer to make it accessible for quantum information processing. After this, quantum information processing routines can be applied and the result of the quantum computation is read out by measuring the quantum system. For example, the outcome of the measurement of a qubit could reveal the result of a binary classification task. While many proposals of quantum machine learning algorithms are still purely theoretical and require a full-scale universal quantum computer to be tested, others have been implemented on small-scale or special purpose quantum devices.

### Linear algebra simulation with quantum amplitudes

A number of quantum algorithms for machine learning are based on the idea of *amplitude encoding*, that is, to associate the amplitudes of a quantum state with the inputs and outputs of computations.^{[22]}^{[23]}^{[24]}^{[25]} Since a state of
qubits is described by
complex amplitudes, this information encoding can allow for an exponentially compact representation. Intuitively, this corresponds to associating a discrete probability distribution over binary random variables with a classical vector. The goal of algorithms based on amplitude encoding is to formulate quantum algorithms whose resources grow polynomially in the number of qubits
, which amounts to a logarithmic growth in the number of amplitudes and thereby the dimension of the input.

Many quantum machine learning algorithms in this category are based on variations of the quantum algorithm for linear systems of equations^{[26]} which, under specific conditions, performs a matrix inversion using an amount of physical resources growing only logarithmically in the dimensions of the matrix. One of these conditions is that a Hamiltonian which entrywise corresponds to the matrix can be simulated efficiently, which is known to be possible if the matrix is sparse^{[27]} or low rank.^{[28]} For reference, any known classical algorithm for matrix inversion requires a number of operations that grows at least quadratically in the dimension of the matrix.

Quantum matrix inversion can be applied to machine learning methods in which the training reduces to solving a linear system of equations, for example in least-squares linear regression,^{[23]}^{[24]} the least-squares version of support vector machines,^{[22]} and Gaussian processes.^{[25]}

A crucial bottleneck of methods that simulate linear algebra computations with the amplitudes of quantum states is state preparation, which often requires one to initialise a quantum system in a state whose amplitudes reflect the features of the entire dataset. Although efficient methods for state preparation are known for specific cases,^{[29]}^{[30]} this step easily hides the complexity of the task.^{[31]}

### Quantum machine learning algorithms based on Grover search

Another approach to improving classical machine learning with quantum information processing uses amplitude amplification methods based on Grover's search algorithm, which has been shown to solve unstructured search problems with a quadratic speedup compared to classical algorithms. These quantum routines can be employed for learning algorithms that translate into an unstructured search task, as can be done, for instance, in the case of the k-medians^{[32]} and the k-nearest neighbors algorithms.^{[6]} Another application is a quadratic speedup in the training of perceptron.^{[33]}

Amplitude amplification is often combined with quantum walks to achieve the same quadratic speedup. Quantum walks have been proposed to enhance Google's PageRank algorithm^{[34]} as well as the performance of reinforcement learning agents in the projective simulation framework.^{[35]}

### Quantum-enhanced reinforcement learning

Reinforcement learning is a third branch of machine learning, distinct from supervised and unsupervised learning, which also admits quantum enhancements.^{[36]}^{[35]}^{[37]}^{[38]} In quantum-enhanced reinforcement learning, a quantum agent interacts with a classical environment and occasionally receives rewards for its actions, which allows the agent to adapt its behavior—in other words, to learn what to do in order to gain more rewards. In some situations, either because of the quantum processing capability of the agent,^{[35]} or due to the possibility to probe the environment in superpositions,^{[21]} a quantum speedup may be achieved. Implementations of these kinds of protocols in superconducting circuits^{[39]} and in systems of trapped ions^{[40]} have been proposed.

### Quantum sampling techniques

Sampling from high-dimensional probability distributions is at the core of a wide spectrum of computational techniques with important applications across science, engineering, and society. Examples include deep learning, probabilistic programming, and other machine learning and artificial intelligence applications.

A computationally hard problem, which is key for some relevant machine learning tasks, is the estimation of averages over probabilistic models defined in terms of a Boltzmann distribution. Sampling from generic probabilistic models is hard: algorithms relying heavily on sampling are expected to remain intractable no matter how large and powerful classical computing resources become. Even though quantum annealers, like those produced by D-Wave Systems, were designed for challenging combinatorial optimization problems, it has been recently recognized as a potential candidate to speed up computations that rely on sampling by exploiting quantum effects.^{[41]}

Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.^{[42]}^{[43]}^{[44]}^{[45]} The standard approach to training Boltzmann machines relies on the computation of certain averages that can be estimated by standard sampling techniques, such as Markov chain Monte Carlo algorithms. Another possibility is to rely on a physical process, like quantum annealing, that naturally generates samples from a Boltzmann distribution. The objective is to find the optimal control parameters that best represent the empirical distribution of a given dataset.

The D-Wave 2X system hosted at NASA Ames Research Center has been recently used for the learning of a special class of restricted Boltzmann machines that can serve as a building block for deep learning architectures.^{[44]} Complementary work that appeared roughly simultaneously showed that quantum annealing can be used for supervised learning in classification tasks.^{[42]} The same device was later used to train a fully connected Boltzmann machine to generate, reconstruct, and classify down-scaled, low-resolution handwritten digits, among other synthetic datasets.^{[43]} In both cases, the models trained by quantum annealing had a similar or better performance in terms of quality. The ultimate question that drives this endeavour is whether there is quantum speedup in sampling applications. Experience with the use of quantum annealers for combinatorial optimization suggests the answer is not straightforward.

Inspired by the success of Boltzmann machines based on classical Boltzmann distribution, a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian was recently proposed.^{[46]} Due to the non-commutative nature of quantum mechanics, the training process of the quantum Boltzmann machine can become nontrivial. This problem was, to some extent, circumvented by introducing bounds on the quantum probabilities, allowing the authors to train the model efficiently by sampling. It is possible that a specific type of quantum Boltzmann machine has been trained in the D-Wave 2X by using a learning rule analogous to that of classical Boltzmann machines.^{[43]}^{[45]}

Quantum annealing is not the only technology for sampling. In a prepare-and-measure scenario, a universal quantum computer prepares a thermal state, which is then sampled by measurements. This can reduce the time required to train a deep restricted Boltzmann machine, and provide a richer and more comprehensive framework for deep learning than classical computing.^{[47]} The same quantum methods also permit efficient training of full Boltzmann machines and multi-layer, fully connected models and do not have well-known classical counterparts. Relying on an efficient thermal state preparation protocol starting from an arbitrary state, quantum-enhanced Markov logic networks exploit the symmetries and the locality structure of the probabilistic graphical model generated by a first-order logic template.^{[48]} This provides an exponential reduction in computational complexity in probabilistic inference, and, while the protocol relies on a universal quantum computer, under mild assumptions it can be embedded on contemporary quantum annealing hardware.

### Quantum neural networks

Quantum analogues or generalizations of classical neural nets^{[49]} are often referred to as quantum neural networks. The term is claimed by a wide range of approaches, including the implementation and extension of neural networks using photons,^{[50]} layered variational circuits^{[10]} or quantum Ising-type models.^{[51]}

### Hidden Quantum Markov Models

Hidden Quantum Markov Models^{[52]} (HQMMs) are a quantum-enhanced version of classical Hidden Markov Models (HMMs), which are typically used to model sequential data in various fields like robotics and natural language processing. Unlike the approach taken by other quantum-enhanced machine learning algorithms, HQMMs can be viewed as models inspired by quantum mechanics that can be run on classical computers as well.^{[53]} Where classical HMMs use probability vectors to represent hidden 'belief' states, HQMMs use the quantum analogue: density matrices. Recent work has shown that these models can be successfully learned by maximizing the log-likelihood of the given data via classical optimization, and there is some empirical evidence that these models can better model sequential data compared to classical HMMs in practice, although further work is needed to determine exactly when and how these benefits are derived.^{[53]} Additionally, since classical HMMs are a particular kind of Bayes net, an exciting aspect of HQMMs is that the techniques used show how we can perform quantum-analogous Bayesian inference, which should allow for the general construction of the quantum versions of probabilistic graphical models.^{[53]}

### Fully quantum machine learning

In the most general case of quantum machine learning, both the learning device and the system under study, as well as their interaction, are fully quantum. This section gives a few examples of results on this topic.

One class of problem that can benefit from the fully quantum approach is that of 'learning' unknown quantum states, processes or measurements, in the sense that one can subsequently reproduce them on another quantum system. For example, one may wish to learn a measurement that discriminates between two coherent states, given not a classical description of the states to be discriminated, but instead a set of example quantum systems prepared in these states. The naive approach would be to first extract a classical description of the states and then implement an ideal discriminating measurement based on this information. This would only require classical learning. However, one can show that a fully quantum approach is strictly superior in this case.^{[54]} (This also relates to work on quantum pattern matching.^{[55]}) The problem of learning unitary transformations can be approached in a similar way.^{[56]}

Going beyond the specific problem of learning states and transformations, the task of clustering also admits a fully quantum version, wherein both the oracle which returns the distance between data-points and the information processing device which runs the algorithm are quantum.^{[57]} Finally, a general framework spanning supervised, unsupervised and reinforcement learning in the fully quantum setting was introduced in,^{[21]} where it was also shown that the possibility of probing the environment in superpositions permits a quantum speedup in reinforcement learning.

## Classical learning applied to quantum systems

The term quantum machine learning is also used for approaches that apply classical methods of machine learning to the study of quantum systems. A prime example is the use of classical learning techniques to process large amounts of experimental data in order to characterize an unknown quantum system (for instance in the context of quantum information theory and for the development of quantum technologies), but there are also more exotic applications.

The ability to experimentally control and prepare increasingly complex quantum systems brings with it a growing need to turn large and noisy data sets into meaningful information. This is a problem that has already been studied extensively in the classical setting, and consequently, many existing machine learning techniques can be naturally adapted to more efficiently address experimentally relevant problems. For example, Bayesian methods and concepts of algorithmic learning can be fruitfully applied to tackle quantum state classification,^{[58]} Hamiltonian learning,^{[59]} and the characterization of an unknown unitary transformation.^{[60]}^{[61]} Other problems that have been addressed with this approach are given in the following list:

- Identifying an accurate model for the dynamics of a quantum system, through the reconstruction of the Hamiltonian;
^{[62]}^{[63]}^{[64]} - Extracting information on unknown states;
^{[65]}^{[66]}^{[67]}^{[58]} - Learning unknown unitary transformations and measurements;
^{[60]}^{[61]} - Engineering of quantum gates from qubit networks with pairwise interactions, using time dependent
^{[68]}or independent^{[69]}Hamiltonians.

However, the characterization of quantum states and processes is not the only application of classical machine learning techniques. Some additional applications include

- Inferring molecular energies;
^{[70]} - Automatic generation of new quantum experiments;
^{[71]}^{[72]} - Solving the many-body, static and time-dependent Schrödinger equation;
^{[73]} - Identifying phase transitions from entanglement spectra;
^{[74]} - Generating adaptive feedback schemes for quantum metrology.
^{[75]}

## Quantum learning theory

Quantum learning theory pursues a mathematical analysis of the quantum generalizations of classical learning models and of the possible speed-ups or other improvements that they may provide. The framework is very similar to that of classical computational learning theory, but the learner in this case is a quantum information processing device, while the data may be either classical or quantum. Quantum learning theory should be contrasted with the quantum-enhanced machine learning discussed above, where the goal was to consider *specific problems* and to use quantum protocols to improve the time complexity of classical algorithms for these problems. Although quantum learning theory is still under development, partial results in this direction have been obtained.^{[76]}

The starting point in learning theory is typically a *concept class*, a set of possible concepts. Usually a concept is a function on some domain, such as
. For example, the concept class could be the set of disjunctive normal form (DNF) formulas on *n* bits or the set of Boolean circuits of some constant depth. The goal for the learner is to learn
(exactly or approximately) an unknown *target concept* from this concept class. The learner may be actively interacting with the target concept, or passively receiving samples from it.

In active learning, a learner can make *membership queries* to the target concept *c*, asking for its value *c(x)* on inputs *x* chosen by the learner. The learner then has to reconstruct the exact target concept, with high probability. In the model of *quantum exact learning*, the learner can make membership queries in quantum superposition. If the complexity of the learner is measured by the number of membership queries it makes, then quantum exact learners can be polynomially more efficient than classical learners for some concept classes, but not more.^{[77]} If complexity is measured by the amount of *time*
the learner uses, then there are concept classes that can be learned efficiently by
quantum learners but not by classical learners (under plausible complexity-theoretic assumptions).^{[77]}

A natural model of passive learning is Valiant's probably approximately correct (PAC) learning. Here the learner receives random examples *(x,c(x))*, where *x* is distributed according to some unknown distribution *D*. The learner's goal is to output a hypothesis function *h* such that *h(x)=c(x)* with high probability when *x* is drawn according to *D*. The learner has to be able to produce such an 'approximately correct' *h* for every *D* and every target concept *c* in its concept class.
We can consider replacing the random examples by potentially more powerful quantum examples
. In the PAC model (and the related agnostic model), this doesn't significantly reduce the number of examples needed: for every concept class, classical and
quantum sample complexity are the same up to constant factors.^{[78]} However, for learning under some
fixed distribution *D*, quantum examples can be very helpful, for example for learning DNF under
the uniform distribution.^{[79]} When considering *time* complexity, there exist concept classes that can be PAC-learned efficiently by quantum learners, even from classical examples, but not by classical learners (again, under plausible complexity-theoretic assumptions).^{[77]}

This passive learning type is also the most common scheme in supervised learning: a learning algorithm typically takes the training examples fixed, without the ability to query the label of unlabelled examples. Outputting a hypothesis *h* is a step of induction. Classically, an inductive model splits into a training and an application phase: the model parameters are estimated in the training phase, and the learned model is applied an arbitrary many times in the application phase. In the asymptotic limit of the number of applications, this splitting of phases is also present with quantum resources.^{[80]}

## Implementations and experiments

The earliest experiments were conducted using the adiabatic D-Wave quantum computer, for instance, to detect cars in digital images using regularized boosting with a nonconvex objective function in a demonstration in 2009.^{[81]} Many experiments followed on the same architecture, and leading tech companies have shown interest in the potential of quantum machine learning for future technological implementations. In 2013, Google Research, NASA, and the Universities Space Research Association launched the Quantum Artificial Intelligence Lab which explores the use of the adiabatic D-Wave quantum computer.^{[82]}^{[83]} A more recent example trained a probabilistic generative models with arbitrary pairwise connectivity, showing that their model is capable of generating handwritten digits as well as reconstructing noisy images of bars and stripes and handwritten digits.^{[43]}

Using a different annealing technology based on nuclear magnetic resonance (NMR), a quantum Hopfield network was implemented in 2009 that mapped the input data and memorized data to Hamiltonians, allowing the use of adiabatic quantum computation.^{[84]} NMR technology also enables universal quantum computing, and it was used for the first experimental implementation of a quantum support vector machine to distinguish hand written number ‘6’ and ‘9’ on a liquid-state quantum computer in 2015.^{[85]} The training data involved the pre-processing of the image which maps them to normalized 2-dimensional vectors to represent the images as the states of a qubit. The two entries of the vector are the vertical and horizontal ratio of the pixel intensity of the image. Once the vectors are defined on the feature space, the quantum support vector machine was implemented to classify the unknown input vector. The readout avoids costly quantum tomography by reading out the final state in terms of direction (up/down) of the NMR signal.

Photonic implementations are attracting more attention,^{[49]} not the least because they do not require extensive cooling. Simultaneous spoken digit and speaker recognition and chaotic time-series prediction were demonstrated at data rates beyond 1 gigabyte per second in 2013.^{[86]} Using non-linear photonics to implement an all-optical linear classifier, a perceptron model was capable of learning the classification boundary iteratively from training data through a feedback rule.^{[87]} A core building block in many learning algorithms is to calculate the distance between two vectors: this was first experimentally demonstrated for up to eight dimensions using entangled qubits in a photonic quantum computer in 2015.^{[88]}

Recently, based on a neuromimetic approach, a novel ingredient has been added to the field of quantum machine learning, in the form of a so-called quantum memristor, a quantized model of the standard classical memristor.^{[89]} This device can be constructed by means of a tunable resistor, weak measurements on the system, and a classical feed-forward mechanism. An implementation of a quantum memristor in superconducting circuits has been proposed,^{[90]} and an experiment with quantum dots performed.^{[91]} A quantum memristor would implement nonlinear interactions in the quantum dynamics which would aid the search for a fully functional quantum neural network.

## See also

## References

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