Challenges and opportunities in quantum machine learning

While there are similarities between classical and quantum ML, there are also some differences. Because QML employs quantum computers, noise from these computers can be a major issue. This includes hardware noise such as decoherence as well as statistical noise (that is, shot noise) that arises from measurements on quantum states. Both of these noise sources can complicate the QML training process. Moreover, nonlinear operations (for example, neural activation functions) that are natural in classical ML require more careful design of QML models due to the linearity of quantum transformations.

For the field of QML, the immediate goal for the near future is demonstrating quantum advantage, that is, outperforming classical methods, in a data science application. Achieving this goal will require keeping an open mind about which applications will benefit most from QML (for example, it may be an application that is inherently quantum mechanical). Understanding how QML methods scale to large problem sizes will also be required, including analysis of trainability (gradient scaling) and prediction error. The availability of high-quality quantum hardware^13,14 will also be crucial.

Finally, we note that QML provides a new way of thinking about established fields, such as quantum information theory, quantum error correction and quantum foundations. Viewing such applications from a data science perspective will likely lead to new breakthroughs.

Data

As shown in Fig. 3, QML can be used to learn from either classical or quantum data, and thus we begin by contrasting these two types of data. Classical data are ultimately encoded in bits, each of which can be in a 0 or 1 state. This includes images, texts, graphs, medical records, stock prices, properties of molecules, outcomes from biological experiments and collision traces from high-energy physics experiments. Quantum data are encoded in quantum bits, called qubits, or higher-dimensional analogs. A qubit can be represented by the states |0〉, |1〉 or any normalized complex linear superposition of these two. Here, the states contain information obtained from some physical process such as quantum sensing¹⁵, quantum metrology¹⁶, quantum networks¹⁷, quantum control¹⁸ or even quantum analog–digital transduction¹⁹. Moreover, quantum data can also be the solution to problems obtained on a quantum computer: for example, the preparation of various Hamiltonians’ ground states.

In principle, all classical data can be efficiently encoded in systems of qubits: a classical bitstring of length n can be easily encoded onto n qubits. However, the same cannot be said for the converse, since one cannot efficiently encode quantum data in bit systems; that is, the state of a general n-qubit system requires (2ⁿ − 1) complex numbers to be specified. Hence, systems of qubits (and more generally the quantum Hilbert space) constitute the ultimate data representation medium, as they can encode not only classical information but also quantum information obtained from physical processes.

In a QML setting, the term quantum data refers to data that are naturally already embedded in a Hilbert space 𝐻. When the data are quantum, they are already in the form of a set of quantum states {|ψ_j〉} or a set of unitaries {U_j} that could prepare these states on a quantum device (via the relation |ψ_j〉 = U_j|0〉). On the other hand, when the data x are classical, they first need to be encoded in a quantum system through some embedding mapping x_j → |ψ(x_j)〉, with |ψ(x_j)〉 in 𝐻. In this case, the hope is that the QML model can solve the learning task by accessing the exponentially large dimension of the Hilbert space^20,21,22,23.

One of the most important and reasonable conjectures to make is that the availability of quantum data will substantially increase in the near future. The mere fact that people will use the quantum computers that are available will logically lead to more quantum problems being solved and quantum simulations being performed. These computations will produce quantum datasets, and hence it is reasonable to expect the rapid rise of quantum data. Note that, in the near term, these quantum data will be stored on classical devices in the form of efficient descriptions of quantum circuits that prepare the datasets.

Finally, as our level of control over quantum technologies progresses, coherent transduction of quantum information from the physical world to digital quantum computing platforms may be achieved¹⁹. This would quantum mechanically mimic the main information acquisition mechanism for classical data from the physical world, this being analog–digital conversion. Moreover, we can expect that the eventual advent of practical quantum error correction²⁴ and quantum memories²⁵ will allow us to store quantum data on quantum computers themselves.

Models

Analyzing and learning from data requires a parameterized model, and many different models have been proposed for QML applications. Classical models such as neural networks and tensor networks (as shown in Fig. 1) are often useful for analyzing data from quantum experiments. However, due to their novelty, we will focus our discussion on quantum models using quantum algorithms, where one applies the learning methodology directly at the quantum level.

Similarly to classical ML, there exist several different QML paradigms: supervised learning (task based)^26,27,28, unsupervised learning (data based)^29,30 and reinforced learning (reward based)^31,32. While each of these fields is exciting and thriving in itself, supervised learning has recently received considerable attention for its potential to achieve quantum advantage^26,33, resilience to noise³⁴ and good generalization properties^35,36,37, which makes it a strong candidate for near-term applications. In what follows we discuss two popular QML models: QNNs and quantum kernels, shown in Fig. 3, with a particular emphasis on QNNs as these are the primary ingredient of several supervised, unsupervised and reinforced learning schemes.

Quantum neural networks

The most basic and key ingredient in QML models is parameterized quantum circuits (PQCs). These involve a sequence of unitary gates acting on the quantum data states |ψ_j〉, some of which have free parameters θ that will be trained to solve the problem at hand³⁸. PQCs are conceptually analogous to neural networks, and indeed this analogy can be made precise: that is, classical neural networks can be formally embedded into PQCs³⁹.

This has led researchers to refer to certain kinds of PQC as QNNs. In practice, the term QNN is used whenever a PQC is employed for a data science application, and hence we will use the term QNN in what follows. QNNs are employed in all three QML paradigms mentioned above. For instance, in a supervised classification task, the goal of the QNN is to map the states in different classes to distinguishable regions of the Hilbert space²⁶. Moreover, in the unsupervised learning scenario of ref. ²⁹, a clustering task is mapped onto a MAXCUT problem and solved by training a QNN to maximize distance between classes. Finally, in the reinforced learning task of ref. ³², a QNN can be used as the Q-function approximator, which can be used to determine the best action for a learning agent given its current state.

Figure 4 gives examples of three distinct QNN architectures where in each layer the number of qubits in the model is increased, preserved or decreased. In Fig. 4a we show a dissipative QNN⁴⁰ which generalizes the classical feedforward network. Here, each node corresponds to a qubit, while lines connecting qubits are unitary operations. The term dissipative arises from the fact that qubits in a layer are discarded after the information forward-propagates to the (new) qubits in the next layer. Figure 4b shows a standard QNN where quantum data states are sent through a quantum circuit, at the end of which some or all of the qubits are measured. Here, no qubits are discarded or added as we go deeper into the QNN. Finally, Fig. 4c depicts a convolutional QNN¹¹, where in each layer qubits are measured to reduce the dimension of the data while preserving its relevant features. Many other QNNs have been proposed^{41,42,43,44,45}, and constructing QNN architectures is currently an active area of research.

To further accommodate the limitation of near-term quantum computers, we can also employ a hybrid approach with models that have both classical and quantum neural networks⁴⁶. Here, QNNs act coherently on quantum states while deep classical neural networks alleviate the need for higher-complexity quantum processing. Such hybridization distributes the representational capacity and computational complexity across both quantum and classical computers. Moreover, since quantum states generally have a mixture of classical correlations and quantum correlations, hybrid quantum–classical models allow for the use of quantum computers as an additive resource to increase the ability of classical models to represent quantum-correlated distributions. Applications of hybrid models include generating⁴⁷ or learning and distilling information⁴⁶ from multipartite-entangled distributions.

Quantum kernels

As an alternative to QNNs, researchers have proposed quantum versions of kernel methods^26,28. A kernel method maps each input to a vector in a high-dimensional vector space, known as the reproducing kernel Hilbert space. Then, a kernel method learns a linear function in the reproducing kernel Hilbert space. The dimension of the reproducing kernel Hilbert space could be infinite, which enables the kernel method to be very powerful in terms of expressiveness. To learn a linear function in a potentially infinite-dimensional space, the kernel trick⁴⁸ is employed, which only requires efficient computation of the inner product between these high-dimensional vectors. The inner product is also known as the kernel⁴⁸. Quantum kernel methods consider the computation of kernel functions using quantum computers. There are many possible implementations. For example, refs. ^26,28 considered a reproducing kernel Hilbert space equal to the quantum state space, which is finite dimensional. Another approach¹³ is to study an infinite-dimensional reproducing kernel Hilbert space that is equivalent to transforming a classical vector using a quantum computer. It then maps the transformed classical vectors to infinite-dimensional vectors.

Inductive bias

For both QNNs and quantum kernels, an important design criterion is their inductive bias. This bias refers to the fact that any model represents only a subset of functions and is naturally biased towards certain types of function (that is, functions relating the input features to the output prediction). One aspect of achieving quantum advantage with QML is to aim for QML models with an inductive bias that is inefficient to simulate with a classical model. Indeed, it was recently shown⁴⁹ that quantum kernels with this property can be constructed, albeit with some subtleties regarding their trainability.

Generally speaking, inductive bias encompasses any assumptions made in the design of the model or the optimization method that bias the search of the potential models to a subset in the set of all possible models. In the language of Bayesian probabilistic theory, we usually call these assumptions our prior. Having a certain parameterization of potential models, such as QNNs, or choosing a particular embedding for quantum kernel methods^13,14,26 is itself a restriction of the search space, and hence a prior. Adding a regularization term to the optimizer or modulating the learning rate to keep searches geometrically local also adds inherently a prior and focuses the search, and thus provides inductive bias.

Ultimately, inductive biases from the design of the ML model, combined with a choice of training process, are what make or break an ML model. The main advantage of QML will then be to have the ability to sample from and learn models that are (at least partially) natively quantum mechanical. As such, they have inductive biases that classical models do not have. This discussion assumes that the dataset to be represented is quantum mechanical in nature, and is one of the reasons why researchers typically believe that QML has greater promise for quantum rather than classical data.

Training and generalization

The ultimate goal of ML (classical or quantum) is to train a model to solve a given task. Thus, understanding the training process of QML models is fundamental for their success.

Consider the training process, whereby we aim to find the set of parameters θ that lead to the best performance. The latter can be accomplished, for instance, by minimizing a loss function θ𝐿(θ) encoding the task at hand. Some methods for training QML models are leveraged from classical ML, such as stochastic gradient descent. However, shot noise, hardware noise and unique landscape features often make off-the-shelf classical optimization methods perform poorly for QML training. (This is due to the fact that extracting information from a quantum state requires computing the expectation values of some observable, which in practice need to be estimated via measurements on a noisy quantum computer. Hence, given a finite number of shots (measurement repetitions), these can only be resolved up to some additive errors. Moreover, such expectation values will be subject to corruption due to hardware noise.) This realization led to development of quantum-aware optimizers, which account for the quantum idiosyncrasies of the QML training process. For example, shot-frugal optimizers^50,51,52,53 can employ stochastic gradient descent while adapting the number of shots (or measurements) needed at each iteration, so as not to waste too many shots during the optimization. Quantum natural gradient^54,55 adjusts the step size according to the local geometry of the landscape (on the basis of the quantum Fisher information metric). These and other quantum-aware optimizers often outperform standard classical optimization methods in QML training tasks.

For the case of supervised learning, we are interested not only in learning from a training dataset but also in making accurate predictions on (generalizing to) previously unseen data. This translates into achieving small training and prediction errors, with the second usually hinging on the first. Thus, let us now consider prediction error, also known as generalization error, which has been studied only very recently for QML^{13,14,35,37,56,57}. Formally speaking, this error measures the extent to which a trained QML model performs well on unseen data. Prediction error depends on the training error as well as the complexity of the trained model. If the training error is large, the prediction error is also typically large. If the training error is small but the complexity of the trained model is large, then the prediction error is likely still large. The prediction error is small only if training error is itself small and the complexity of the trained model is moderate (that is, sufficiently smaller than training data size)^14,35. The notion of complexity depends on the QML model. We have a good understanding of the complexity of quantum kernel methods^13,14, while more research is needed on QNN complexity. Recent theoretical analysis of QNNs shows that their prediction performance is closely linked to the number of independent parameters in the QNN, with good generalization obtained when the number of training data is roughly equal to the number of parameters³⁵. This gives the exciting prospect of using only a small number of training data to obtain good generalization.