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Challenges and Opportunities in Quantum Machine Learning

M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, Patrick J. Coles

arXiv:2303.09491v1quant-phcs.LGstat.ML

TL;DR

QML asks how quantum mechanics can improve data analysis and whether quantum advantage is achievable, especially when learning from quantum data. The paper reviews QML methods and applications, emphasizing quantum neural networks, kernels, trainability, and the prospects for quantum advantage. It concludes that quantum data offers a promising route to advantage, while noise, embedding choices, scarce datasets, and trainability remain important constraints.

  • Problem

    QML seeks the ultimate limits of data analysis under physics, but quantum advantage for classical data remains uncertain and truly quantum datasets are scarce.

  • Method

    The paper reviews QML applications and methods, focusing on quantum neural networks, quantum kernels, data embeddings, inductive bias, and training approaches.

  • Results

    Quantum data provides a more promising route toward quantum advantage, including a rigorously established exponential sample-complexity advantage under quantum access assumptions.

  • Takeaways & Limitations

    QML may become especially important for learning, inference, and prediction from quantum data as quantum hardware progresses toward fault tolerance.

Abstract

from arXiv · show

At the intersection of machine learning and quantum computing, Quantum Machine Learning (QML) has the potential of accelerating data analysis, especially for quantum data, with applications for quantum materials, biochemistry, and high-energy physics. Nevertheless, challenges remain regarding the trainability of QML models. Here we review current methods and applications for QML. We highlight differences between quantum and classical machine learning, with a focus on quantum neural networks and quantum deep learning. Finally, we discuss opportunities for quantum advantage with QML.

I. INTRODUCTION

QML embeds machine learning in quantum mechanics to pursue improved data analysis and quantum advantage, while its applications and practical benefits remain uncertain. The field spans classical and quantum data, multiple algorithmic approaches, and applications from science to quantum computing.

  • QML embeds machine learning and learning theory into quantum mechanics to study the limits of data analysis allowed by physics.
  • Quantum computing may provide substantial speedups, but quantum advantage for data science remains theoretically uncertain.
  • QML encompasses classical or quantum algorithms applied to classical or quantum data, including quantum-inspired classical methods and quantum neural networks.
  • Potential applications include chemistry, materials science, sensing, metrology, classical data analysis, quantum error correction, and quantum algorithm design.
  • Quantum hardware noise, including decoherence and shot noise, complicates QML training, while quantum linearity makes nonlinear model operations require careful design.
  • Demonstrating quantum advantage will require scalable, trainable models, suitable prediction-error analysis, and high-quality quantum hardware.
  • Viewing quantum information, error correction, and foundations through data science may lead to new breakthroughs.

II. FRAMEWORK

QML distinguishes classical data, which must be embedded into quantum states, from quantum data that is already represented in Hilbert space. This framework supports classification and other learning tasks while motivating the development of quantum datasets.

  • Classical data includes images, texts, graphs, medical records, financial data, molecular properties, biological outcomes, and high-energy physics traces.
  • Quantum data is encoded in qubits or higher-dimensional quantum systems and can represent states or unitaries associated with physical processes.
  • Classical bitstrings can be efficiently encoded on qubits, whereas general n-qubit states cannot generally be efficiently encoded in classical bits.
  • For classical inputs, an embedding maps x_j to quantum states |ψ(x_j)⟩ in Hilbert space; quantum inputs are already represented as states or state-preparing unitaries.
  • The expected growth of quantum simulations and quantum computing may produce more quantum datasets, initially stored classically as efficient circuit descriptions.
  • Improved quantum control, transduction, error correction, and quantum memories could support direct acquisition and storage of quantum information.

B. Models

QML includes supervised, unsupervised, and reinforcement learning, with models chosen according to the data-analysis task. The framework includes both classical models for quantum data and quantum algorithms operating directly at the quantum level.

  • Classical neural and tensor networks can analyze data from quantum experiments, while quantum models apply learning methods directly to quantum systems.
  • QML uses supervised, unsupervised, and reinforcement learning paradigms for task-based, data-based, and reward-based learning, respectively.
  • Supervised learning has attracted attention for potential quantum advantage, noise resilience, and generalization properties, making it a candidate for near-term applications.

1. Quantum neural networks

Quantum neural networks use parameterized quantum circuits to process quantum data, with architectures that alter, preserve, or reduce the number of qubits. Hybrid quantum-classical designs distribute representational capacity and computational complexity across both types of hardware.

  • 1. Quantum neural networks: Parameterized quantum circuits apply trainable unitary gates to quantum data states and provide the basic ingredient of QNN models.
  • 1. Quantum neural networks: QNN circuits propagate encoded information through layers of quantum gates before measuring some or all qubits for outputs.
  • 1. Quantum neural networks: QNNs can map different classes to distinguishable Hilbert-space regions and support supervised, unsupervised, and reinforcement learning.
  • 1. Quantum neural networks: Dissipative QNNs discard qubits and replace them across layers, standard QNNs preserve qubits, and convolutional QNNs measure qubits to reduce dimensionality.
  • 1. Quantum neural networks: Hybrid models combine coherent QNN processing with deep classical networks to accommodate limitations of near-term quantum computers.

2. Quantum kernels

Quantum kernel methods use quantum computers to compute kernels, while their effectiveness depends strongly on inductive bias and trainability. The paper emphasizes quantum-native modeling, quantum-aware optimization, and generalization to unseen data.

  • Quantum kernels: Quantum kernel methods compute inner products using quantum computers, providing an alternative to quantum neural networks.Kernel methods learn linear functions in a potentially high-dimensional reproducing kernel Hilbert space.
  • Inductive bias: Inductive bias restricts the model search space toward selected functions and is a key design criterion for both quantum kernels and QNNs.Choosing a parameterization or embedding imposes assumptions about the functions the model can represent.
  • Inductive bias: Quantum advantage may arise when a QML model has an inductive bias that is inefficient for classical models to simulate.Quantum kernels with this property have been constructed, although their trainability involves subtleties.
  • Inductive bias: Quantum-native inductive biases are considered more promising for quantum data than for classical data.The paper connects this promise to models that can sample from and learn partially native quantum-mechanical structures.
  • Training and generalization: QML training commonly minimizes a loss function over parameters, but shot noise, hardware noise, and quantum landscape features complicate optimization.Quantum-aware methods include shot-frugal optimizers and quantum natural gradient.
  • Training and generalization: Generalization error measures performance on unseen data and is small when training error is small and model complexity is moderate.The paper notes that quantum-kernel complexity is relatively well understood.

III. CHALLENGES IN QML

QML faces technical and data-related challenges that can impede progress, especially in training, encoding classical data, and establishing useful quantum benchmarks. The review frames rapid identification and resolution of these challenges as a way to avoid stagnation.

  • III. CHALLENGES IN QML: QML research aims to identify technical challenges quickly and focus effort on addressing them to avoid periods of stagnation.Figure 5 organizes model components alongside associated challenges.
  • III. CHALLENGES IN QML: QML models require datasets, classical-data encoding schemes, parameterized models, loss functions, and classical optimizers.The figure presents these ingredients as interconnected components of model construction.
  • Embedding schemes and quantum datasets: High-quality standardized datasets are expected to be important for advancing QML, as they have been for classical machine learning.The passage presents this as a conjecture rather than an established result.
  • Embedding schemes and quantum datasets: Most QML architectures are benchmarked on accessible classical datasets, but encoding classical information into quantum states remains unclear.Examples include MNIST, Dogs vs Cats, and Iris.
  • Embedding schemes and quantum datasets: Useful embeddings must combine classical hardness of simulating output-state inner products with distinguishable states for classification.An embedding satisfying one property does not necessarily satisfy the other.
  • Embedding schemes and quantum datasets: Encoding schemes with dataset-informed inductive bias are an active research area.The review identifies such bias as a desirable property for embedding schemes.
  • Embedding schemes and quantum datasets: Quantum advantage appears less straightforward with classical data, while quantum-data models have a more promising route but lack sufficient standardized datasets.The paper calls for datasets containing easily preparable quantum states for benchmarking.

B. Quantum landscapes

Quantum landscapes are generally non-convex and can contain local minima and barren plateaus, making QML training difficult. Proposed responses include adaptive architectures, sharper inductive priors, and training strategies designed around landscape structure.

  • B. Quantum landscapes: Training QML parameters typically minimizes a loss function over a non-convex landscape in search of a global minimum.Quantum landscape theory studies these properties and how to engineer them.
  • B. Quantum landscapes: Many local minima can make overall quantum optimization NP-hard, as in the classical case.Variable-structure QNNs can adapt the model prior and turn some local minima into saddle points.
  • B. Quantum landscapes: Overparametrization can produce a computational phase transition in which spurious local minima disappear beyond a critical parameter count.This phenomenon has been observed in QML.
  • B. Quantum landscapes: In a barren plateau, the loss landscape becomes exponentially flat with problem size and the global-minimum valley shrinks into a narrow gorge.Navigating such a landscape requires exponential resources such as numbers of shots.
  • B. Quantum landscapes: Barren plateaus can increase algorithmic complexity and even destroy quantum speedup by imposing exponential navigation resources.This conflicts with the goal of avoiding classical exponential complexity.
  • B. Quantum landscapes: Deep hardware-efficient QNNs develop barren plateaus partly because their highly expressive architectures use relatively uninformed priors.The phenomenon is linked to insufficient inductive bias.
  • B. Quantum landscapes: Clever initialization, pre-training, parameter correlation, and sharper architectural priors are proposed strategies for mitigating barren plateaus.These methods constrain the search over over-expressive parameterizations.

4. Barren plateaus from global observables

Barren plateaus can arise from global observables, excessive entanglement, or both, but their mechanisms differ. Local observables and controlled entanglement are presented as routes to avoid these trainability problems.

  • 4. Barren plateaus from global observables: Global-observable loss functions can cause barren plateaus even in shallow circuits with sharp priors.The issue arises when observables measure all qubits.
  • 4. Barren plateaus from global observables: Local observables avoid the global-observable barren-plateau issue by comparing quantum states at the single-qubit level.The distinction reflects the difficulty of comparing objects in exponentially large Hilbert spaces.
  • 4. Barren plateaus from global observables: Excessive entanglement between visible output qubits and many hidden qubits can also produce barren plateaus.Information becomes distributed across non-local correlations, leaving visible-qubit reduced states near maximally mixed.
  • 4. Barren plateaus from global observables: Taming entanglement across the QNN is proposed as a way to solve this entanglement-induced barren plateau.This mechanism differs from barren plateaus caused by inadequate inductive bias.

C. QNN architecture design

QNN architecture design increasingly centers on sharp, problem-informed inductive biases to improve trainability and generalization. Quantum geometric deep learning is proposed as a route toward systematically constructing such architectures from quantum-system symmetries.

  • Architecture principles: Sharp priors are a central design goal because QNNs support supervised, unsupervised, and reinforcement learning.The paper identifies architecture design as crucial across these learning settings.
  • Architecture principles: QCNNs restrict architectures to hierarchical and translationally invariant structures, reducing expressivity and parameter-space dimensionality.These sharper priors have been shown to avoid barren-plateau issues altogether.
  • Symmetry-informed models: Embedding dataset and problem knowledge, including graph-permutation symmetries, can provide useful inductive bias for QML models.Quantum graph neural networks and graph convolutional networks are examples of this strategy.
  • Symmetry-informed models: Classical geometric deep learning offers group-theoretic concepts of invariance and equivariance as inspiration for quantum architectures.This direction seeks to move beyond inductive biases based only on temporal or translational invariance.
  • Quantum geometric deep learning: Quantum geometric deep learning could prescribe architectures whose transformation-space priors encode symmetries and principles of quantum physical systems.The intended benefits are improved trainability and generalization, with related approaches recently explored.

D. Effect of quantum noise

Quantum noise is a defining obstacle for QML because it corrupts circuit information and can undermine model trainability. Proposed responses include better hardware, partial error correction, shallow architectures, and error mitigation, while quantum-data applications remain promising.

  • Noise effects: Hardware noise and measurement shot noise complicate QML training, while quantum linearity makes nonlinear operations require careful model design.The relevant hardware effects include decoherence and statistical noise from measurements.
  • Noise effects: Noise propagates through quantum circuits, disproportionately affecting deeper and longer-running circuits and potentially inducing barren plateaus in QNNs.It can also affect data preparation and quantum-kernel computation.
  • Mitigation strategies: Mitigating QML noise may require lower hardware error rates, partial quantum error correction, or shallow QNNs such as QCNNs.Error mitigation can improve performance but may not resolve noise-induced trainability problems.
  • Potential advantages: The first QML advantages are expected from hidden-parameter extraction in quantum sensing and quantum state classification or regression.Non-local measurements can extract hidden parameters using fewer samples, and QML can search over parameterized measurement hypotheses.
  • Potential advantages: QML may also advantage quantum error-correcting-code discovery and generative modeling of ground states, equilibrium states, or quantum dynamics.Generative advantage requires distributions that cannot be sampled classically and improved prediction or generalization relative to classical ML.

B. What will quantum advantage look like?

Quantum advantage is expected to be most plausible when QML processes data originating from quantum-mechanical systems, especially through sample-complexity gains. Time-complexity advantages remain conditional, while purely classical data currently lacks a known exponential advantage.

  • Quantum-origin data: Quantum-mechanical data from chemistry, materials science, biology, and physics is considered more likely to yield exponential QML advantage.Potential gains may concern either sample complexity or time complexity.
  • Quantum-origin data: An exponential sample-complexity advantage has been rigorously shown when quantum sensors, memories, and computers retrieve, store, and process experimental quantum information.The passage reports that this advantage has also been demonstrated on the Sycamore processor.
  • Time complexity: Exponential time-complexity advantage is more subtle because data availability can make classical ML algorithms computationally more powerful.The paper notes no rigorous proofs yet for an exponential advantage in practical physical systems.
  • Classical-origin data: For purely classical data, no exponential QML advantage is known, although polynomial and rigorously proven quadratic advantages remain possible.Potential long-term impact is constrained by the overhead of quantum error correction in fault-tolerant schemes.

C. Transition to the fault-tolerant era and beyond

QML is projected to remain useful across post-NISQ development, first through partial error correction and later through fault-tolerant processing of quantum data. In the longer term, direct access to networked quantum data could support widespread QML use.

  • Post-NISQ eras: The partial-error-corrected era combines a small number of logical qubits with non-error-corrected physical qubits, enabling a trade-off between the two.The fault-tolerant era begins when quantum hardware has many error-corrected qubits.
  • Partial error correction: High-fidelity circuits in the partial-error-corrected era could improve performance, mitigate noise-induced barren plateaus, and reduce noise-induced classification errors.These effects are presented as reasons QML may be useful before fully fault-tolerant hardware.
  • Fault tolerance: During the fault-tolerant era, quantum simulation and quantum memories could prepare and store quantum data for QML models to learn directly from it.The paper describes QML as a natural model for learning, inference, and prediction from such data.
  • Beyond fault tolerance: Further-term QML could receive quantum data captured from nature through quantum analog-digital interconversion and distributed over quantum networks.This scenario is compared with contemporary classical ML pipelines that relay sensor data to centralized processing.
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