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Parameterized quantum circuits as machine learning models

Marcello Benedetti, Erika Lloyd, Stefan Sack, Mattia Fiorentini

arXiv:1906.07682v2quant-phcs.LG

TL;DR

Near-term quantum hardware remains limited by noise and scale, creating a need for useful hybrid approaches. This Review presents parameterized quantum circuits as machine-learning models and surveys their components, training, and applications. It concludes that their flexibility and expressive power could support automated forecasting and decision-making if they scale to realistic tasks.

  • Problem

    Near-term quantum hardware has few tens of physical qubits and can execute only short noisy gate sequences, limiting access to large-scale fault-tolerant algorithms.

  • Method

    The Review synthesizes hybrid quantum-classical machine-learning models built from parameterized quantum circuits, covering their components, learning algorithms, and applications.

  • Results

    The reviewed PQC models exhibit expressive power across supervised and generative learning, including quantum generative models with greater expressive power than some classical models.

  • Takeaways & Limitations

    PQC models may become part of automated forecasting and decision-making systems if they scale well to realistic machine-learning tasks.

Abstract

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Hybrid quantum-classical systems make it possible to utilize existing quantum computers to their fullest extent. Within this framework, parameterized quantum circuits can be regarded as machine learning models with remarkable expressive power. This Review presents the components of these models and discusses their application to a variety of data-driven tasks, such as supervised learning and generative modeling. With an increasing number of experimental demonstrations carried out on actual quantum hardware and with software being actively developed, this rapidly growing field is poised to have a broad spectrum of real-world applications.

I. INTRODUCTION

Parameterized quantum circuits support hybrid quantum-classical machine learning on near-term hardware, where classical optimization compensates for limited quantum resources. This Review surveys their components, learning procedures, and applications across data-driven tasks.

  • Motivation: Existing quantum hardware has few tens of physical qubits and supports only short gate sequences before noise overwhelms computation.Consequently, fault-tolerant algorithms such as Shor’s remain out of reach, while NISQ applications are being investigated.
  • Hybrid approach: Hybrid algorithms assign classically tractable subroutines to classical hardware, reducing requirements for qubit count, circuit depth, and coherence time.The quantum device focuses on the classically intractable part of the problem.
  • Hybrid approach: A hybrid machine-learning loop combines human model selection, classical preprocessing and optimization, quantum-state preparation and measurement, and classical post-processing.Updated parameters define a new circuit that is repeatedly sent back to the quantum hardware.
  • Applications: PQC-based hybrid systems have been applied to classification, regression, and generative modeling, partly because PQCs resemble kernels and neural networks.The Review focuses on these machine-learning applications rather than earlier quantum-algorithm-learning proposals.
  • Review scope: The Review introduces PQC model components and learning algorithms, surveys classical and quantum tasks, and discusses advantages and future directions.Its exposition is pedagogical and assumes basic familiarity with machine learning and quantum computing.

II. FRAMEWORK

PQC machine-learning models encode data into quantum states, apply trainable variational circuits, measure observables, and classically post-process those measurements into task outputs. The resulting components can be trained jointly or adapted to different model families.

  • Quantum computation: A quantum computation prepares a simple basis state, applies a unitary operator, and measures Hermitian observables to obtain their values.Measurement outcomes follow the Born-rule probability distribution over the observable’s eigenvalues.
  • Model framework: PQC models use quantum measurements to define machine-learning outputs while providing adjustable unitary operators for model construction.Their probabilistic measurements support multiple machine-learning model types.
  • Model framework: A supervised PQC model preprocesses data, encodes it with Uφ(x), applies a variational circuit Uθ, estimates observables, and post-processes them into an output.The variational circuit may act on an extended register containing ancilla qubits.
  • Learning: All model components, including pre-processing and post-processing functions, can in principle be parameterized and trained.For regression, post-processing may use a weighted linear combination of measured expectation values.
  • Implementation: Estimating each observable term requires repeated measurements determined by the desired precision and the term’s variance.The Review does not discuss estimation methods.

A. The encoder circuit Uφ(x)

The encoder circuit maps classical data into quantum states as a feature map, with encoding choices controlling expressive power and computational demands. Proposed maps include local, redundant, random, and amplitude encodings, but practical quantum advantage remains unresolved.

  • The encoder circuit Uφ(x): Encoding classical data as x → Uφ(x)|0⟩^⊗n acts as a quantum feature map into the high-dimensional state space of n qubits.The preprocessing function φ transforms data vectors into circuit parameters.
  • Encoding choices: Different encodings provide different expressive power and can alter the effective kernel used by downstream machine-learning models.Quantum kernels can be evaluated with the SWAP test and used in support vector machines, Gaussian processes, and principal component analysis.
  • Encoding choices: Redundant component encodings populate the wavefunction with higher-order terms that can fit nonlinear data functions, with required redundancy bounded logarithmically in function complexity.These constructions use multiple qubits for each data component.
  • Encoding choices: Random linear-map encoders implicitly implement kernels whose form depends on the encoder-circuit layout, and not every layout yields useful kernels.Quanvolutional networks provide another random-encoder approach inspired by convolutional filters.
  • Encoding choices: Amplitude encoding can provide compact state representations, but generic encoder depth is expected to scale exponentially with qubit number, impeding coherent data loading.The favorable memory scaling therefore comes with a circuit-depth constraint.
  • Quantum advantage: Feature maps inspired by quantum-supremacy proposals may make kernel estimation classically intractable, while their practical machine-learning advantage remains an open empirical question.The Review identifies learning kernel forms and parameters from data as largely unexplored.

B. The variational circuit Uθ

Variational circuits offer expressive quantum models, but practical circuit design must balance representational power with hardware constraints and differs fundamentally from neural-network computation.

  • Quantum circuits can approximate target functions arbitrarily closely, although the required circuit may be exponentially deep and impractical.
  • Fixed circuit structures reduce model complexity so the number of free parameters scales polynomially with the qubit count.
  • Hardware-efficient circuits alternate native entangling gates and single-qubit rotations to accommodate sparse connectivity, limited qubits, and simple gate sets.
  • Tensor-network-inspired circuits reduce computational cost by representing states with constrained entanglement in a smaller portion of the vector space.
  • Qubit-efficient tensor-network schemes trace out qubits, reinitialize them, and reuse them as inputs to later unitaries.
  • Unlike neural networks, variational circuits use unitary linear operations and cannot access intermediate quantum states, complicating nonlinear processing and backpropagation-like learning.

C. Circuit learning

Circuit learning trains parameterized quantum circuits by minimizing a loss function with classical optimization, while circuit design and gradient estimation are adapted to NISQ constraints. The review compares gradient-based and gradient-free methods, emphasizing hardware compatibility, query costs, and optimization limitations.

  • Optimization: PQC circuit learning minimizes a loss function L(θ) with respect to the circuit parameter vector θ.Both gradient-based and gradient-free optimization algorithms can be used.
  • Circuit design: Hardware-efficient circuits alternate native entangling gates and single-qubit rotations to match sparse connectivity, simple gates, and hardware constraints.Heuristics can reduce the number of costly entangling gates.
  • Circuit design: Qubit-efficient TTNs reinitialize traced-out qubits for later operations, reducing qubit requirements while potentially increasing circuit depth.The qubit-efficient scheme implements the same coarse-graining operation as the standard binary TTN.
  • Gradient estimation: Analytical gradients for expectation-value outputs can be estimated with the parameter-shift rule using two circuit executions.An alternative Hadamard-test method uses one circuit but requires an ancilla qubit and controlled operations.
  • Gradient estimation: O(n^2/ϵ) oracle queries suffice for analytical gradients near a convex optimum, whereas finite differences require at least Ω(n^3/ϵ^2) calls.Analytical gradients provide an unbiased estimator and outperform finite-difference methods in the cited comparison.
  • Optimization limitations: Random circuits can produce exponentially large vanishing-gradient plateaus, while structured circuits may alleviate this optimization problem.Unitary quantum circuits naturally avoid exploding gradients, but random initialization can limit variational-circuit effectiveness.
  • Optimization: Gradient-free optimization is useful when noisy losses are non-smooth or objectives are black-box, with hardware studies reporting optimizer-dependent outcomes.One study found zeroth-order optimization achieved the lowest average loss, while another found Bayesian optimization outperformed particle swarm optimization in its setting.
  • Optimization: Genetic algorithms can improve the fidelity of pre-trained PQC compression models on superconducting hardware.They evolve candidate circuits using recombination, mutation, and natural selection.

III. APPLICATIONS

PQC models are applied to supervised and unsupervised learning with classical data, can process quantum data when efficient preparation is available, and have been proposed for reinforcement learning. The review notes that hybrid PQC-based reinforcement-learning systems were not yet available to the authors' knowledge.

  • Classical data: Applications with classical data include supervised classification and regression, while unsupervised-learning coverage focuses on generative modeling.The review frames these applications as efforts to obtain an advantage over classical models.
  • Quantum data: PQC models can process quantum inputs and outputs already in superposition when the input state can be prepared efficiently.Quantum inputs may originate remotely or be prepared locally with an encoder circuit.
  • Reinforcement learning: Reinforcement learning addresses tasks without available best-action data by learning through interaction with an environment and repeated experience.The review gives a video-game character learning a successful strategy as an example.
  • Reinforcement learning: The review states that proposed quantum reinforcement-learning algorithms were not based on hybrid systems and PQC models.

A. Supervised learning

Supervised PQC models learn mappings from classical inputs to targets through either quantum-kernel estimation or variational quantum processing. These approaches have been demonstrated experimentally, but comparisons with established classical models remain incomplete.

  • Supervised learning trains a model function to map each input x to its corresponding target y.
  • Quantum-kernel estimators use a SWAP test to evaluate a potentially intractable kernel, followed by classical post-processing.
  • A coherent SWAP test may be non-trivial to implement on NISQ computers, and systematic comparisons with classical models are still needed.
  • Variational quantum models process data directly with a parameterized circuit and post-process measured expectation values into model outputs.
  • Experiments have demonstrated QKE and VQM classifiers on two superconducting qubits, using error mitigation in the QKE study.

The variational circuit

Variational circuits can represent highly expressive classification models, but their usefulness depends on circuit complexity, dataset scale, and hardware constraints. Existing demonstrations report both competitive performance and important practical limitations.

  • For every possible binary label function on n-bit strings, some variational circuit achieves zero classification error.Some functions require exponentially deep circuits, making them impractical.
  • Tensor-network-inspired TTN and MERA circuits have been used for binary classification of classical images and quantum states.Entanglement entropy can characterize contextual information in image data.
  • Amplitude-encoded classifiers can combine low circuit depth, high expressivity, and polynomially many parameters.Simulations on benchmark datasets reported performance comparable to off-the-shelf classical models.
  • All supervised experiments to date used scaled-down, often trivial datasets because available quantum hardware is limited.Demonstrations at a more realistic scale remain desirable, and PQC-specific regularization is underdeveloped.

B. Generative modeling

Generative PQC models learn target distributions and produce synthetic samples through direct sampling, discrepancy minimization, or adversarial training. They offer expressive modeling capabilities, while hardware noise, hyperparameter selection, and loss design remain practical challenges.

  • Generative modeling learns an unknown probability distribution and generates synthetic data, with applications including imaging, speech, denoising, and chemical design.
  • Quantum circuit Born machines encode distributions in quantum states and output binary samples by measuring a variational circuit in the computational basis.
  • QCBMs are implicit models: sampling is easy, while estimating the generated likelihood can be difficult.
  • QCBMs have been reported as strictly more expressive than classical deep Boltzmann machines under a polynomial-parameter restriction, with some circuits not efficiently classically simulable.
  • Experiments include convergence to target distributions on four trapped-ion qubits, while other hardware studies found performance significantly worse than numerical simulations.
  • Generative performance depends on hyperparameter choices, and suitable loss functions can be difficult to optimize or design.
  • QGANs use a discriminator to distinguish real from generated data, providing a differentiable loss for the generator during adversarial training.

C. Quantum learning tasks

PQC models extend machine learning to quantum-native tasks including algorithm compilation, state compression, error correction, classification, tomography, and generative modeling. These applications show promise on near-term hardware, but their scaling, noise robustness, and sample complexity remain open questions.

  • C. Quantum learning tasks: Quantum learning tasks apply PQCs to algorithm implementation, state classification, tomography, diagonalization, compression, error correction, and compilation.
  • C. Quantum learning tasks: Experimental studies are still needed to assess scaling under realistic NISQ noise, while theory must establish training sample complexity.
  • C. Quantum learning tasks: Quantum state tomography can require measurements scaling linearly with qubit number for practical purposes, supported by an experimental photonic demonstration.
  • C. Quantum learning tasks: QGANs for quantum data train quantum generators and discriminators together, including models that generate states conditioned on labels or approximate pure states.
  • C. Quantum learning tasks: Quantum autoencoders compress an n-qubit state into k qubits by encoding, tracing out n − k qubits, and decoding to maximize expected input-output fidelity.
  • C. Quantum learning tasks: Learned encoding and recovery circuits can optimize average code fidelity to reduce error-correction overhead in NISQ devices.
  • C. Quantum learning tasks: Hardware-efficient variational circuits can be trained to approximately reproduce target unitaries despite gate-set and qubit-connectivity constraints.

IV. OUTLOOK

The Review highlights PQCs' flexibility, experimental and software progress, and their potential for quantum learning and real-world applications, while identifying scaling to realistic datasets as an open question.

  • PQC-based hybrid systems have moved beyond theory and simulation through experimental demonstrations on scaled-down problems.
  • Software platforms now support PQC models, automatic differentiation, and interfaces to simulators and existing quantum computers.
  • PQCs can implement classically intractable feature maps and kernel functions, but their performance benefits for established kernel methods require further study.
  • For quantum learning, PQC models may provide an advantage when quantum data can be efficiently loaded or prepared and the underlying system exhibits quantum supremacy.
  • PQC models naturally support unitary operations and gradient computation for discrete measured data, while their broader statistical and learning properties remain largely unexplored.
  • The framework supports incremental algorithm development, with classical resources potentially replaced by quantum resources as hardware and software improve.
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