Parameterized quantum circuits as machine learning models

I. INTRODUCTION

• NISQ devices offer a near-term research opportunity despite having only a few tens of physical qubits and tolerating only short gate sequences before noise overwhelms them.Large-scale fault-tolerant universal machines may take years, leaving algorithms such as Shor’s out of reach on current hardware.Source: §I · p. 1
• PQC classes can generate non-trivial outputs even at low depth, and some are believed impossible to simulate efficiently classically, but useful application-level quantum advantage is the practical goal.The cited complexity-theoretic example is instantaneous quantum polynomial-time, under well-believed assumptions.Source: §I · p. 1
• Hybrid algorithms divide work between classical and quantum hardware, reducing required qubits, circuit depth, and coherence time while reserving quantum resources for classically intractable parts.A closed loop uses classical learning to update PQC parameters from measurement outcomes and improve forecasts.Source: §I · p. 1, Figure 1 · p. 2
• This Review focuses on hybrid machine-learning models that treat quantum circuits as components for data-driven tasks, including classification, regression, and generative modeling.The authors relate PQCs to classical kernel methods and neural networks; the Review covers model components and learning algorithms, applications, advantages, and outlook.Source: §I · pp. 1–2

Framework

• Quantum computation prepares a simple basis state, applies a unitary, and measures a Hermitian observable whose eigenvalue outcomes follow a probability distribution.The framework assumes a closed quantum system and describes its state as a unit vector in a complex inner-product space.Source: §II · p. 3
• A supervised PQC model preprocesses a sampled data vector, encodes it with Uφ(x), applies variational circuit Uθ, and estimates expectation values {⟨Mk⟩x,θ}.The variational circuit may act on an extended qubit register.Source: §II · p. 3
• Post-processing maps the measured expectation values to a task output; for regression, it may use a linear combination with additional weights wk.Pre- and post-processing functions can also be parameterized and trained.Source: §II · p. 3
• Measurement repetitions depend on desired precision and observable variance, while this Review does not discuss estimation methods.Source: §II · p. 3

A. The encoder circuit Uφ(x)

• Encoding maps x through user-defined preprocessing φ into an n-qubit state, whose inner products define kernels usable in support vector machines, Gaussian processes, and principal component analysis.The SWAP test can evaluate the inner-product quantity.Source: A. The encoder circuit Uφ(x) · p. 3
• Local qubit encoding produces product states and highly non-linear kernels, while encoding each input component into multiple qubits adds higher-order terms for fitting non-linear functions.Vidal and Theis [17] found redundancy lower bounds logarithmic in function complexity under a linear-algebraic complexity measure.Source: A. The encoder circuit Uφ(x) · p. 3
• Random linear preprocessing creates a quantum random-kitchen-sink approach whose implicit kernel depends on encoder layout, and some layouts do not yield useful kernels.The quanvolutional network [20] is another random-encoder proposal, inspired by neural-network convolutional filters.Source: A. The encoder circuit Uφ(x) · p. 3
• Amplitude encoding maps unit vectors into n-qubit states, offering exponential memory savings and a linear kernel, while generic-input loading may require exponentially deep circuits.Preparing copies of this feature map can implement arbitrary polynomial kernels.Source: A. The encoder circuit Uφ(x) · pp. 3–4
• Havlíček et al. propose feature maps whose kernels are classically hard to estimate, conjecturing that two encoder layers yield this hardness.The conjecture is motivated by similarity to hidden-shift circuits for Boolean bent functions, which are classically hard to simulate.Source: A. The encoder circuit Uφ(x) · p. 4
• Whether feature maps inspired by quantum-supremacy proposals provide practical machine-learning advantage remains an open empirical question.Learning kernel forms and parameters from data is largely unexplored and could reduce kernel-selection bias or discover unknown feature maps with quantum-supremacy properties.Source: A. The encoder circuit Uφ(x) · p. 4

B. The variational circuit Uθ

• A variational circuit aims to approximate its target function with scalable parameter count and depth, since an arbitrarily accurate circuit may otherwise be exponentially deep and impractical.Lin et al. argue that physical datasets exhibit symmetry and locality, motivating cheaper models that can still achieve satisfactory results.Source: B. The variational circuit Uθ · p. 4
• A fixed gate structure reduces model complexity, keeping the number of free parameters polynomial in the qubit count despite the exponentially growing state space.Source: B. The variational circuit Uθ · p. 4
• Hardware-efficient circuits alternate native entangling gates and single-qubit rotations suited to sparse NISQ connectivity; Liu and Wang place entangling gates using a Chow-Liu tree.Their heuristic estimates pairwise mutual information from the dataset and places gates to represent most of it.Source: B. The variational circuit Uθ · pp. 4–5
• Tensor-network-inspired circuits, including matrix product states, TTNs and MERA, reduce computational cost by representing states with constrained entanglement and have been tested for classification and generative modeling [28].Source: B. The variational circuit Uθ · p. 5
• In a TTN, each unitary is followed by tracing out half the qubits; Huggins et al. instead reinitialize and reuse traced qubits, reducing qubit requirements for some NISQ hardware.Source: B. The variational circuit Uθ · p. 5
• Unlike neural-network activations, quantum circuit operations are unitary and linear, although nonlinear operations can be constructed coherently or non-coherently to implement artificial neurons [34].Source: B. The variational circuit Uθ · p. 5
• Because the full quantum state cannot be accessed mid-computation, circuit execution is not a neural-network forward pass and backpropagation-like training is difficult; proposed approaches use parameter superposition [39] or quantum neurons [40].Verdon et al. [39] exploit phase kickback and tunneling, while Beer et al. [40] define a quantum neuron as a completely positive map and describe efficient training resembling backpropagation.Source: B. The variational circuit Uθ · p. 5

C. Circuit learning

• PQC learning minimizes a data-driven loss over circuit parameters, and classical updates let hybrid systems use established optimizers such as stochastic gradient descent and adaptive momentum.These heuristics address large datasets, noisy gradient estimates, and adaptive learning rates; the passage reports they can reduce successful training time from days to hours.Source: C. Circuit learning · pp. 5, 7
• The parameter-shift rule estimates expectation-value derivatives with two circuit executions; a Hadamard-test alternative uses one circuit and an ancilla but can require challenging controlled gates and coherence.Direct measurements can replace some indirect measurements with simpler circuits suitable for NISQ implementations, and Hadamard tests can extend to higher-order derivatives.Source: C. Circuit learning · p. 7
• For n qubits and precision ϵ, query cost for convex optimization near the optimum scales as O(n^2/ϵ) with analytical gradients, versus at least Ω(n^3) with finite differences.Source: C. Circuit learning · p. 7
• In reported generative-model experiments, Bayesian optimization outperformed particle swarm on trapped ions, while zeroth-order optimization achieved the lowest average loss on superconducting hardware.The studies identify large parameter counts and loss variance—from initialization, hardware noise, and finite measurements—as optimization challenges.Source: C. Circuit learning · p. 8
• Genetic algorithms evolve gate-defined candidate circuits through recombination, mutation, and selection; an experiment found that a subsequent genetic algorithm improved a pretrained model’s fidelity.The review concludes that optimization algorithms should be tailored to PQC models to improve scalability.Source: C. Circuit learning · p. 8

• The section surveys PQC applications aimed at advantage over classical models, covering supervised classification and regression and unsupervised generative modeling with classical data.Source: §III · p. 8
• With efficient data preparation, PQC models can extend supervised and unsupervised learning to quantum states and quantum information.Quantum inputs may arrive remotely from other quantum computers or be prepared locally with an encoder circuit.Source: §III · p. 8
• Although quantum reinforcement-learning generalizations and algorithms have been proposed, the authors know of none based on hybrid systems and PQC models.Source: §III · p. 8

• A regularized loss penalizes undesired parameter values to reduce overfitting, which can otherwise let a model memorize a small training set instead of generalizing.Source: A. Supervised learning · p. 9
• QKE estimates kernels with a coherent SWAP test and learns classical post-processing weights, but implementing the test may be non-trivial on NISQ computers.The model function is expressed as a kernel expansion using the representer theorem [70].Source: A. Supervised learning · p. 9
• VQM processes data directly in feature space with a variational circuit, then post-processes estimated expectation values; its parameters define quantum operations requiring circuit learning.The section identifies QKE and VQM as the two main options after setting up the encoder circuit.Source: A. Supervised learning · p. 9
• With error mitigation, Havlíček et al. found classification success increased with circuit depth on two superconducting qubits.Their experiments used IBM Q5 Yorktown; the review notes that future comparisons with established classical models should assess accuracy and training efficiency.Source: A. Supervised learning · pp. 9–10
• Farhi and Neven [46] propose a VQM binary classifier for bitstrings, encoded as computational-basis states using identity and NOT gates at almost no cost.Source: A. Supervised learning · p. 10

• Variational circuits can achieve zero classification error for any possible label function, although some require impractically exponential depth, paralleling the universal approximation theorem [24].The theorem is compared with neural networks requiring an exponentially large hidden layer to represent any Boolean function.Source: The variational circuit · p. 10
• Mitarai et al. encode real-valued data nonlinearly and entangle qubits so a local observable can extract relevant nonlinear features for classification and regression.Tensor-network-inspired construction is one approach to designing highly entangling circuits.Source: The variational circuit · p. 10
• Grant et al. use TTN and MERA circuits for binary classification on Iris and MNIST; in simulations, MERA always outperforms TTN, and a simple model is deployed on IBM Q5 Tenerife with noise resilience.The deployed model is efficiently trained classically before use on the quantum computer.Source: The variational circuit · p. 10
• Stoudenmire et al. [28] use a TTN for pairwise MNIST classification and entanglement entropy to quantify information about image details gained from their context.The passage presents this as an example of using quantum properties to characterize classical-data complexity.Source: The variational circuit · p. 10
• Schuld et al. [48] design a low-depth, expressive classifier for amplitude-encoded data using systematic entangling gates and polynomially many parameters; simulations match off-the-shelf classical models with significantly fewer parameters.The low-depth design addresses the potentially high cost of the encoder circuit.Source: The variational circuit · p. 10
• Supervised-learning experiments so far use scaled-down, often trivial datasets because of hardware limits, while realistic-scale demonstrations and PQC-specific regularization remain underdeveloped.The passage identifies regularization techniques designed specifically for PQC models as an interesting research area.Source: The variational circuit · p. 10

B. Generative modeling

• A QCBM prepares a variational quantum state and samples its computational-basis measurements as binary outputs; real-valued targets require post-processing.Because its wave function is not directly accessible, a QCBM is an implicit model: sampling is easy, but estimating sample likelihoods can be hard.Source: B. Generative modeling · p. 10
• QCBMs are more expressive than deep Boltzmann machines with polynomially many parameters, and some QCBMs resist efficient classical simulation, including during training.These results are reported by Du et al. [74] and Coyle et al. [75], respectively.Source: B. Generative modeling · p. 11
• Low-depth QCBMs trained with particle swarms fit Bars-and-Stripes and Boltzmann distributions in simulation, while a four-qubit trapped-ion experiment demonstrated convergence to a target distribution.Benedetti et al. report the simulations; Zhu et al. [59] report the hardware experiment.Source: B. Generative modeling · p. 11
• Sample-estimated maximum mean discrepancy enables gradient-descent training on discrete data; simulations fit Bars-and-Stripes and discretized Gaussians, but hardware performance was usually worse than simulation.An Aspen experiment also emphasizes that performance depends on correctly setting hyperparameters across the hybrid system’s components.Source: B. Generative modeling · p. 11
• Adversarial training addresses QCBM loss-design challenges by using a discriminator to distinguish real from generated data and provide the generator with a differentiable loss.Both models train at the same time.Source: B. Generative modeling · p. 11
• QGANs with neural-network discriminators reproduce discrete and continuous target statistics; a proposed amplitude-encoding circuit uses polynomially many gates versus exponential complexity for the best known generic method.The amplitude-encoding approach has potential if training cost and required precision remain low.Source: B. Generative modeling · p. 11
• Grover-based inference offers a quadratic speedup over naïve methods but remains exponentially complex; applying it to QCBMs may exceed NISQ capabilities, leaving near-term inference open.Inference conditions on known (“clamped”) variables and estimates other variables from the conditional distribution.Source: B. Generative modeling · p. 11

Quantum algorithm learning

• PQC models have been trained to learn quantum algorithms, with demonstrations including improved Grover operators for three and four qubits, recovery of Simon’s algorithm, factoring, and smaller-gate SWAP tests.Morales et al. report improved operators for the specific three- and four-qubit cases; Wan et al. [87] recover Simon’s algorithm with equal performance in simulations; Anschuetz et al. train for integer factoring; Cincio et al. find SWAP-test solutions with fewer gates than known circuits.Source: C. Quantum learning tasks · pp. 11–12
• Practical use of quantum algorithm learning requires experimental scaling studies under NISQ constraints and noise, plus theoretical analysis to avoid exponential sample complexity.The required sample complexity is the number of training samples needed to learn a target algorithm successfully.Source: C. Quantum learning tasks · p. 12

Quantum state classification

• Quantum-state classification depends on circuit design: simple TTN operations perform no better than random assignment, while ancilla-based operations provide some accuracy; measurement-conditioned gates can add nonlinearity.Chen et al. [91] simulate classifying states as pure or mixed, with an inconclusive output; Grant et al. study pure states with different entanglement levels.Source: C. Quantum learning tasks · p. 12

Quantum state learning

• Quantum state learning offers an alternative to full tomography’s exponentially growing measurement requirement: practical PAC learning needs linearly many measurements in the number of qubits, and variational methods can reconstruct or diagonalize states.Rocchetto et al. experimentally verify linear scaling on a custom photonic computer; Lee et al. use a variational circuit to map an unknown state to a fiducial state, while LaRose et al. [96] train a circuit to diagonalize mixed states.Source: C. Quantum learning tasks · p. 12

Quantum generative modeling

• Quantum generative adversarial networks model quantum data by having a discriminator implement an optimal distinguishability measurement and a generator make target and synthetic states harder to distinguish.The circuits are optimized in tandem; proposals generate label-conditioned states or approximate pure states, with generator and discriminator depth affecting approximation quality.Source: C. Quantum learning tasks · p. 12

Quantum-information applications

• PQC models also target quantum-information tasks: autoencoders compress n-qubit states into k-qubit states, learned encoding and recovery circuits seek higher code fidelity, and variational circuits approximate hardware-constrained target unitaries.The autoencoder traces out n − k qubits and trains for expected input-output fidelity; encoding and recovery circuits aim to reduce NISQ overhead [101], while compiler circuits approximate U = e−iHt.Source: C. Quantum learning tasks · p. 13

IV. OUTLOOK

• The field has moved beyond theory and simulation to experimental demonstrations on scaled-down problems, while dedicated hybrid machine-learning platforms offer PQC models and simulator and hardware interfaces.The Review recommends open-source libraries and numerical analysis to support faster experimentation, fair comparison, and replicability; its software summary is not comprehensive.Source: §IV · pp. 13–14
• Researchers are beginning to explore connections between quantum-supremacy proposals and machine learning, alongside links between quantum supremacy and optimization algorithms [109], [111] [112], [75], [113].The Review describes these connections as a potential route toward practical utility if key requirements are met.Source: §IV · p. 14
• Meaningful-scale real-world datasets should be prioritized to test whether PQC models improve established kernel methods or offer practical generative-modeling advantages.Although PQCs can implement classically intractable feature maps and kernels, greater circuit expressivity may make optimization harder to explore.Source: §IV · p. 14
• For quantum learning, PQCs can outperform classical methods when quantum data can be efficiently loaded or prepared and the system’s statistics require exponentially scaling classical resources to reproduce.The passage conditions this advantage on efficient quantum-data preparation.Source: §IV · p. 14
• PQC models naturally implement unitary operations and permit gradient computation with discrete measurement data, while opening research on VC dimensions, regularization, Bayesian inference, and reinforcement learning.The Review also notes unitary constraints as a way to prevent exploding gradients in recurrent neural networks.Source: §IV · p. 14
• Hybrid PQC algorithms can develop incrementally, starting with heavy classical-resource use and replacing it with quantum resources as hardware improves.Examples include interpolating from variational quantum eigensolvers to quantum phase estimation, replacing destructive tests with non-destructive variants, and moving toward tensor-network-driven parameterizations supported by quantum compilers.Source: §IV · p. 14
• Closer integration of quantum and classical components calls for hybrid data centers, better cloud interfaces, and native hybrid software frameworks to facilitate research, real-world demonstrations, and production implementation.The Review presents these as goals for future development.Source: §IV · p. 14
• If PQC models scale well to realistic machine-learning tasks, they may become part of automated forecasting and decision-making systems.The Review frames this as a possibility, not an established outcome.Source: §IV · p. 14