Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.14962 · Quantum

Backpropagation scaling in parameterised quantum circuits

Joseph Bowles, David Wierichs, Chae-Yeun Park

Published Oct 2, 2025• 15 citations
arXivHTMLPDF

Abstract

The discovery of the backpropagation algorithm ranks among one of the most important moments in the history of machine learning, and has made possible the training of large-scale neural networks through its ability to compute gradients at roughly the same computational cost as model evaluation. Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the most popular method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that are not known to be classically simulable and admit gradient estimation with significantly fewer circuits. In the simplest case – for which the parameters feed into commuting quantum gates – these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix. Moreover, specific families of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.

DOI
10.22331/q-2025-10-02-1873

Related papers

Sample-based training of quantum generative models

Maria Demidik, Cenk Tüysüz, Michele Grossi +1 more

Quantum computers can efficiently sample from probability distributions that are believed to be classically intractable, providing a foundation for quantum generative modeling. However, practical training of such models remains challenging, as gradient evaluation via the parameter-shift rule scales linearly with the number of parameters and requires repeated expectation-value estimation under finite-shot noise. We introduce a training framework that extends the principle of contrastive divergence to quantum models. By deriving the circuit structure and providing a general recipe for constructing it, we obtain quantum circuits that generate the samples required for parameter updates, yielding constant scaling with respect to the cost of a forward pass, analogous to backpropagation in classical neural networks. Numerical results demonstrate that it attains comparable accuracy to likelihood-based optimization while requiring substantially fewer samples. The framework thereby establishes a scalable route to training expressive quantum generative models directly on quantum hardware.

quant-pharXiv:2511.11802•Nov 18, 2025•0 citations

Gradient Estimation with Constant Scaling for Hybrid Quantum Machine Learning

Thomas Hoffmann, Douglas Brown

We present a novel method for determining gradients of parameterised quantum circuits (PQCs) in hybrid quantum-classical machine learning models by applying the multivariate version of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The gradients of PQC layers can be calculated with an overhead of two evaluations per circuit per forward-pass independent of the number of circuit parameters, compared to the linear scaling of the parameter shift rule. These are then used in the backpropagation algorithm by applying the chain rule. We compare our method to the parameter shift rule for different circuit widths and batch sizes, and for a range of learning rates. We find that, as the number of qubits increases, our method converges significantly faster than the parameter shift rule and to a comparable accuracy, even when considering the optimal learning rate for each method.

arXiv:2211.13981•Nov 25, 2022•2 citations

Classical Pre-Training of Large Shallow Parametrized Quantum Circuits through Pauli Propagation and deployment on Quantum Hardware

Saverio Monaco, Jamal Slim, Ran Xue +2 more

Scaling parametrized quantum circuits to a larger number of qubits is a central challenge in Quantum Machine Learning: gradient estimation via the parameter-shift rule quickly becomes prohibitively expensive, while gradient-free optimizers offer limited convergence guarantees. We apply the Pauli propagation framework to a broad class of Quantum Machine Learning problems in which the loss is expressed as a function of expectation values of the circuit's observables, while keeping the relevant quantity of interest, such as the quantum state or its samples, classically intractable to compute. By propagating the observables backward through the circuit, we obtain an explicit parametric expression for the loss that can be minimized entirely on classical hardware. We also present complementary strategies to make propagation tractable at large scale, including weight-based and degree-based truncations and pruning heuristics. We demonstrate the approach on two benchmarks run on 64-qubit circuits: Variational Quantum Eigensolver on a two-dimensional Ising model, and training of a discrete-type generative model for binary images. In both cases, the classically pre-trained parameters are deployed on the IBM quantum processor ibm_berlin. For generative learning, we additionally show that the circuits considered here require substantially fewer parametrized gates than IQP-style ansätze, reducing transpilation overhead and making deployment on current quantum hardware feasible.

quant-pharXiv:2610.32571•Oct 26, 2026•0 citations

Train on classical, deploy on quantum: scaling generative quantum machine learning to a thousand qubits

Erik Recio-Armengol, Shahnawaz Ahmed, Bowles, Joseph

We propose an approach to generative quantum machine learning that overcomes the fundamental scaling issues of variational quantum circuits. The core idea is to use a class of generative models based on instantaneous quantum polynomial circuits, which we show can be trained efficiently on classical hardware. Although training is classically efficient, sampling from these circuits is widely believed to be classically hard, and so computational advantages are possible when sampling from the trained model on quantum hardware. By combining our approach with a data-dependent parameter initialisation strategy, we do not encounter issues of barren plateaus and successfully circumvent the poor scaling of gradient estimation that plagues traditional approaches to quantum circuit optimisation. We investigate and evaluate our approach on a number of real and synthetic datasets, training models with up to one thousand qubits and hundreds of thousands of parameters. We find that the quantum models can successfully learn from high dimensional data, and perform surprisingly well compared to simple energy-based classical generative models trained with a similar amount of hyperparameter optimisation. Overall, our work demonstrates that a path to scalable quantum generative machine learning exists and can be investigated today at large scales.

arXiv:2503.02934•Mar 4, 2025•0 citations

Adaptive directional gradients for parameterised quantum circuits

Brian Coyle, Snehal Raj, Virag Umathe +2 more

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.

quant-pharXiv:2606.09734•Jun 9, 2026•0 citations

Bayesian learning of parameterised quantum circuits

Samuel Duffield, Marcello Benedetti, Matthias Rosenkranz

Abstract Currently available quantum computers suffer from constraints including hardware noise and a limited number of qubits. As such, variational quantum algorithms that utilise a classical optimiser in order to train a parameterised quantum circuit have drawn significant attention for near-term practical applications of quantum technology. In this work, we take a probabilistic point of view and reformulate the classical optimisation as an approximation of a Bayesian posterior. The posterior is induced by combining the cost function to be minimised with a prior distribution over the parameters of the quantum circuit. We describe a dimension reduction strategy based on a maximum a posteriori point estimate with a Laplace prior. Experiments on the Quantinuum H1-2 computer show that the resulting circuits are faster to execute and less noisy than the circuits trained without the dimension reduction strategy. We subsequently describe a posterior sampling strategy based on stochastic gradient Langevin dynamics. Numerical simulations on three different problems show that the strategy is capable of generating samples from the full posterior and avoiding local optima.

arXiv:2206.07559•Machine Learning Science and Technology•Mar 29, 2023•10 citations
The discovery of the backpropagation algorithm ranks among one of the most important moments in the history of machine learning, and has made possible the training of large-scale neural networks through its ability to compute gradients at roughly the same computational cost as model evaluation. Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the most popular method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that are not known to be classically simulable and admit gradient estimation with significantly fewer circuits. In the simplest case – for which the parameters feed into commuting quantum gates – these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix. Moreover, specific families of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.
Quantum computers can efficiently sample from probability distributions that are believed to be classically intractable, providing a foundation for quantum generative modeling. However, practical training of such models remains challenging, as gradient evaluation via the parameter-shift rule scales linearly with the number of parameters and requires repeated expectation-value estimation under finite-shot noise. We introduce a training framework that extends the principle of contrastive divergence to quantum models. By deriving the circuit structure and providing a general recipe for constructing it, we obtain quantum circuits that generate the samples required for parameter updates, yielding constant scaling with respect to the cost of a forward pass, analogous to backpropagation in classical neural networks. Numerical results demonstrate that it attains comparable accuracy to likelihood-based optimization while requiring substantially fewer samples. The framework thereby establishes a scalable route to training expressive quantum generative models directly on quantum hardware.
We present a novel method for determining gradients of parameterised quantum circuits (PQCs) in hybrid quantum-classical machine learning models by applying the multivariate version of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The gradients of PQC layers can be calculated with an overhead of two evaluations per circuit per forward-pass independent of the number of circuit parameters, compared to the linear scaling of the parameter shift rule. These are then used in the backpropagation algorithm by applying the chain rule. We compare our method to the parameter shift rule for different circuit widths and batch sizes, and for a range of learning rates. We find that, as the number of qubits increases, our method converges significantly faster than the parameter shift rule and to a comparable accuracy, even when considering the optimal learning rate for each method.
Scaling parametrized quantum circuits to a larger number of qubits is a central challenge in Quantum Machine Learning: gradient estimation via the parameter-shift rule quickly becomes prohibitively expensive, while gradient-free optimizers offer limited convergence guarantees. We apply the Pauli propagation framework to a broad class of Quantum Machine Learning problems in which the loss is expressed as a function of expectation values of the circuit's observables, while keeping the relevant quantity of interest, such as the quantum state or its samples, classically intractable to compute. By propagating the observables backward through the circuit, we obtain an explicit parametric expression for the loss that can be minimized entirely on classical hardware. We also present complementary strategies to make propagation tractable at large scale, including weight-based and degree-based truncations and pruning heuristics. We demonstrate the approach on two benchmarks run on 64-qubit circuits: Variational Quantum Eigensolver on a two-dimensional Ising model, and training of a discrete-type generative model for binary images. In both cases, the classically pre-trained parameters are deployed on the IBM quantum processor ibm_berlin. For generative learning, we additionally show that the circuits considered here require substantially fewer parametrized gates than IQP-style ansätze, reducing transpilation overhead and making deployment on current quantum hardware feasible.
We propose an approach to generative quantum machine learning that overcomes the fundamental scaling issues of variational quantum circuits. The core idea is to use a class of generative models based on instantaneous quantum polynomial circuits, which we show can be trained efficiently on classical hardware. Although training is classically efficient, sampling from these circuits is widely believed to be classically hard, and so computational advantages are possible when sampling from the trained model on quantum hardware. By combining our approach with a data-dependent parameter initialisation strategy, we do not encounter issues of barren plateaus and successfully circumvent the poor scaling of gradient estimation that plagues traditional approaches to quantum circuit optimisation. We investigate and evaluate our approach on a number of real and synthetic datasets, training models with up to one thousand qubits and hundreds of thousands of parameters. We find that the quantum models can successfully learn from high dimensional data, and perform surprisingly well compared to simple energy-based classical generative models trained with a similar amount of hyperparameter optimisation. Overall, our work demonstrates that a path to scalable quantum generative machine learning exists and can be investigated today at large scales.
Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.
Abstract Currently available quantum computers suffer from constraints including hardware noise and a limited number of qubits. As such, variational quantum algorithms that utilise a classical optimiser in order to train a parameterised quantum circuit have drawn significant attention for near-term practical applications of quantum technology. In this work, we take a probabilistic point of view and reformulate the classical optimisation as an approximation of a Bayesian posterior. The posterior is induced by combining the cost function to be minimised with a prior distribution over the parameters of the quantum circuit. We describe a dimension reduction strategy based on a maximum a posteriori point estimate with a Laplace prior. Experiments on the Quantinuum H1-2 computer show that the resulting circuits are faster to execute and less noisy than the circuits trained without the dimension reduction strategy. We subsequently describe a posterior sampling strategy based on stochastic gradient Langevin dynamics. Numerical simulations on three different problems show that the strategy is capable of generating samples from the full posterior and avoiding local optima.