[Paper Review] Gradient Estimation with Constant Scaling for Hybrid Quantum Machine Learning

Thomas Hoffmann, Douglas Brown|arXiv (Cornell University)|Nov 25, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces Simultaneous Perturbation for Stochastic Backpropagation (SPSB), a novel gradient estimation method for hybrid quantum machine learning that achieves constant scaling with respect to circuit parameters—requiring only two circuit evaluations per forward pass regardless of parameter count. SPSB outperforms the parameter shift rule in convergence speed by up to 90% fewer circuit evaluations, even when using optimal learning rates, particularly for larger models and batch sizes.

ABSTRACT

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.

Motivation & Objective

• To address the high computational cost of gradient estimation in parameterized quantum circuits (PQCs), especially with increasing parameter counts.
• To develop a scalable, general-purpose method for backpropagation in hybrid quantum-classical machine learning models.
• To enable efficient training on NISQ-era quantum hardware by minimizing circuit evaluations per optimization step.
• To evaluate the performance of the proposed method against the standard parameter shift rule across varying model sizes, batch sizes, and learning rates.
• To demonstrate that SPSB maintains comparable accuracy while significantly improving training speed and scalability.

Proposed method

• The method applies the multivariate form of the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm to estimate gradients in reverse-mode automatic differentiation for PQCs.
• It computes Jacobians using only two circuit evaluations per forward pass, independent of the number of parameters, enabling constant scaling.
• The gradients are integrated into the backpropagation framework via the chain rule, allowing end-to-end optimization of hybrid models.
• The approach leverages statistical perturbations in parameter space to approximate gradients stochastically, reducing reliance on exact analytical shifts.
• The method is compatible with standard optimizers like Adam and SGD, and can be applied to both quantum-only and hybrid quantum-classical models.
• The implementation supports batching and can mitigate noise through averaging over multiple Jacobian evaluations.

Experimental results

Research questions

• RQ1 Can a gradient estimation method be developed that scales linearly with circuit depth but independently of the number of parameters?
• RQ2 How does the proposed SPSB method compare to the parameter shift rule in terms of convergence speed and final accuracy across different model sizes and batch sizes?
• RQ3 What is the optimal learning rate for SPSB, and how does it compare to the optimal learning rate for the parameter shift rule?
• RQ4 Does SPSB maintain performance advantages under varying noise levels and larger parameter counts typical of future NISQ devices?
• RQ5 Can SPSB be effectively combined with advanced optimizers like Adam or quantum natural gradient methods?

Key findings

• SPSB converges significantly faster than the parameter shift rule, achieving comparable accuracy with up to 90% fewer circuit evaluations when using optimal learning rates.
• For the random data classification task, SPSB with a learning rate of 0.01 outperforms the parameter shift rule with a learning rate of 0.5 in terms of evaluation efficiency.
• In the 'quanvolutional' MNIST task, SPSB converges more quickly than the parameter shift rule even when both use their respective optimal learning rates.
• The method shows robustness to statistical noise and is well-suited for large batch sizes, which may help escape barren plateaus in the loss landscape.
• SPSB performs best with lower learning rates (e.g., 0.01–0.05), while higher learning rates (e.g., 0.5) cause instability and non-convergence.
• The method is compatible with existing quantum machine learning libraries and can be readily integrated into frameworks supporting differentiable quantum circuits.