Adaptive directional gradients for parameterised quantum circuits

2026-06-08

AI Summary

This work addresses the high measurement cost of gradient estimation in hardware-efficient training of parameterized quantum circuits, where the parameter-shift rule incurs a linear overhead in the number of parameters, limiting scalability. The authors propose an unbiased gradient estimation framework based on forward-mode automatic differentiation, which efficiently approximates gradients by averaging directional derivatives along a small number of random directions—requiring neither ancillary qubits nor controlled gates. This framework unifies stochastic perturbation methods such as SPSA, random coordinate descent, and the parameter-shift rule, and introduces QUIVER, an adaptive optimizer that optimally allocates measurement resources. Theoretical analysis establishes convergence guarantees for stochastic quantum forward gradient descent. Experiments demonstrate successful training of a 60-qubit, 1,770-parameter quantum neural network on ECG5000 and MNIST datasets, achieving orders-of-magnitude speedup over the parameter-shift rule; QUIVER also significantly outperforms iCANS and gCANS in VQE and QAOA tasks.

Abstract

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.

Problem

• parameterised quantum circuits
• gradient estimation
• measurement cost
• quantum training
• shot budget

Innovation

• parameterised quantum circuits
• forward gradient estimation
• adaptive optimisation
• measurement efficiency
• QUIVER