Adaptive directional gradients for parameterised quantum circuits

• URL: http://arxiv.org/abs/2606.09734v1
• Date: Mon, 08 Jun 2026 16:59:58 GMT
• Title: Adaptive directional gradients for parameterised quantum circuits
• Abstract summary: Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation.<n>We propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation.<n>We show numerically that forward optimisation train Hamming-weight-preserving quantum neural networks with up to 60 qubits and 1770 parameters more efficiently than the parameter-shift rule.
• Score: 1.5549963606717787
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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