Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule

TL;DR

This paper introduces a stochastic parameter shift rule for efficiently estimating gradients of quantum evolutions directly from measurements, applicable to noisy near-term quantum devices, enhancing optimization in quantum algorithms.

Contribution

The authors derive an exact stochastic gradient estimation formula for multi-qubit quantum evolutions that does not require ancillary qubits or Hamiltonian simulation, and works under noise.

Findings

Provides a mathematically exact stochastic gradient estimation method.

Applicable to noisy quantum devices with all Pauli rotations.

Simplifies and generalizes existing gradient estimation approaches.

Abstract

Hybrid quantum-classical optimization algorithms represent one of the most promising application for near-term quantum computers. In these algorithms the goal is to optimize an observable quantity with respect to some classical parameters, using feedback from measurements performed on the quantum device. Here we study the problem of estimating the gradient of the function to be optimized directly from quantum measurements, generalizing and simplifying some approaches present in the literature, such as the so-called parameter-shift rule. We derive a mathematically exact formula that provides a stochastic algorithm for estimating the gradient of any multi-qubit parametric quantum evolution, without the introduction of ancillary qubits or the use of Hamiltonian simulation techniques. The gradient measurement is possible when the underlying device can realize all Pauli rotations in the expansion of the Hamiltonian whose coefficients depend on the parameter. Our algorithm continues to work, although with some approximations, even when all the available quantum gates are noisy, for instance due to the coupling between the quantum device and an unknown environment.