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

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.

Automated summary

Hybrid quantum-classical optimization algorithms aim to optimize observable quantities with respect to classical parameters using feedback from quantum device measurements. This study introduces a stochastic algorithm for estimating gradients of multi-qubit parametric quantum evolutions directly from quantum measurements, without the need for ancillary qubits or Hamiltonian simulation techniques. The algorithm can handle noise in quantum gates and still provide accurate approximations for gradient estimation.