Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

TL;DR

The paper introduces CoVaR, a novel variational quantum algorithm that leverages classical shadows to efficiently find eigenstates by estimating numerous covariance functions, significantly improving convergence speed over traditional methods.

Contribution

We propose CoVaR, a new method that exploits classical shadows to estimate many covariances simultaneously, enabling faster eigenstate finding in variational quantum algorithms.

Findings

CoVaR estimates over 10^4 covariance functions efficiently.

Numerical simulations show orders of magnitude faster convergence.

Quantum resource cost per iteration is comparable to standard gradient methods.

Abstract

Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge. Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimising a single classical (energy) surface which is sampled from by a quantum computer. Here we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice. The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques, i.e., we simultaneously estimate a very large number $>10^4-10^7$ of covariances. We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer. We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation, however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed. CoVaR is directly analogous to stochastic gradient-based optimisations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.