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

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.

Exploring ab initio machine synthesis of quantum circuits

“…Therefore, if a different method for optimising the parameters proves more suitable, it can be straightforwardly substituted for the imaginary time evolution used in the present paper. One promising candidate is introduced in [71] named CoVaR, where an eigenstate of the system is prepared by a root-finding algorithm similar to Newton's method. By tweaking the spectrum of our synthesis Hamiltonians such that only product states are eigenstates, this method could be used to find the appropriate parameters in each iteration.…”