The theory of variational hybrid quantum-classical algorithms

TLDR

Quantum algorithms typically demand resources beyond current hardware, prompting the development of the quantum variational eigensolver, a hybrid scheme that leverages limited quantum resources alongside classical routines. The study extends the theory of the quantum variational eigensolver and proposes practical algorithmic improvements. The authors introduce a variational adiabatic ansatz, relate second‑order unitary coupled cluster to universal gate sets through relaxed exponential splitting, propose quantum variational error suppression for pre‑threshold devices, and analyze truncation and correlated sampling in Hamiltonian averaging to lower computational cost. They demonstrate that modern derivative‑free optimization can reduce computational cost by up to three orders of magnitude compared to earlier methods.

Abstract

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as "the quantum variational eigensolver" was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through relaxation of exponential splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.