Theory of variational quantum simulation

TL;DR

This paper develops a comprehensive theory for variational quantum simulation, extending it to mixed states, stochastic evolution, and imaginary time, with practical implications for near-term quantum hardware.

Contribution

It introduces a unified framework for variational quantum simulation of both pure and mixed states, including real and imaginary time evolution, applicable to near-term quantum devices.

Findings

Extended variational principles to mixed states and stochastic evolution.

Unified approach for real and imaginary time simulation.

Applicable to near-term quantum hardware.

Abstract

The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.