Variational ansatz-based quantum simulation of imaginary time evolution

Abstract

Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground state energy of many-particle systems; specifically molecular Hydrogen and Lithium Hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation, and can exploit shallow quantum circuits, it can be implemented with current quantum computers.

Paired electron motion in interacting chains of quantum dots

“…This substitution, known as Wick rotation [20] is used in the diffusion Monte Carlo methods [21]. The evolution in the imaginary time [22,23] brings the system to the ground-state. In our calculation the rate of the energy relaxation is controlled by the α parameter which indicates the ratio of imaginary time steps to the total number of steps.…”

QuESTlink—Mathematica embiggened by a hardware-optimised quantum emulator*

“…Next, we choose a random initial assignment of the ansatz parameters, and measure the energy of the resulting quantum state. The variational imaginary-time algorithm [12] involves repeatedly measuring a matrix and vector of quantities,…”

A Quantum Algorithm for Linear PDEs Arising in Finance

“…We now describe a strategy to solve the linear Schrödinger equation along its imaginary time axis using a quantum computer. Our approach relies on the hybrid algorithm developed in [10], where part of the computation is performed on a quantum computer, and part on a classical machine. Consider first a timeindependent Hamiltonian H with evolution operator (or propagator) exp(i Hξ) evolving along real values ξ.…”

“…Along the imaginary axis however, the corresponding evolution operator exp( Hτ ) is represented by a non-unitary matrix; It can be easily simulated on a classical computer, however as the dimension of the wave function grows exponentially, it rapidly becomes infeasible; its Trotter decomposition using unitary gates is not straightforward, making its implementation on a quantum computer more challenging. We follow instead a recent idea [10] to solve an equivalent normalised imaginary time evolution…”

“…One of the main advantages of this method is that both A and C can be measured efficiently using a quantum computer [2,10]. In order to build the hybrid classical-quantum scheme, we rely on the following assumptions: Assumption 3.1(i) is in fact purely for convenience, as unitary gates with multiple parameters can be easily decomposed into single-parameter ones.…”