Improved amplitude amplification strategies for the quantum simulation of classical transport problems

TL;DR

This paper analyzes the limitations of oblivious amplitude amplification in quantum simulations of classical transport problems and proposes a new strategy to reduce errors caused by non-unitary dynamics.

Contribution

It provides an analytical understanding of amplitude amplification errors in non-unitary quantum dynamics and introduces a mitigation strategy for improved quantum simulation accuracy.

Findings

Oblivious amplitude amplification distorts non-unitary quantum states.

An analytical upper bound for amplification error is derived.

A new amplification strategy reduces distortion while increasing success probability.

Abstract

The quantum simulation of classical fluids often involves the use of probabilistic algorithms that encode the result of the dynamics in the form of the amplitude of the selected quantum state. In most cases, however, the amplitude probability is too low to allow an efficient use of these algorithms, thereby hindering the practical viability of the quantum simulation. The oblivious amplitude amplification algorithm is often presented as a solution to this problem, but to no avail for most classical problems, since its applicability is limited to unitary dynamics. In this paper, we show analytically that oblivious amplitude amplification when applied to non-unitary dynamics leads to a distortion of the quantum state and to an accompanying error in the quantum update. We provide an analytical upper bound of such error as a function of the degree of non-unitarity of the dynamics and we test it against a quantum simulation of an advection-diffusion-reaction equation, a transport problem of major relevance in science and engineering. Finally, we also propose an amplification strategy that helps mitigate the distortion error, while still securing an enhanced success probability.