Accuracy vs run time in adiabatic quantum search

TL;DR

This paper investigates the relationship between accuracy and run time in adiabatic quantum search algorithms, revealing two error regimes and their impact on quantum speedup.

Contribution

It provides rigorous analysis linking error behavior to run time, identifying regimes and strategies for optimizing adiabatic quantum search performance.

Findings

Error decreases exponentially at short times

Error decreases polynomially at longer times

Quadratic speedup occurs only in the exponential error regime

Abstract

Adiabatic quantum algorithms are characterized by their run time and accuracy. The relation between the two is essential for quantifying adiabatic algorithmic performance, yet is often poorly understood. We study the dynamics of a continuous time, adiabatic quantum search algorithm, and find rigorous results relating the accuracy and the run time. Proceeding with estimates, we show that under fairly general circumstances the adiabatic algorithmic error exhibits a behavior with two discernible regimes: the error decreases exponentially for short times, then decreases polynomially for longer times. We show that the well known quadratic speedup over classical search is associated only with the exponential error regime. We illustrate the results through examples of evolution paths derived by minimization of the adiabatic error. We also discuss specific strategies for controlling the adiabatic error and run time.