Grover Adaptive Search with Problem-Specific State Preparation

TL;DR

This paper develops heuristic quantum state preparation routines inspired by classical heuristics to enhance Grover's search for combinatorial problems like TSP, aiming for efficient approximate solutions with polynomial iterations.

Contribution

It introduces problem-specific heuristic state preparation methods for Grover's algorithm tailored to TSP, building on prior work and exploring algorithmic settings for unknown solution counts.

Findings

Heuristic state preparation improves amplitude amplification for TSP.

Polynomial number of Grover iterations suffices for approximate solutions.

Analysis of termination criteria enhances algorithm robustness.

Abstract

Grover's search algorithm is one of the basic building block in the world of quantum algorithms. Successfully applying it to combinatorial optimization problems is a subtle challenge. As a quadratic speedup is not enough to naively search an exponentially large space, the search has to be complemented with a state preparation routine which increases the amplitudes of promising states by exploiting the problem structure. In this paper, we build upon previous work by Baertschi and Eidenbenz to construct heuristic state preparation routines for the Traveling Salesperson Problem (TSP), mimicking the well-known classical Lin-Kernighan heuristic. With our heuristic, we aim to achieve a reasonable approximation ratio with only a polynomial number of Grover iterations. Further, we compare several algorithmic settings relating to termination criteria and the choice of Grover iterations when the number of marked solutions is unknown.