Guided quantum walk

TL;DR

This paper introduces guided quantum walks (GQW), a novel approach that uses local amplitude transfer theory to improve quantum optimization by systematically guiding probability amplitudes towards solutions, potentially outperforming traditional adiabatic methods.

Contribution

The paper develops the concept of guided quantum walks, integrating QW and QA, and demonstrates their effectiveness on combinatorial problems, revealing the existence of optimal, non-adiabatic annealing schedules.

Findings

GQW can be guided using a time-dependent hopping rate based on the energy spectrum.

Optimal annealing schedules exist beyond adiabatic evolution.

GQW shows promising results on problems with 9 to 30 qubits.

Abstract

We utilize the theory of local amplitude transfers (LAT) to gain insights into quantum walks (QWs) and quantum annealing (QA) beyond the adiabatic theorem. By representing the eigenspace of the problem Hamiltonian as a hypercube graph, we demonstrate that probability amplitude traverses the search space through a series of local Rabi oscillations. We argue that the amplitude movement can be systematically guided towards the ground state using a time-dependent hopping rate based solely on the problem's energy spectrum. Building upon these insights, we extend the concept of multi-stage QW by introducing the guided quantum walk (GQW) as a bridge between QW-like and QA-like procedures. We assess the performance of the GQW on exact cover, traveling salesperson and garden optimization problems with 9 to 30 qubits. Our results provide evidence for the existence of optimal annealing schedules, beyond the requirement of adiabatic time evolutions. These schedules might be capable of solving large-scale combinatorial optimization problems within evolution times that scale linearly in the problem size.