How to Make the Quantum Adiabatic Algorithm Fail

TL;DR

This paper demonstrates that certain poor choices of Hamiltonians in the quantum adiabatic algorithm can cause it to fail to find the minimum efficiently, highlighting pitfalls in algorithm design.

Contribution

It identifies specific nonlocal Hamiltonian choices that hinder the quantum adiabatic algorithm's performance, providing guidance on what to avoid in design.

Findings

Poor Hamiltonian choices can cause failure for run times less than sqrt(N)

Nonlocal Hamiltonians eliminate problem structure, reducing speedup to Grover's

Guidelines for avoiding ineffective Hamiltonian designs in quantum adiabatic algorithms

Abstract

The quantum adiabatic algorithm is a Hamiltonian based quantum algorithm designed to find the minimum of a classical cost function whose domain has size N. We show that poor choices for the Hamiltonian can guarantee that the algorithm will not find the minimum if the run time grows more slowly than square root of N. These poor choices are nonlocal and wash out any structure in the cost function to be minimized and the best that can be hoped for is Grover speedup. These failures tell us what not to do when designing quantum adiabatic algorithms.