Discrepancies between Asymptotic and Exact Spectral Gap Analyses of Quantum Adiabatic Barrier Tunneling

TL;DR

This paper reveals that asymptotic spectral gap analyses in quantum annealing do not accurately predict finite-size behavior, especially for practical system sizes, due to fundamental discrepancies in simple tunneling models.

Contribution

It demonstrates that asymptotic spectral gap predictions are unreliable for realistic system sizes in quantum annealing, highlighting the need for exact finite-size analysis.

Findings

Asymptotic analysis fails for system sizes up to 10^{12}

Discrepancies are inherent to simple tunneling models

Asymptotic methods may not be suitable for practical quantum annealing studies

Abstract

We study the asymptotic behavior of the spectral gap of simple barrier tunneling problems, which are related to using quantum annealing to find the global optimum of cost functions defined over n bits. Specifically we look at the problem of having an n qubit system tunnel through a barrier of width and height proportional to n^a. We show that with these quantum annealing problems, the asymptotic, $n\to\infty$, behavior of the spectral gap does not accurately describe the behavior of the gap at finite n until extremely large values of n (n>10^{12}). We prove that this deficiency of the asymptotic expression is a feature of simple one-dimensional tunneling problems themselves, casting doubt on whether asymptotic analysis is an appropriate tool for studying tunneling problems in quantum annealing for reasonably sized systems.