Spectral Gap Analysis for Efficient Tunneling in Quantum Adiabatic Optimization

TL;DR

This paper analyzes how the spectral gap affects the efficiency of quantum adiabatic optimization in tunneling scenarios, providing polynomial and exponential scaling results for different barrier sizes.

Contribution

It offers a rigorous analysis of spectral gap scaling in quantum adiabatic optimization with symmetric cost functions, extending previous folklore and recent findings.

Findings

Spectral gap scales polynomially as n^{1/2 - a - b} when 2a + b < 1.

Spectral gap scales exponentially as n^{-b/2} exp(-C n^{(2a + b - 1)/2}) when 1 < 2a + b.

The analysis confirms and extends existing results on exponential gap scaling in quantum tunneling scenarios.

Abstract

We investigate the efficiency of Quantum Adiabatic Optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at n qubit systems with symmetric cost functions f:{0, 1}^n->R where the ground state must tunnel through a potential barrier of width n^a and height n^b. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with 1/2 < a + b and a < 1/2 and show that the minimum gap scales polynomially as n^{1/2-a-b} when 2a+b < 1 and exponentially as n^{-b/2} exp(-C n^{(2a+b-1)/2} ) when 1 < 2a+b. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson and Jiang et al. about the exponential gap scaling.