Unstructured Randomness, Small Gaps and Localization

TL;DR

This paper analyzes a quantum adiabatic algorithm with a random, unstructured cost function, revealing exponential decay of the minimum gap, ground state localization, and a lack of near-degenerate levels at the end of evolution.

Contribution

It provides rigorous results on the ground state energy, gap behavior, and localization transition for an unstructured random Hamiltonian in the quantum adiabatic context.

Findings

Minimum gap decreases exponentially with system size

Ground state exhibits localization transition

No near-degenerate levels near the end of evolution

Abstract

We study the Hamiltonian associated with the quantum adiabatic algorithm with a random cost function. Because the cost function lacks structure we can prove results about the ground state. We find the ground state energy as the number of bits goes to infinity, show that the minimum gap goes to zero exponentially quickly, and we see a localization transition. We prove that there are no levels approaching the ground state near the end of the evolution. We do not know which features of this model are shared by a quantum adiabatic algorithm applied to random instances of satisfiability since despite being random they do have bit structure.