Exponential vanishing of the ground-state gap of the QREM via adiabatic quantum computing

TL;DR

This paper proves that the energy gap of the quantum random energy model (QREM) decreases exponentially with system size, impacting the efficiency of adiabatic quantum algorithms for this model.

Contribution

It provides a simple proof confirming the conjecture that the QREM's energy gap vanishes exponentially, using ideas linking run time bounds to energy gaps.

Findings

Energy gap of QREM vanishes exponentially with system size

Implications for the efficiency of adiabatic quantum algorithms

Provides a rigorous proof of a long-standing conjecture

Abstract

In this note we use ideas of Farhi, Goldstone, Gosset, Gutmann, Nagaj and Shor who link a lower bound on the run time of their quantum adiabatic search algorithm to an upper bound on the energy gap above the ground-state of the generators of this algorithm. We apply these ideas to the quantum random energy model (QREM). Our main result is a simple proof of the conjectured exponential vanishing of the energy gap of the QREM.