First Order Quantum Phase Transition in Adiabatic Quantum Computation

TL;DR

This paper explores how local minima in problem Hamiltonians can cause first order quantum phase transitions with exponentially small energy gaps, affecting adiabatic quantum computation efficiency.

Contribution

It derives an analytical formula linking local minima properties to the quantum gap and demonstrates how to control the gap size through Hamiltonian parameters.

Findings

Analytical formula predicts gap behavior accurately.

Small gaps are linked to local minima properties.

Parameter tuning can control the quantum gap.

Abstract

We investigate the connection between local minima in the problem Hamiltonian and first order quantum phase transitions during an adiabatic quantum computation. We demonstrate how some properties of the local minima can lead to an extremely small gap that is exponentially sensitive to the Hamiltonian parameters. Using perturbation expansion, we derive an analytical formula that can not only predict the behavior of the gap, but also provide insight on how to controllably vary the gap size by changing the parameters. We show agreement with numerical calculations for a weighted maximum independent set problem instance.