The Effects of the Problem Hamiltonian Parameters on the Minimum Spectral Gap in Adiabatic Quantum Optimization

TL;DR

This paper investigates how the parameters of the problem Hamiltonian in adiabatic quantum optimization influence the minimum spectral gap, introducing a new anti-crossing definition and demonstrating parameter effects on the gap through the MIS problem.

Contribution

It introduces a new parametrization of anti-crossings, showing how Hamiltonian parameters can drastically alter the spectral gap in quantum annealing.

Findings

Changing the energy penalty J can switch the system between having and not having an anti-crossing.

Adjusting parameters can significantly increase or decrease the minimum spectral gap.

The new anti-crossing definition is scale-invariant and applicable to any Ising problem.

Abstract

We study the relation between the Ising problem Hamiltonian parameters and the minimum spectral gap (min-gap) of the system Hamiltonian in the Ising-based quantum annealer. The main argument we use in this paper to assess the performance of a QA algorithm is the presence or absence of an anti-crossing during quantum evolution. For this purpose, we introduce a new parametrization definition of the anti-crossing. Using the Maximum-weighted Independent Set (MIS) problem in which there are flexible parameters (energy penalties J between pairs of edges) in an Ising formulation as the model problem, we construct examples to show that by changing the value of J, we can change the quantum evolution from one that has an anti-crossing (that results in an exponential small min-gap) to one that does not have, or the other way around, and thus drastically change (increase or decrease) the min-gap. However, we also show that by changing the value of $J$ alone, one can not avoid the anti-crossing. We recall a polynomial reduction from an Ising problem to an MIS problem to show that the flexibility of changing parameters without changing the problem to be solved can be applied to any Ising problem. As an example, we show that by such a reduction alone, it is possible to remove the anti-crossing and thus increase the min-gap. Our anti-crossing definition is necessarily scaling invariant as scaling the problem Hamiltonian does not change the nature (i.e. presence or absence) of an anti-crossing. As a side note, we show exactly how the min-gap is scaled if we scale the problem Hamiltonian by a constant factor.