Minimal Constraints in the Parity Formulation of Optimization Problems

TL;DR

This paper investigates the minimal strength of constraints needed in lattice gauge models for quantum optimization, providing bounds and scaling laws that depend on problem class and size.

Contribution

It introduces a method to determine the minimal constraint strength in lattice gauge models for quantum optimization problems, with derived bounds and scaling behaviors.

Findings

Minimal constraint strength bounds are derived.

Constraint strength scales from linear to quadratic with qubit number.

Scaling depends on the specific problem class.

Abstract

As a means to solve optimization problems using quantum computers, the problem is typically recast into a Ising spin model whose ground-state is the solution of the optimization problem. An alternative to the Ising formulation is the Lechner-Hauke-Zoller model, which has the form of a lattice gauge model with nearest neighbor 4-body constraints. Here we introduce a method to find the minimal strength of the constraints which are required to conserve the correct ground-state. Based on this, we derive upper and lower bounds for the minimal constraints strengths. We find that depending on the problem class, the exponent ranges from linear $\alpha \propto 1$ to quadratic $\alpha \propto 2$ scaling with the number of logical qubits.