Weaker Assumptions for the Short Path Optimization Algorithm

TL;DR

This paper extends the short path quantum optimization algorithm by removing the unique ground state assumption, broadening its applicability and demonstrating potential speedups for more general problem instances.

Contribution

It removes the unique ground state assumption from the short path algorithm and analyzes its performance on arbitrary MAX-D-LIN-2 problems and random instances.

Findings

Super-Grover speedup achieved without density of states assumptions.

Applicable to arbitrary MAX-D-LIN-2 problems.

Heuristic suggests more significant improvements for random instances.

Abstract

The short path algorithm gives a super-Grover speedup for various optimization problems under the assumption of a unique ground state and under an assumption on the density of low-energy states. Here, we remove the assumption of a unique ground state; this uses the same algorithm but a slightly different analysis and holds for arbitrary MAX-$D$-LIN-$2$ problems. Then, specializing to the case $D=2$, we show that for certain values of the objective function we can always achieve a super-Grover speedup (albeit a very slight one) without any assumptions on the density of states. Finally, for random instances, we give a heuristic treatment suggesting a more significant improvement.