# The Short Path Algorithm Applied to a Toy Model

## TL;DR

This paper numerically evaluates the short path optimization algorithm on a simplified model, demonstrating potential polynomial speedups over Grover search and exploring parameter choices and phase transition behaviors.

## Contribution

It extends previous work by analyzing the algorithm's performance on a broader class of problems with multiple minima and larger system sizes.

## Key findings

- The algorithm can achieve polynomial speedups over Grover search.
- Performance depends on parameter choices and phase transition scaling.
- The study broadens understanding of the algorithm's applicability to complex potentials.

## Abstract

We numerically investigate the performance of the short path optimization algorithm on a toy problem, with the potential chosen to depend only on the total Hamming weight to allow simulation of larger systems. We consider classes of potentials with multiple minima which cause the adiabatic algorithm to experience difficulties with small gaps. The numerical investigation allows us to consider a broader range of parameters than was studied in previous rigorous work on the short path algorithm, and to show that the algorithm can continue to lead to speedups for more general objective functions than those considered before. We find in many cases a polynomial speedup over Grover search. We present a heuristic analytic treatment of choices of these parameters and of scaling of phase transitions in this model.