# Obstacles to quantum annealing in a planar embedding of XORSAT

## TL;DR

This paper introduces a planar embedding of the XORSAT problem that avoids certain phase transitions but still exhibits slow glassy dynamics, making quantum annealing inefficient for solving it despite the polynomial solvability of the original problem.

## Contribution

The study presents a novel planar embedding of XORSAT that bypasses thermodynamic phase transitions but demonstrates that quantum annealing remains inefficient due to energy gap scaling.

## Key findings

- No finite-temperature phase transition in the embedding.
- Quantum Monte Carlo shows polynomial and exponential gap scaling for k=2 and k=3.
- Quantum annealing does not efficiently solve the embedded XORSAT.

## Abstract

We introduce a planar embedding of the k-regular k-XORSAT problem, in which solutions are encoded in the ground state of a classical statistical mechanics model of reversible logic gates arranged on a square grid and acting on bits that represent the Boolean variables of the problem. The special feature of this embedding is that the resulting model lacks a finite-temperature phase transition, thus bypassing the first-order thermodynamic transition known to occur in the random graph representation of XORSAT. In spite of this attractive feature, the thermal relaxation into the ground state displays remarkably slow glassy behavior. The question addressed in this paper is whether this planar embedding can afford an efficient path to solution of k-regular k-XORSAT via quantum adiabatic annealing. We first show that our model bypasses an avoided level crossing and consequent exponentially small gap in the limit of small transverse fields. We then present quantum Monte Carlo results for our embedding of the k-regular k-XORSAT that strongly support a picture in which second-order and first-order transitions develop at a finite transverse field for k = 2 and k = 3, respectively. This translates into power-law and exponential dependences in the scaling of energy gaps with system size, corresponding to times-to-solution which are, respectively, polynomial and exponential in the number of variables. We conclude that neither classical nor quantum annealing can efficiently solve our reformulation of XORSAT, even though the original problem can be solved in polynomial time by Gaussian elimination.