Duality approach to quantum annealing of the 3-XORSAT problem

TL;DR

This paper introduces a duality method to analyze quantum annealing in 3-XORSAT problems with highly degenerate ground states, revealing how hypergraph structure influences quantum algorithm performance and phase transitions.

Contribution

It develops a duality approach to study quantum annealing in degenerate 3-XORSAT models, linking classical degeneracy, symmetries, and quantum phase transitions.

Findings

Tree hypergraphs have a constant energy gap during quantum annealing.

Closed hypergraphs exhibit a second-order phase transition with a gap closing as a power-law.

Duality provides insights into quantum models with degenerate energy landscapes.

Abstract

Classical models with complex energy landscapes represent a perspective avenue for the near-term application of quantum simulators. Until now, many theoretical works studied the performance of quantum algorithms for models with a unique ground state. However, when the classical problem is in a so-called clustering phase, the ground state manifold is highly degenerate. As an example, we consider a 3-XORSAT model defined on simple hypergraphs. The degeneracy of classical ground state manifold translates into the emergence of an extensive number of $Z_2$ symmetries, which remain intact even in the presence of a quantum transverse magnetic field. We establish a general duality approach that restricts the quantum problem to a given sector of conserved $Z_2$ charges and use it to study how the outcome of the quantum adiabatic algorithm depends on the hypergraph geometry. We show that the tree hypergraph which corresponds to a classically solvable instance of the 3-XORSAT problem features a constant gap, whereas the closed hypergraph encounters a second-order phase transition with a gap vanishing as a power-law in the problem size. The duality developed in this work provides a practical tool for studies of quantum models with classically degenerate energy manifold and reveals potential connections between glasses and gauge theories.