Topological Quantum Spin Glass Order and its realization in qLDPC codes

TL;DR

This paper introduces topological quantum spin glass (TQSG) order, combining spin glass complexity with topological quantum properties, demonstrated in quantum LDPC codes on expander graphs, with implications for quantum information and statistical mechanics.

Contribution

It defines TQSG order and shows it exists as a low-temperature phase in quantum LDPC codes on expander graphs, linking spin glasses and topological order.

Findings

TQSG order exhibits both spin glass and topological features.

Quantum LDPC codes on expander graphs realize TQSG phases.

The work bridges statistical mechanics and quantum error correction.

Abstract

Ordered phases of matter have close connections to computation. Two prominent examples are spin glass order, with wide-ranging applications in machine learning and optimization, and topological order, closely related to quantum error correction. Here, we introduce the concept of topological quantum spin glass (TQSG) order which marries these two notions, exhibiting both the complex energy landscapes of spin glasses, and the quantum memory and long-range entanglement characteristic of topologically ordered systems. Using techniques from coding theory and a quantum generalization of Gibbs state decompositions, we show that TQSG order is the low-temperature phase of various quantum LDPC codes on expander graphs, including hypergraph and balanced product codes. Our work introduces a topological analog of spin glasses that preserves quantum information, opening new avenues for both statistical mechanics and quantum computer science.