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. In this talk, I will introduce the concept of topological quantum spin glass (TQSG) order [1] which marries these two notions, exhibiting both the complex energy landscapes typical of many 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 realized at low-temperatures in various quantum error correcting codes, namely so-called low density parity check (LDPC) codes on expander graphs. This extends previous results on the connection between classical LDPC codes and spin glasses to the quantum realm. 

 

[1] B Placke, T Rakovszky, NP Breuckmann, V Khemani: "Topological Quantum Spin Glass Order and its realization in qLDPC codes", arXiv 2412.13248