Topological Quantum Spin Glass Order and its realization in qLDPC codes

Ordered phases of matter have close connections to computation. I will discuss some highlights of this deep relationship between quantum matter and quantum computing. 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. I will then introduce the new 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. This phase is a topological analog of spin glasses that preserves quantum information and displays robust many-body entanglement even at finite temperatures, opening new avenues for both statistical mechanics and quantum computer science. Notably this phase is realized in certain families of quantum low density parity check codes which live on non-Euclidean expander graphs, and which have been a topic of much recent interest in quantum error correction. We show how the property of ``code expansion" can be used to derive rigorous results about energy landscapes and spin-glass order, which have been notoriously difficult to realize even in the context of classical spin glasses. Our work opens new prospects in the study of many-body phases in non-local geometries inspired by novel families of error correcting codes, which are increasingly accessible to modern day quantum simulators.