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arXiv:2412.13248

Topological Quantum Spin Glass Order and its realization in qLDPC codes

Benedikt Placke, Tibor Rakovszky, Nikolas P. Breuckmann, Vedika Khemani

Published Dec 17, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2412.13248

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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.
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We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
Sensitivity to noise makes most of the current quantum computing schemes prone to error and nonscalable, allowing only for small proof-of-principle devices. Topologically-protected quantum computing aims at solving this problem by encoding quantum bits and gates in topological properties of the hardware medium that are immune to noise that does not impact the entire system at once. There are different approaches to achieve topological stability or active error correction, ranging from quasiparticle braidings to spin models and topological colour codes. The stability of these proposals against noise can be quantified by their error threshold. This figure of merit can be computed by mapping the problem onto complex statistical-mechanical spin-glass models with local disorder on nontrival lattices that can have many-body interactions and are sometimes described by lattice gauge theories. The error threshold for a given source of error then represents the point in the temperature-disorder phase diagram where a stable symmetry-broken phase vanishes. An overview of the techniques used to estimate the error thresholds is given, as well as a summary of recent results on the stability of different topologically-protected quantum computing schemes to different error sources.