Related papers: Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes

Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes

URL: http://arxiv.org/abs/2503.04699v1
Date: Thu, 06 Mar 2025 18:46:14 GMT
Title: Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes
Authors: Keyang Chen, Yuanting Liu, Yiming Zhang, Zijian Liang, Yu-An Chen, Ke Liu, Hao Song,
Abstract summary: Quantum low-density parity-check (QLDPC) codes present a promising route to low-overhead fault-tolerant quantum computation.<n>We establish a topological framework for studying the bivariate-bicycle codes, a prominent class of QLDPC codes tailored for real-world quantum hardware.<n>Novel phenomena are unveiled, including topological frustration, where ground-state degeneracy on a torus deviates from the total anyon number.
Score: 12.383649662360302
License: http://creativecommons.org/licenses/by/4.0/
Abstract: Quantum low-density parity-check (QLDPC) codes present a promising route to low-overhead fault-tolerant quantum computation, yet systematic strategies for their exploration remain underdeveloped. In this work, we establish a topological framework for studying the bivariate-bicycle codes, a prominent class of QLDPC codes tailored for real-world quantum hardware. Our framework enables the investigation of these codes through universal properties of topological orders. Besides providing efficient characterizations for demonstrations using Gr\"obner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein--Khovanskii--Kushnirenko theorem, allowing us to analytically determine how the topological order varies with the generic choice of bivariate-bicycle codes under toric layouts. Novel phenomena are unveiled, including topological frustration, where ground-state degeneracy on a torus deviates from the total anyon number, and quasi-fractonic mobility, where anyon movement violates energy conservation. We demonstrate their inherent link to symmetry-enriched topological orders and offer an efficient method for searching for finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our work provides a rigorous theoretical basis for exploring the fault tolerance of QLDPC codes and deepens the interplay among topological order, quantum error correction, and advanced mathematical structures.

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