Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911111839744000 |
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| author | Chen, Keyang Liu, Yuanting Zhang, Yiming Liang, Zijian Chen, Yu-An Liu, Ke Song, Hao |
| author_facet | Chen, Keyang Liu, Yuanting Zhang, Yiming Liang, Zijian Chen, Yu-An Liu, Ke Song, Hao |
| contents | Quantum low-density parity-check (QLDPC) codes offer a promising path to low-overhead fault-tolerant quantum computation but lack systematic strategies for exploration. In this Letter, 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. In addition to efficient characterizations using Gröbner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein--Khovanskii--Kushnirenko theorem. This approach allows us to analytically determine how the topological order varies with the generic choices 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 intrinsic link to symmetry-enriched topological orders and derive an efficient method for generating finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our Letter 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 algebraic structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes Chen, Keyang Liu, Yuanting Zhang, Yiming Liang, Zijian Chen, Yu-An Liu, Ke Song, Hao Quantum Physics Strongly Correlated Electrons Mathematical Physics Quantum low-density parity-check (QLDPC) codes offer a promising path to low-overhead fault-tolerant quantum computation but lack systematic strategies for exploration. In this Letter, 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. In addition to efficient characterizations using Gröbner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein--Khovanskii--Kushnirenko theorem. This approach allows us to analytically determine how the topological order varies with the generic choices 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 intrinsic link to symmetry-enriched topological orders and derive an efficient method for generating finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our Letter 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 algebraic structures. |
| title | Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes |
| topic | Quantum Physics Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2503.04699 |