Algebraic Topology Principles behind Topological Quantum Error Correction
X. Zou and H. Lo · 2025 · arXiv
| 编号 QE093 | 主题 拓扑码 | 链接 arXiv:2505.06082 |
摘要
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Quantum error correction (QEC) is crucial for realizing scalable quantum technologies, and topological quantum error correction (TQEC) has emerged as the most experimentally advanced paradigm of QEC. Existing homological and topological code constructions, however, are largely confined to orientable two-manifolds with simple boundary conditions. In this work, we develop a unified algebraic-topological framework for TQEC based on homology, cohomology, and intersection theory, which characterizes exactly when an arbitrary-dimensional manifold (with or without boundary) can serve as a quantum memory, thereby extending the standard 2D homological-code picture to arbitrary dimension and to manifolds with boundary via Poincar’e-Lefschetz duality. Building on this classification, we introduce concrete code families that exploit nontrivial topology beyond the planar and toric settings. These include 3-torus code''and higher-dimensional volume codes”on compact manifolds with mixed - and -type boundaries. We further give a topological construction of qudit TQEC codes on general two-dimensional cell complexes using group presentation complexes, which unifies and extends several kn
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