Algebraic Topology Principles behind Topological Quantum Error Correction

TL;DR

This paper develops a comprehensive algebraic-topological framework for topological quantum error correction, extending code constructions to arbitrary dimensions and boundaries, and demonstrating improved performance through topology modifications.

Contribution

It introduces a unified algebraic-topological approach to TQEC, generalizes code constructions to higher dimensions and boundaries, and provides design principles for topology-aware quantum architectures.

Findings

Extended TQEC to arbitrary-dimensional manifolds with boundaries.

Constructed new code families like 3-torus and volume codes.

Showed topology changes can improve logical performance.

Abstract

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 $X$- and $Z$-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 known quantum LDPC and homological-product-like constructions within a single geometric language. Finally, we combine the theoretical framework with numerical simulations to demonstrate that changing only the global topology can yield improved logical performance at fixed entanglement resources. Taken together, our results provide a systematic set of topological design principles for constructing and analyzing TQEC codes across dimensions and boundaries, and they open new avenues for topology-aware fault-tolerant quantum architectures.