Higher-dimensional quantum hypergraph-product codes

Abstract

We describe a family of quantum error-correcting codes which generalize both the quantum hypergraph-product (QHP) codes by Tillich and Zémor, and all families of toric codes on $m$-dimensional hypercubic lattices. Similar to the latter, our codes form $m$-complexes ${\mathcal{K}}_m$, with $m\ge 2$. These are defined recursively, with ${\mathcal{K}}_m$ obtained as a tensor product of a complex ${\mathcal{K}}_{m-1}$ with a $1$-complex parameterized by a binary matrix. Parameters of the constructed codes are given explicitly in terms of those of binary codes associated with the matrices used in the construction.

arXiv comment6 pages, no figures. In version 2, a hole in the proof and several typos are corrected

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• Improved single-shot decoding of higher dimensional hypergraph product codes
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• Connectivity constrains quantum codes
• Cellular automaton decoders for topological quantum codes with noisy measurements and beyond
• Quantum LDPC Codes for Modular Architectures
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