Campbell double homological product code[1]
Description
A multi-dimensional HGP code derived from two applications of the hypergraph product to a classical code, resulting in a length-\(4\) chain complex. The construction method allows for the use of two different classical codes as inputs, with Ref. [1] assuming identical input codes for simplicity.
Explicitly, the resulting chain complex is \begin{align} A_{-2}\xrightarrow{\partial_{-2}}A_{-1}\xrightarrow{\partial_{-1}}A_{0}\xrightarrow{\partial_{0}}A_{1}\xrightarrow{\partial_{1}}A_{2}\,. \tag*{(1)}\end{align} The boundary maps \(\partial_j\) are constructed using tensor products of the original boundary maps, ensuring the chain condition \(\partial_{j+1} \partial_j = 0\). The additional parts of the chain complex yields metachecks to detect measurement errors, enabling single-shot error correction.
Protection
Given a classical \([n, k, d]\) code, the double homological product yields a quantum code with parameters \([[n^4 + 4n^2(n-k)^2 + (n-k)^4, k^4, \geq d]]\).
The Campbell double homological product code is a single-shot code. It is \((d, f)\)-sound with \(f(x) = x^3/4\), meaning that small syndromes can be corrected by small errors. The check redundancy is tightly bounded by one, and the construction preserves LDPC properties if the original code is LDPC.
Decoding
The minimum-weight decoder optimizes the recovery operation \( E_{\text{rec}} \) to minimize the residual error \( E_{\text{rec}} \cdot E \) given a noisy syndrome \( s = \sigma(E) + u \). The decoder’s performance is intrinsically tied to the code’s soundness: when the code is \((t, f)\)-sound, the minimum-weight decoder guarantees that the residual error’s min-weight scales as \( f(2|u|) \) for measurement errors \( |u| < t/2 \) [1]. This property is particularly robust in double homological codes, where soundness follows a cubic scaling (\( f(x) \sim x^3 \)).A meta-check-based decoder operates through a two-stage process: first, it identifies a minimal correction \( s_{\text{rec}} \) to the syndrome \( s \) such that the repaired syndrome \( s + s_{\text{rec}} \) satisfies all metachecks (\( H(s + s_{\text{rec}}) = 0 \)). Second, it computes a minimal-weight physical error \( E_{\text{rec}} \) consistent with the repaired syndrome. This approach uniquely tolerates up to \( \lfloor (d_{\textit{ss}} - 1)/2 \rfloor \) measurement errors in a single round (where \(d_{\textit{ss}}\) is the single-shot distance), eliminating the need for repeated syndrome measurements.Cousin
- \((2,2)\) Loop toric code— The 4D loop planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [2]. As such, it is a particular Campbell double homological product code [1; table I].
Primary Hierarchy
References
- [1]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [2]
- W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
Page edit log
- Feroz Ahmed Mian (2025-07-28) — most recent
Cite as:
“Campbell double homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/double_homological_product