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 bounded (\(\breve{\upsilon} < 2\)), 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 \( \lfloor (d_{ss} - 1)/2 \rfloor \) measurement errors in a single round, 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