## Description

A member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [3].

The \([[n,k,d]]\) CSS code's construction is based on two binary linear seed codes, \(i\in\{1,2\}\) \(C_i\) with parameters \([n_i, k_i, d_i]\) defined as the kernel of \(r_i \times n_i\) check matrices \(H_i\) of rank \(n_i - k_i\). The hypergraph product yields two classical codes \(C_{X,Z}\) with parity-check matrices \begin{align} H_{X}&=\begin{pmatrix}H_{1}\otimes I_{n_{2}} & \,\,I_{r_{1}}\otimes H_{2}^{T}\end{pmatrix}\tag*{(1)}\\ H_{Z}&=\begin{pmatrix}I_{n_{1}}\otimes H_{2} & \,\,H_{1}^{T}\otimes I_{r_{2}}\end{pmatrix}~, \tag*{(2)}\end{align} where \(I_m\) is the \(m\)-dimensional identity matrix. These two codes then yield a hypergraph product code via the CSS construction.

## Protection

## Transversal Gates

## Gates

## Decoding

## Code Capacity Threshold

## Threshold

## Parents

- Homological product code — A homological product of chain complexes corresponding to two classical codes is a hypergraph product code [16].
- Lifted-product (LP) code — Lifted-product codes for trivial lift are hypergraph-product codes.
- Quantum spatially coupled (SC-QLDPC) code — Hypergraph-product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [17].
- Locally testable code (LTC) — Applying the hypergraph product of an LTC yields a code which provides an explicit example of No Low-Error Trivial States (NLETS) [18].

## Children

- Sierpinsky fractal spin-liquid (SFSL) code — The Sierpinsky fractal spin-liquid code is the hypergraph product of the repetition code and the Newman-Moore code [19,20].
- La-cross code — La-cross codes are constructed using the hypergraph product a cyclic LDPC code with itself.
- Long-range enhanced surface code (LRESC) — LRESCs are constructed using the hypergraph product a concatenated LDPC-repetition code with itself.
- Quantum expander code
- Toric code — The toric code can be obtained from a hypergraph product of two repetition codes [2; Ex. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [2; Exam. 5].

## Cousins

- XYZ product code — Hypergraph (XYZ) product codes are constructed out of hypergraph products of two (three) classical linear codes.
- Linear binary code — Hypergraph product codes are constructed out of two classical linear binary codes.
- Single-shot code — Two-fold application of the hypergraph product to a pair of binary linear codes yields single-shot QLDPC codes that exploit redundancy in their stabilizer generators [21].
- High-dimensional expander (HDX) code — Ramanujan codes utilize the hypergraph product with a twist, which is an automorphism on one of the complexes in the tensor product, in order to increase distance [22].
- Rotated surface code — Rotated code can be obtained from hypergraph product of two cyclic binary cyclic codes with palindromic generator polynomial ([2], Ex. 7).
- Fractal surface code — The fractal product code is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [23].
- Subsystem homological product code — SP codes are projected higher-dimensional HGP codes [24].
- Subsystem hypergraph product (SHP) code — Two SHP codes can be gauge-fixed to yield an HGP code [25; Sec. III]. The SHP and HGP code constructions yield the same dimension and minimum distance, but the former does not yield QLDPC codes; see [1; pg. 18].
- Generalized bicycle (GB) code — An arbitrary qubit GB code of length \(2\ell\) is equivalent [26] to a rotated quantum hypergraph-product code with periodicity vectors \(\vec{L}_{1}\) and \(\vec{L}_{2}\) such that \(\lvert{\vec{L}_{1}\times\vec{L}_{2}\rvert=\ell}\).

## References

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- [2]
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- [3]
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- N. Connolly et al., “Fast erasure decoder for a class of quantum LDPC codes”, (2023) arXiv:2208.01002
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- [15]
- A. A. Kovalev et al., “Numerical and analytical bounds on threshold error rates for hypergraph-product codes”, Physical Review A 97, (2018) arXiv:1804.01950 DOI
- [16]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber Bundle Codes: Breaking the \(N^{1/2} \operatorname{polylog}(N)\) Barrier for Quantum LDPC Codes”, (2020) arXiv:2009.03921
- [17]
- S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137
- [18]
- L. Eldar and A. W. Harrow, “Local Hamiltonians Whose Ground States Are Hard to Approximate”, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (2017) arXiv:1510.02082 DOI
- [19]
- Y. Tan et al., “Fracton models from product codes”, (2024) arXiv:2312.08462
- [20]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [21]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [22]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
- [23]
- C. G. Brell, “A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)”, New Journal of Physics 18, 013050 (2016) arXiv:1411.7046 DOI
- [24]
- W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
- [25]
- M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
- [26]
- R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216

## Page edit log

- Christopher A. Pattison (2023-10-25) — most recent
- Victor V. Albert (2022-08-02)
- Victor V. Albert (2022-01-20)
- Joschka Roffe (2021-11-04)

## Cite as:

“Hypergraph product (HGP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hypergraph_product