## Description

A family of \([[n,k,d]]\) CSS codes whose 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

If \([n_i, k_i, d_i]\) (\([r_i, k^T_i, d^T_i]\)) are the parameters of the codes \(\mathrm{ker}H_i\) (\(\mathrm{ker}H_i^T\), taking (\(d=\infty\) if \(k=0\)), the hypergraph product has parameters \([[n_1 n_2 + r_1 r_2, k_1 k_2 + k_1^T k_2^T, \min(d_1, d_2, d_1^T, d_2^T)]]\)

## Transversal Gates

Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [3].

## Gates

Code deformation techniques yield Clifford gates [4].

## Decoding

ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [5].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [6].Improved BP-OSD decoder [7].Erasure-correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [8] of the peeling and pruned peeling decoders [9], with a probabilistic version running in \(O(n^{1.5})\) operations.Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [10].Generalization [11] of Viderman's algorithm for expander codes [12].

## Code Capacity Threshold

Some thresholds were determined in Ref. [13].Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [14]. For example, a threshold of \(7\%\) was obtained under independent \(X\) and \(Z\) noise for codes obtained from random \((3,4)\)-regular Gallager codes.

## Threshold

Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits [6]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.

## Parents

- Homological product code — A homological product of chain complexes corresponding to two classical codes is a hypergraph product code [15].
- Lifted-product (LP) code — Lifted-product codes for trivial group \(G\) 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 [16].

## Children

- Quantum expander code
- Kitaev surface code — Planar (toric) code can be obtained from hypergraph product of two repetition (cyclic) codes [2; Ex. 6].

## Cousins

- 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 [17].
- XYZ product code — The XYZ product code is based on a hypergraph product of three classical codes.
- 3D surface code — The 3D surface code is a hypergraph product of three repetition codes [18; Exam. A.1].
- Rotated surface code — Rotated code can be obtained from hypergraph product of two cyclic binary cyclic codes with palindromic generator polynomial ([2], Ex. 7).
- Bacon-Casaccino subsystem code — The Bacon-Casaccino subsystem and hypergraph-product code constructions yield the same dimension and minimum distance, but the former does not yield QLDPC codes; see [1; pg. 18]. Two generalized Shor codes can be gauge-fixed to yield a hypergraph product code [19; Sec. III].
- Generalized bicycle (GB) code — An arbitrary GB code of length \(2\ell\) is equivalent [20] 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

- [1]
- J.-P. Tillich and G. Zemor, “Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength”, IEEE Transactions on Information Theory 60, 1193 (2014) arXiv:0903.0566 DOI
- [2]
- A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0928 DOI
- [3]
- A. O. Quintavalle, P. Webster, and M. Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”, Quantum 7, 1153 (2023) arXiv:2204.10812 DOI
- [4]
- A. Krishna and D. Poulin, “Fault-Tolerant Gates on Hypergraph Product Codes”, Physical Review X 11, (2021) arXiv:1909.07424 DOI
- [5]
- A. O. Quintavalle and E. T. Campbell, “ReShape: a decoder for hypergraph product codes”, (2022) arXiv:2105.02370
- [6]
- N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
- [7]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [8]
- N. Connolly et al., “Fast erasure decoder for a class of quantum LDPC codes”, (2023) arXiv:2208.01002
- [9]
- M. G. Luby et al., “Efficient erasure correcting codes”, IEEE Transactions on Information Theory 47, 569 (2001) DOI
- [10]
- A. G. Manes and J. Claes, “Distance-preserving stabilizer measurements in hypergraph product codes”, (2023) arXiv:2308.15520
- [11]
- A. Krishna, I. L. Navon, and M. Wootters, “Viderman’s algorithm for quantum LDPC codes”, (2023) arXiv:2310.07868
- [12]
- M. Viderman, “Linear-time decoding of regular expander codes”, ACM Transactions on Computation Theory 5, 1 (2013) DOI
- [13]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
- [14]
- 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
- [15]
- 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
- [16]
- S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137
- [17]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
- [18]
- L. Berent et al., “Analog information decoding of bosonic quantum LDPC codes”, (2023) arXiv:2311.01328
- [19]
- M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
- [20]
- 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