Alternative names: Quantum hypergraph (QHG) code, Tillich-Zemor product code.
Description
A member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear binary codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [3].
More technically, the \(x\)- and \(Z\)-type stabilizer generator matrices of a hypergraph product code are, respectively, the boundary and coboundary operators of the 2-complex obtained from the tensor product of a chain complex and cochain complex corresponding to two classical linear binary seed codes. Let the two seed codes be \(C_i\) for \(i\in\{1,2\}\) 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.
In general, the stabilizer generator matrices of an \(m\)-dimensional hypergraph product code are the boundary and co-boundary operators of a 2-dimensional chain complex contained within an \(m\)-complex that is recursively constructed by taking the tensor product of an \((m-1)\)-complex and a 1-complex, with the 1-complex corresponding to some classical linear binary code.
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 \([[n_1 n_2 + r_1 r_2, k_1 k_2 + k_1^T k_2^T, \min(d_1, d_2, d_1^T, d_2^T)]]\)Encoding
Fault-tolerant state preparation via dimension jump [4].Transversal Gates
Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [5].Patch-transversal gates inherited from the automorphism group of the underlying classical codes [6; Appx. D].Gates
Code deformation techniques yield Clifford gates [7].Targeted logical gates [8].Logical gates via Dehn twists for hypergraph products of cyclic codes [9].Decoding
Single-ancilla syndrome extraction circuits do not admit hook errors [10].ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction [11].2D geometrically local syndrome extraction circuits with depth order \(O(\sqrt{n})\) using order \(O(n)\) ancilla qubits [12].Improved BP-OSD decoder [13].Erasure correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations [14] of the peeling and pruned peeling decoders [15], with a probabilistic version running in \(O(n^{1.5})\) operations. Other nearly optimal erasure decoders exist [16,17].Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [18].Generalization [19] of Viderman's algorithm for expander codes [20].Fault Tolerance
Single-ancilla syndrome extraction circuits do not admit hook errors [10].Code Capacity Threshold
Some thresholds were determined in Ref. [21].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 [22]. 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 [12] and DiVincenzo-Aliferis syndrome extraction circuits [23] combined with non-local gates [24]. No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.Cousins
- Locally testable code (LTC)— Applying the hypergraph product to an LTC yields a code which provides an explicit example of No Low-Error Trivial States (NLETS) [25].
- 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 linear binary codes.
- Neural network quantum code— Hypergraph product codes have been optimized against the erasure channel using reinforcement learning [26].
- Tiger surface code— The tiger surface code is constructed from a hypergraph product of two repetition codes over the integers.
- 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 [27].
- Self-correcting quantum code— There are bounds on the energy barrier of hypergraph product codes [28].
- Quantum locally recoverable code (QLRC)— A variant of the hypergraph product can be used to define QLRCs with intersecting recovery sets [29].
- Quantum rainbow code— Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy. In particular, utilizing this construction for quasi-hyperbolic color codes yields an \([[n,O(n),O(\log n)]]\) triorthogonal code family with magic-state yield parameter \(\gamma\to 0\) [30].
- 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 [31].
- Rotated surface code— The rotated code can be obtained from hypergraph product of two cyclic linear binary codes with palindromic generator polynomial [2; Exam. 7].
- Fractal surface code— The fractal product code is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [32].
- Subsystem homological product code— SP codes are projected higher-dimensional HGP codes [33].
- Subsystem hypergraph product (SHP) code— Two SHP codes can be gauge-fixed to yield an HGP code [34; 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 [35] to a rotated quantum hypergraph-product code with periodicity vectors \(\vec{L}_{1}\) and \(\vec{L}_{2}\) such that \(\lvert{\vec{L}_{1}\times\vec{L}_{2}\rvert=\ell}\).
Member of code lists
- Hamiltonian-based codes
- Locally testable codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with code capacity thresholds
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum codes with transversal gates
- Quantum CSS codes
- Quantum LDPC codes
- Self-correcting quantum codes and friends
- Single-shot codes
- Stabilizer codes
Primary Hierarchy
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Fiber-bundle codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
A homological product of chain complexes corresponding to two linear binary codes is a hypergraph product code [36].
Quantum spatially coupled (SC-QLDPC) codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum
Hypergraph-product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [37].
Galois-qudit HGP codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product codes are Galois-qudit hypergraph-product codes for qudit dimension \(q=2\).
Hypergraph product (HGP) code
Children
The two-foliated fracton code is a hypergraph product of the repetition code and the plaquette Ising code on a square lattice with periodic boundary conditions [40].
La-cross codes are constructed using a hypergraph product a cyclic LDPC code with itself.
LRESCs are constructed using a hypergraph product a concatenated LDPC-repetition code with itself.
The toric code can be obtained from a hypergraph product of two repetition codes [2; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [2; Exam. 5].
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 348 (2012) arXiv:1202.0928 DOI
- [3]
- W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
- [4]
- Y. Hong, “Single-shot preparation of hypergraph product codes via dimension jump”, (2024) arXiv:2410.05171
- [5]
- 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
- [6]
- Y. Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, “Long-range-enhanced surface codes”, Physical Review A 110, (2024) arXiv:2309.11719 DOI
- [7]
- A. Krishna and D. Poulin, “Fault-Tolerant Gates on Hypergraph Product Codes”, Physical Review X 11, (2021) arXiv:1909.07424 DOI
- [8]
- A. Patra and A. Barg, “Targeted Clifford logical gates for hypergraph product codes”, (2024) arXiv:2411.17050
- [9]
- R. Tiew and N. P. Breuckmann, “Low-Overhead Entangling Gates from Generalised Dehn Twists”, (2024) arXiv:2411.03302
- [10]
- S. J. S. Tan and L. Stambler, “Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes”, (2024) arXiv:2409.02193
- [11]
- A. O. Quintavalle and E. T. Campbell, “ReShape: a decoder for hypergraph product codes”, (2022) arXiv:2105.02370
- [12]
- 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
- [13]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [14]
- N. Connolly, V. Londe, A. Leverrier, and N. Delfosse, “Fast erasure decoder for hypergraph product codes”, Quantum 8, 1450 (2024) arXiv:2208.01002 DOI
- [15]
- M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Efficient erasure correcting codes”, IEEE Transactions on Information Theory 47, 569 (2001) DOI
- [16]
- M. Gökduman, H. Yao, and H. D. Pfister, “Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation”, 2024 60th Annual Allerton Conference on Communication, Control, and Computing 1 (2024) arXiv:2411.08177 DOI
- [17]
- H. Yao, M. Gökduman, and H. D. Pfister, “Cluster Decomposition for Improved Erasure Decoding of Quantum LDPC Codes”, (2024) arXiv:2412.08817
- [18]
- A. G. Manes and J. Claes, “Distance-preserving stabilizer measurements in hypergraph product codes”, (2023) arXiv:2308.15520
- [19]
- A. Krishna, I. L. Navon, and M. Wootters, “Viderman’s algorithm for quantum LDPC codes”, (2023) arXiv:2310.07868
- [20]
- M. Viderman, “Linear-time decoding of regular expander codes”, ACM Transactions on Computation Theory 5, 1 (2013) DOI
- [21]
- 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
- [22]
- A. A. Kovalev, S. Prabhakar, I. Dumer, and L. P. Pryadko, “Numerical and analytical bounds on threshold error rates for hypergraph-product codes”, Physical Review A 97, (2018) arXiv:1804.01950 DOI
- [23]
- D. P. DiVincenzo and P. Aliferis, “Effective Fault-Tolerant Quantum Computation with Slow Measurements”, Physical Review Letters 98, (2007) arXiv:quant-ph/0607047 DOI
- [24]
- O. Chandra, G. Muraleedharan, and G. K. Brennen, “Non-local resources for error correction in quantum LDPC codes”, (2024) arXiv:2409.05818
- [25]
- 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) 427 (2017) arXiv:1510.02082 DOI
- [26]
- B. C. A. Freire, N. Delfosse, and A. Leverrier, “Optimizing hypergraph product codes with random walks, simulated annealing and reinforcement learning”, (2025) arXiv:2501.09622
- [27]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [28]
- G. Zhao, A. C. Doherty, and I. H. Kim, “On the energy barrier of hypergraph product codes”, (2024) arXiv:2407.20526
- [29]
- K. Bu, W. Gu, and X. Li, “Quantum locally recoverable code with intersecting recovery sets”, (2025) arXiv:2501.10354
- [30]
- T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
- [31]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
- [32]
- 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
- [33]
- 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
- [34]
- M. Li and T. J. Yoder, “A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes”, (2020) arXiv:2002.06257
- [35]
- R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216
- [36]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing 1276 (2021) arXiv:2009.03921 DOI
- [37]
- S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137
- [38]
- Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, (2024) arXiv:2312.08462
- [39]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [40]
- N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2024) arXiv:2410.16250
Page edit log
- Victor V. Albert (2024-08-19) — most recent
- Shi Jie Samuel Tan (2024-08-19)
- Christopher A. Pattison (2023-10-25)
- 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.), 2024. https://errorcorrectionzoo.org/c/hypergraph_product