Alternative names: Sparse stabilizer code.
Description
Member of a family of \([[n,k,d]]_{\Sigma}\) stabilizer codes over alphabet \(\Sigma\) for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\). The code can be denoted by \([[n,k,d,w]]_{\Sigma}\), where possible alphabets include \(\Sigma \in \{\mathbb{Z}_q,\mathbb{F}_q,\mathbb{R},\mathbb{Z}\}\) corresponding to modular qudits, Galois qudits, oscillators, or rotors, respectively. Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\).
A geometrically local stabilizer code is a QLDPC code where the sites involved in any syndrome value are contained in a fixed volume that does not scale with \(n\). As opposed to general stabilizer codes, syndrome extraction of the constant-weight check operators of a QLDPC codes can be done using a constant-depth circuit.
Strictly speaking, the term parity check describes only bitwise qubit error syndromes. Nevertheless, qudit and bosonic stabilizer codes satisfying the above criteria are also called QLDPC codes. This entry includes general code constructions which are intended to yield QLDPC codes, but may include specific instances that are non-QLDPC.
Protection
Detects errors on \(d-1\) sites, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) sites. Code distance may not be a reliable marker of code performance.Decoding
Non-binary decoding algorithm for CSS-type QLDPC codes [1].GD-CSS Decoder for Galois-qudit CSS QLDPC codes [2,3]Notes
Infleqtion QLDPC software library for estimating distance and creating various qubit and Galois-qudit QLDPC CSS codes [4]LDPC Python software library for decoding LDPC and QLDPC codes [5,6].Reviews of QLDPC codes provided in Refs. [1,7,8].Cousins
- \(q\)-ary LDPC code— Galois-qudit QLDPC codes are quantum analogues of \(q\)-ary LDPC codes.
- Topological code— Topological codes are not generally defined using Pauli strings or their qudit and bosonic generalizations. However, for appropriate tesselations, the codespace is the ground-state subspace of a geometrically local Hamiltonian. In this sense, topological codes are QLDPC codes. Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [9–13]. This notion can be extended to semi-hyperbolic manifolds [14] and non-geometrically local QLDPC codes exhibiting check soundness [15] (see also [16]).
- Dynamically generated QECC— QLDPC codes can arise from a dynamical process [17].
- Hamiltonian-based code— QLDPC code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional Hamiltonians with non-commuting terms whose interactions scale with \(n\) [18].
- Quantum locally testable code (QLTC)— Stabilizer LTCs are QLDPC. More general QLTCs are not defined using Pauli strings, but the codespace is the ground-state subspace of a local Hamiltonian. In this sense, QLTCs are QLDPC codes.
- Self-correcting quantum code— Linear confinement of QLDPC (LDPC) codes implies (classical) self-correction [19].
- Sparse subsystem code— Sparse subsystem codes reduce to QLDPC codes when there are no gauge qudits.
- Galois-qudit BCH code— Some Galois-qudit BCH codes are QLDPC [20,21].
- Two-block CSS code— When matrices \(A\) and \(B\) have row and column weights bounded by \(W\), a two-block CSS code is a quantum LDPC code with stabilizer generators bounded by \(2W\).
- Two-block group-algebra (2BGA) codes— Given group algebra elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) (i.e., number of nonzero terms in the expansion), the 2BGA code LP\((a,b)\) has stabilizer generators of uniform weight \(W_a+W_b\).
- Generalized bicycle (GB) code— Stabilizer generators of the code GB\((a,b)\) have weights given by the sum of weights of polynomials \(a(x)\) and \(b(x)\). The GB code ansatz is convenient for designing QLDPC codes and several extensions exist [22].
- Distance-balanced code— Lattice surgery techniques for QLDPC codes [23,24] utilize weight reduction. Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [25].
Primary Hierarchy
Parents
QLDPC codes are QLWC codes for which the number of stabilizer generators that each site participates in is bounded by a constant as \(n\to\infty\).
QLDPC codes are stabilizer QLRCs whose locality \(r \leq w\), the maximum number of subsystems that a stabilizer generator participates in [26].
QLDPC code
Children
Homological products are a primary tool for generating QLDPC codes with favorable parameters. Typically, whenever the input codes are LDPC or QLDPC, the resulting code will be QLDPC with non geometrically local stabilizer generators.
References
- [1]
- Z. Babar, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015) DOI
- [2]
- D. Komoto and K. Kasai, “Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding”, (2025) arXiv:2507.11534
- [3]
- Kasai, Kenta. “GD-CSS Decoder (Quantum Error Correction using Non-binary LDPC over GF(q))” (2025). https://github.com/kasaikenta/gd-css-decoder
- [4]
- Michael A. Perlin. qLDPC. https://github.com/Infleqtion/qLDPC, 2023.
- [5]
- J. Roffe, D. R. White, S. Burton, and E. Campbell, “Decoding across the quantum low-density parity-check code landscape”, Physical Review Research 2, (2020) arXiv:2005.07016 DOI
- [6]
- Roffe, Joschka. “LDPC: Python tools for low density parity check codes.” (2022).
- [7]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
- [8]
- B. Vasic, V. Savin, M. Pacenti, S. Borah, and N. Raveendran, “Quantum Low-Density Parity-Check Codes”, (2025) arXiv:2510.14090
- [9]
- S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
- [10]
- S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
- [11]
- S. Michalakis and J. P. Zwolak, “Stability of Frustration-Free Hamiltonians”, Communications in Mathematical Physics 322, 277 (2013) arXiv:1109.1588 DOI
- [12]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms”, Journal of Mathematical Physics 60, (2019) arXiv:1810.02428 DOI
- [13]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States”, Annales Henri Poincaré 23, 393 (2021) arXiv:2010.15337 DOI
- [14]
- A. Lavasani, M. J. Gullans, V. V. Albert, and M. Barkeshli, “On stability of k-local quantum phases of matter”, (2024) arXiv:2405.19412
- [15]
- C. Yin and A. Lucas, “Low-Density Parity-Check Codes as Stable Phases of Quantum Matter”, PRX Quantum 6, (2025) arXiv:2411.01002 DOI
- [16]
- W. De Roeck, V. Khemani, Y. Li, N. O’Dea, and T. Rakovszky, “Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations”, PRX Quantum 6, (2025) arXiv:2411.02384 DOI
- [17]
- M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, “Entanglement Phase Transitions in Measurement-Only Dynamics”, Physical Review X 11, (2021) arXiv:2004.09560 DOI
- [18]
- H. Apel and N. Baspin, “Simulating LDPC code Hamiltonians on 2D lattices”, (2023) arXiv:2308.13277
- [19]
- Y. Hong, J. Guo, and A. Lucas, “Quantum memory at nonzero temperature in a thermodynamically trivial system”, Nature Communications 16, (2025) arXiv:2403.10599 DOI
- [20]
- S. A. Aly, “Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets”, (2008) arXiv:0802.4079
- [21]
- S. A. Aly, “On Quantum and Classical Error Control Codes: Constructions and Applications”, (2008) arXiv:0812.5104
- [22]
- N. Koukoulekidis, F. Šimkovic, M. Leib, and F. R. F. Pereira, “Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes”, (2024) arXiv:2401.07583
- [23]
- L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
- [24]
- Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [25]
- S. J. S. Tan and L. Stambler, “Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes”, Quantum 9, 1897 (2025) arXiv:2409.02193 DOI
- [26]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
Page edit log
- Victor V. Albert (2026-03-24) — most recent
Cite as:
“QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/general_qldpc