# Quantum low-density parity-check (QLDPC) code[1]

## Description

Family of \([[n,k,d]]\) stabilizer codes for which the number of sites (either qubit or qudit) participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). A geometrically local stabilizer code is a QLDPC code where the sites involved in any syndrome bit 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.

Notable \([[n,k,d]]\) QLDPC codes are summarized in Table I, demonstrating the steady improvement in code parameters that culminated in the first asymptotically good QLDPC codes.

\(k\) | \(d\) | Code |
---|---|---|

\(2\) | \(\sqrt{n}\) | |

\(2\) | \(\sqrt{n\sqrt{\log n}}\) | |

\(n\) | \(\sqrt{n}\) | |

\(n\) | \(\sqrt{n} \log n\) | |

\(\sqrt{n}\) | \(\sqrt{n} \log^c n\) | |

\(n^{3/5}/\text{polylog}(n)\) | \(n^{3/5}/\text{polylog}(n)\) | |

\(\log n\) | \(n/\log n\) | |

\(\Theta(n)\) | \(\Theta(n)\) | |

\(\Theta(n)\) | \(\Theta(n)\) | |

\(\Theta(n)\) | \(\Theta(n)\) |

Strictly speaking, the term parity check describes only bitwise qubit error syndromes. Nevertheless, qudit stabilizer codes satisfying the above criteria are also called QLDPC codes.

## Protection

Detects errors on \(d-1\) sites, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) sites. Asymptotic scaling of \(k\) and \(d\) with \(n\) is often of interest, and this depends heavily on the code construction. Geometrically local qubit codes are limited by the BPT bound [2], which states that \(d=O(n^{1-1/D})\) for \(D\)-dimensional lattice geometries. For general graphs, distance is limited by graphs' connectivity, and a constant relative minimum distance can be achieved only for graphs that contain expanders [3]. Conversely, a code with parameters \(k\) and \(d\) requires a graph with \(\Omega(d)\) edges of length \(\Omega(d/n^{1/D})\) [4].

Code distance may not be a reliable marker of code performance. QLDPC codes with generator weights bounded by some constant can correct many stochastic errors far beyond the distance, which may not scale as favorably. Together with more accurate, faster, and easier-to-parallelize measurements than those of general stabilizer codes, this property makes QLDPC codes interesting in practice.

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Notes

## Parents

- Stabilizer code
- Hamiltonian-based code — QLDPC codespaces are ground-state subspaces of a local Hamiltonain consisting of commuting terms.

## Children

- Abelian topological code — All abelian topological orders can be realized as geometrically local modular-qudit stabilizer codes [19], and topological-code Hamiltonians are geometrically local for appropriate tesselations.
- Fracton code — Fracton codes admit geometrically local stabilizer generators on a cubic lattice.
- Generalized homological product code — 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.
- Good QLDPC code
- Translationally-invariant stabilizer code — Translationally-invariant stabilizer codes are geometrically local.

## Cousins

- Low-density parity-check (LDPC) code
- Topological code — Topological codes are not generally defined using Pauli strings. However, for appropriate tesselations, the codespace is the ground-state subspace of a geometrically local Hamiltonain. In this sense, topological codes are QLDPC codes. On the other hand, chain complexes describing some QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [20]. This opens up the possibility that some QLDPC codes, despite not being geometrically local, can in fact be associated with a geometrically local theory described by a category.
- Dynamically-generated quantum error-correcting code — QLDPC codes can arise from a dynamical process [21].
- Honeycomb code — The Floquet check operators are weight-two, and each qubit participates in one check each round.
- 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 Hamiltonain. In this sense, QLTCs are QLDPC codes.

## References

- [1]
- D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse-Graph Codes for Quantum Error Correction”, IEEE Transactions on Information Theory 50, 2315 (2004). DOI; quant-ph/0304161
- [2]
- S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009). DOI; 0810.1983
- [3]
- N. Baspin and A. Krishna, “Connectivity constrains quantum codes”, Quantum 6, 711 (2022). DOI; 2106.00765
- [4]
- N. Baspin and A. Krishna, “Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive”, Physical Review Letters 129, (2022). DOI; 2109.10982
- [5]
- David Poulin and Yeojin Chung, “On the iterative decoding of sparse quantum codes”. 0801.1241
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- Z. Babar et al., “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015). DOI
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- L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022). DOI; 2110.10794
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- Daniel Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”. 1310.2984
- [13]
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- Alexey A. Kovalev and Leonid P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”. 1311.7688
- [17]
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- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309
- [19]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022). DOI; 2112.11394
- [20]
- Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249
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- M. Ippoliti et al., “Entanglement Phase Transitions in Measurement-Only Dynamics”, Physical Review X 11, (2021). DOI; 2004.09560

## Page edit log

- Victor V. Albert (2022-10-02) — most recent
- Eugene Tang (2022-10-02)
- Victor V. Albert (2022-09-16)
- Victor V. Albert (2022-05-13)
- Victor V. Albert (2022-01-24)
- Xiaozhen Fu (2021-12-12)

## Zoo code information

## Cite as:

“Quantum low-density parity-check (QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qldpc

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qldpc/qldpc.yml.