# 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 \(n\).

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.

## Rate

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Notes

## Parent

## Children

- Abelian topological code — All abelian topological orders can be realized as geometrically local modular-qudit stabilizer codes [17], 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.
- 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 forms a 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 [18]. 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 [19].
- Hamiltonian-based code — Hamiltonian-based codes are not generally defined using Pauli strings. However, codes forming the ground-state subspace of a local Hamiltonain consisting of commuting terms are QLDPC codes in the sense that they satisfy the QLDPC locality requirements.
- Honeycomb code — The Floquet check operators are weight-two, and each qubit participates in one check each round.

## Zoo code information

## 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]
- Nouédyn Baspin and Anirudh Krishna, “Connectivity constrains quantum codes”. 2106.00765
- [4]
- Nouédyn Baspin and Anirudh Krishna, “Quantifying nonlocality: how outperforming local quantum codes is expensive”. 2109.10982
- [5]
- Z. Babar et al., “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015). DOI
- [6]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021). DOI; 1904.02703
- [7]
- Nithin Raveendran et al., “Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors”. 2205.02341
- [8]
- Julien du Crest, Mehdi Mhalla, and Valentin Savin, “Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes”. 2205.06125
- [9]
- Lawrence Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”. 2110.10794
- [10]
- Daniel Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”. 1310.2984
- [11]
- Omar Fawzi, Antoine Grospellier, and Anthony Leverrier, “Constant overhead quantum fault-tolerance with quantum expander codes”. 1808.03821
- [12]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
- [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). DOI; 1208.2317
- [14]
- Alexey A. Kovalev and Leonid P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”. 1311.7688
- [15]
- I. Dumer, A. A. Kovalev, and L. P. Pryadko, “Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes”, Physical Review Letters 115, (2015). DOI; 1412.6172
- [16]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309
- [17]
- Tyler D. Ellison et al., “Pauli stabilizer models of twisted quantum doubles”. 2112.11394
- [18]
- Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249
- [19]
- M. Ippoliti et al., “Entanglement Phase Transitions in Measurement-Only Dynamics”, Physical Review X 11, (2021). DOI; 2004.09560

## 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.