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


Also called a sparse quantum code. Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes 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 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.






Kitaev toric


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




hypergraph product

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

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

high-dimensional expander (HDX)


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

tensor-product HDX




\(\log n\)

\(n/\log n\)

lifted-product (LP)



expander LP



quantum Tanner




Table I: Notable QLDPC codes; \(c\) is a positive integer.

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.


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. 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.


Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction.

Geometrically local qubit codes are limited by the Bravyi-Poulin-Terhal (BPT) bound [2] (see also [35]), which states that \(d \leq O(n^{1-1/D})\) and \(k d^{2/D-1} = O(n)\) for \(D\)-dimensional lattice geometries. Codes on a \(D\)-dimensional Riemannian manifold with diameter \(L\) satisfy \(k = O(L^{D-2})\) [6].

For general graphs, distance is limited by graphs' connectivity, and a constant relative minimum distance can be achieved only for graphs that contain expanders [7]. Conversely, a code with parameters \(k\) and \(d\) requires a graph with \(\Omega(d)\) edges of length \(\Omega(d/n^{1/D})\) [8].


Logical gates implemented via constant-depth quantum circuits of \(D\)-dimensional geometrically qubit stabilizer codes whose distance increases with \(n\) lie in the \(D\)th level of the Clifford hierarchy [9].


Belief-propagation (BP) decoder [10] and neural BP decoder [11] for qubit codes. Partially and fully decoupled BP decoders, which uses the decoupling representation, yield improvements against depolarizing noise [12].Non-binary decoding algorithm for CSS-type QLDPC codes [13].BP-OSD decoder adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [14].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [15].Soft (i.e., analog) syndrome iterative belief propagation for CSS-type QLDPC codes, utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [16].Message-passing decoder utilizing stabilizer inactivation (MP-SI) for CSS-type QLDPC qubit codes [17].Extension of the union-find decoder for qubit QLDPC codes, as well as a related heuristic decoder [18].

Fault Tolerance

Lattice surgery techniques with ancilla qubits [19].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [20], e.g., quantum expander codes [21].GHz state distillation for Steane error correction [22].

Code Capacity Threshold

Bounds on code capacity thresholds using maximum-likelihood (ML) decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [2325].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [26].


QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [20].


Links to code tables of notable QLDPC codes [27].Reviews of QLDPC codes provided in Refs. [13,27].




  • 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 Hamiltonian. 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 [28]. 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 QECC — QLDPC codes can arise from a dynamical process [29].
  • 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.
  • Honeycomb Floquet code — The Floquet check operators are weight-two, and each qubit participates in one check each round.
  • Generalized bicycle (GB) code — A code GB\((a,b)\) is given by the sum of weights of polynomials \(a(x)\) and \(b(x)\). The GB code ansatz is convenient for designing quantum LDPC codes.


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“Quantum low-density parity-check (QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qldpc
  title={Quantum low-density parity-check (QLDPC) code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Quantum low-density parity-check (QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qldpc

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