Alternative names: Sparse quantum code.
Description
Member of a family of \([[n,k,d]]\) 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 \(w\) as \(n\to\infty\). The code can be denoted by \([[n,k,d,w]]\) or, for \(q\)-dimensional Galois-qudit QLDPC codes, \([[n,k,d]]_{(w,q)}\). 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 \(\leq w\). For finite size codes, the notation QLDPC codes 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.
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.
Notable qubit 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 |
|---|---|---|
\(2\) | \(\sqrt{n}\) | |
\(2\) | \(\sqrt{n\sqrt{\log n}}\) | |
\(\Theta(n)\) | \(\sqrt{n}\) | |
\(\sqrt{n}/\log n\) | \(\sqrt{n} \log n\) | |
\(\sqrt{n}\) | \(\sqrt{n} \log^c n\) | |
\(n^{3/5}/\text{polylog}(n)\) | \(n^{3/5}/\text{polylog}(n)\) | |
\(\log n\) | \(n/\log n\) | |
\(\Theta(n)\) | \(\Theta(n)\) | |
\(\Theta(n)\) | \(\Theta(n)\) | |
\(\Theta(n)\) | \(\Theta(n)\) |
Strictly speaking, the term parity check describes only bitwise qubit error syndromes. Nevertheless, qudit and bosonic 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. Code distance may not be a reliable marker of code performance.
Since QLDPC codes are stabilizer QLRCs whose locality \(r = w\), their relative distance is bounded by [1; Thm. 4] \begin{align} \delta = \frac{d}{n} \leq \frac{1}{2} - \Omega\left(\frac{1}{r}\right)~. \tag*{(1)}\end{align}
Rate
Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction. Bounds generalizing the BPT bound to qubit QLDPC codes depend on the separation profile of the code's underlying connectivity graph [2,3]. A constant relative minimum distance can be achieved only for graphs that contain expanders [2]. Conversely, a code with parameters \(k\) and \(d\) requires a graph with order \(\Omega(d)\) edges of length of order \(\Omega(d/n^{1/D})\) [4]. Random QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [5].
Qubit QLDPC codes cannot attain the capacity of the erasure channel [6], but this capacity can be attained by code families with weight \(w = O(\text{polylog}n)\) [7]. There are bounds on their performance against erasure noise [6].
Encoding
Fault-tolerant encoders utilizing pre-shared entanglement for qubit QLDPC codes [8].Fault-tolerant constant-depth encoder and unencoder for qubit QLDPC codes [9].Gates
Fault-tolerant logical measurements [10] that generalize a previous construction [11] and that require an order \(O(d/\beta)\) ancilla qubits, where \(\beta\) is the Cheeger constant of the Tanner subgraph supporting the logical operator to be measured. This can be used for a generalization of lattice surgery for CSS QLDPC codes [12].Repetition-code adapter for logical Pauli measurements and logical CNOT gates via Dehn twists [13].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [14].Decoding
Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [15].Quantum belief propagation (BP) decoder [16–18] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [19]. Various post-processing algorithms have been proposed (see below and also Refs. [20,21]).BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder [22]. For an open-source implementation, see LDPC Python software library [23,24].Neural BP decoder [25,26] and GNN decoders [27,28] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [29].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [30].BP localized statistics decoding (BP-LSD) that exploits error clustering [31].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [32].BP guided decimation (BPGD) decoder [33].SymBreak decoder, which adaptively modifies the decoding graph to break the degeneracy of the BP decoder [34].Ambiguity clustering (AC) decoder, in which measurement data is divided into clusters and decoded independently [35].Non-binary decoding algorithm for CSS-type QLDPC codes [36].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [37]. For CSS codes, syndrome extraction can be implemented in constant depth [38].Soft (i.e., analog) syndrome iterative BP 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) [39].The MWPM decoder for surface codes may be generalizable to QLDPC codes [40].Extensions of the union-find decoder for qubit QLDPC codes [41,42].Sliding-window decoding [43].Closed-branch decoder [44].BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes [45].Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code [46].Fault-tolerant constant-depth encoder and unencoder [9].BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder [47].Cluster decoder [48].BP approximate degenerate OSD (BP+ADOSD) decoder [49].Decision tree decoders (DTDs), one that provably finds the minimum-weight correction, and one that is heuristic [50].AutDEC decoder for codes with large automorphism groups [51].Tesseract ML decoder [52].Fault Tolerance
Lattice surgery techniques with ancilla qubits [11,53,54]. In one such technique, one first performs a logical measurement by code switching into a code whose stabilizer group includes the original stabilizers together with the logical Paulis that are to be measured. Then, one can reduce the weight of the output code using weight reduction.Fault-tolerance with constant overhead can be performed on certain QLDPC codes [46], e.g., quantum expander codes [55].GHZ state distillation for Steane error correction [56].Fault-tolerant logical measurements [10] that generalize a previous construction [11] and that require an order \(O(d/\beta)\) ancilla qubits, where \(\beta\) is the Cheeger constant of the Tanner subgraph supporting the logical operator to be measured. This can be used for a generalization of lattice surgery for CSS QLDPC codes [12].Fault-tolerant constant-depth encoder and unencoder [9].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [14].Code Capacity Threshold
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 [57–59].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [60].Threshold
QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [46].Notes
Infleqtion QLDPC software library for estimating distance and creating various qubit and Galois-qudit QLDPC CSS codes [61]Links to code tables of notable QLDPC codes [62].Collection of QLDPC qubit codes based on hyperbolic tilings in the QEC-Pages software library [63].Reviews of QLDPC codes provided in Refs. [36,62].There exist distance-dependent [64; Thm. 1] and rate-dependent [64; Thm. 3ii] lower bounds on the degree of entanglement of a qubit QLDPC code.LDPC Python software library for decoding LDPC and QLDPC codes [23,24].High-rate QLDPC codes can be used for Bell-pair distillation [65].Qldpc code circUIT Simulator (QUITS) Python software library for simulating QLDPC code circuits [66,67].Cousins
- Low-density parity-check (LDPC) code
- 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 [68–72]. This notion can be extended to semi-hyperbolic manifolds [73] and non-geometrically local QLDPC codes exhibiting check soundness [74] (see also [75]).
- Dynamically generated QECC— QLDPC codes can arise from a dynamical process [76].
- 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 \(n\) [77].
- Commuting-projector Hamiltonian code— Qubit QLDPC codes with check soundness, meaning that every weight-\(m\) stabilizer can be written as a product of order \(O(m)\) stabilizer generators, are robust against few-body perturbations. This means that phases of matter can be defined from certain non-geometrically local QLDPC code Hamiltonians [74].
- Single-shot code— Qubit QLDPC codes satisfying linear confinement are single shot [78]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [53].
- Low-density generator-matrix (LDGM) code— LDGM codes can yield CSS [79–82] and non-CSS [83,84] qubit QLDPC codes. Some of the LDGM-based CSS codes have \(n\)-independent minimum distance and no code capacity threshold [85; Sec. 4.2].
- Random stabilizer code— Random qubit QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [5].
- Algebraic LDPC code— Algebraic LDPC codes made from Latin squares can be used to make QLDPC codes [86; Ch. 15].
- 2D lattice stabilizer code— Chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [87].
- Lattice stabilizer code— Chain complexes describing some QLDPC codes [88,89], and, more generally, CSS codes [90] can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality. In addition, chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [87].
- 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 [91].
- Sparse subsystem code— Sparse subsystem codes reduce to QLDPC codes when there are no gauge qudits.
- Honeycomb Floquet code— The Floquet check operators are weight-two, and each qubit participates in one check each round.
- Spacetime circuit code— General spacetime circuit codes can be sparsified to yield QLDPC spacetime circuit codes [92].
- EA QLDPC code
- Concatenated Steane code— The combination of the concatenated Steane code and QLDPC codes with non-vanishing rate yield fault-tolerant quantum computation with constant space and polylogarithmic time overheads, even when classical computation time is taken into account [93].
- Bicycle code— Bicycle codes are the first QLDPC codes [94].
- \(D\)-dimensional twisted toric code— It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [95].
- Galois-qudit BCH code— Some Galois-qudit BCH codes are QLDPC [86,96].
- 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 [97].
- Distance-balanced code— Lattice surgery techniques for QLDPC codes [11,53] utilize weight reduction. Single-ancilla syndrome extraction circuits that, for the most part, preserve the effective distance of weight-reduced qLDPC codes [98].
Member of code lists
- Dynamically generated quantum codes
- Hamiltonian-based codes
- Quantum codes
- Quantum codes with a rate
- Quantum codes with code capacity thresholds
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum LDPC codes
- Quantum locally testable codes and friends
- Self-correcting quantum codes and friends
- Single-shot codes
- Stabilizer codes
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 [1].
Quantum LDPC (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.
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Page edit log
- Austin Yubo He (2025-04-01) — most recent
- Victor V. Albert (2022-10-02)
- 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)
Cite as:
“Quantum LDPC (QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/qldpc