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Qubit QLDPC code

Alternative Names: Sparse qubit stabilizer code.

Description

Member of a family of \([[n,k,d]]\) qubit 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 \([[n,k,d,w]]\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular qubit QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\).

Qubit 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 qubit stabilizer code is a qubit 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 code can be done using a constant-depth circuit.

The first code proposed as a quantum analogue of classical LDPC codes was a \([[15,4,2]]\) FG-QLDPC code [1][2; Sec. I][3; Sec. III].

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 qubit QLDPC codes are stabilizer QLRCs whose locality \(r \leq w\), their relative distance is bounded by [4; Thm. 35] \begin{align} \delta = \frac{d}{n} \leq \frac{1}{2} - \Omega\left(\frac{1}{r}\right)~.\tag*{$(1)$} \end{align}

Rate

Suppose each qubit takes part in \(\bar{J}\) stabilizer generators on average and each generator has average weight \(\bar{L}\). Counting qubit-generator incidences gives \(n\bar{J}/\bar{L}\) generators, so the rate is the design rate \(1-\bar{J}/\bar{L}\) plus the number of generator dependencies divided by \(n\) [5].

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 [6,7] or the intrinsic locality dimension [8] of the code’s connectivity graph. A constant relative minimum distance can be achieved only for graphs that contain expanders [6]. Conversely, a code with distance \(d\) embedded in \(D\) spatial dimensions requires a graph with order \(\Omega(d)\) edges of length of order \(\Omega(d/(n^{(D-1)/D}\log n))\) [9]. Random qubit QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [10].

Qubit QLDPC codes cannot attain the capacity of the erasure channel [11], but this capacity can be attained by code families with weight \(w = O(polylogn)\) [12]. There are bounds on their performance against erasure noise [11].

Encoding

Fault-tolerant encoders utilizing pre-shared entanglement for qubit QLDPC codes [13].Any logical state of an \(s\)-sparse qubit QLDPC code with \(d > s^4 2^{5h}\) has depth-\(h\) geometric entanglement \(\Omega(d)\), equivalently exponentially small overlap with any depth-\(h\) topologically trivial state, even allowing arbitrary ancillas [14; Thm. 1].For an \(s\)-sparse qubit QLDPC code with parameters \((n,k,d)\) and \(d > s^4 2^{5h}\), any logical state has depth-\(h\) geometric entanglement \(\Omega\!\left(n H^{-1}(k/n)/2^{4h}\right)\); in particular, constant-rate families require \(\Omega(n/2^{4h})\) entanglement [14; Thm. 3(ii)]. Here, \(H^{-1}\) is the inverse of the binary entropy function \(H(p) = -p\log_2 p - (1-p)\log_2(1-p)\).Fault-tolerant state preparation can be done in an overhead that is constant with the number of qubits \(n\) [15].

Transversal and Permutation-Based Gates

There are recipes to determine transversal gates for asymmetric qubit QLDPC codes [16].

Gates

Fault-tolerant logical measurements by gauging logical operators have worst-case qubit overhead \(O(W\log^{2}W)\) for a weight-\(W\) Pauli and improve earlier QLDPC measurement schemes [17]. This can be used for a generalization of lattice surgery for CSS QLDPC codes [18]. There are conditions on when fault-tolerant surgery can be performed with constant-time overhead [19].Parsimonious surgery measures a weight-\(W\) logical Pauli using an ancilla of order \(O(W\log W)\) qubits, yielding a deformed \([[n+O(W\log W),k-1,\Omega(d)]]\) QLDPC code [20; Sec. II.C]. It replaces the decongestion and thickening steps of gauging measurement [17] with a parsimonious cone of the measurement graph [21,22]. Repetition-code adapter for logical Pauli measurements and logical CNOT gates via Dehn twists [23].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [24], with space overhead \(O(n\log n)\) when built from parsimonious cones [20; App. A].Fault-tolerant batched gadgets for CSS QLDPC codes with constant spacetime overhead [25].

Decoding

Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [26].Quantum belief propagation (BP) decoder [27–29] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [30]. Various post-processing algorithms have been proposed (see below and also Refs. [31,32]).BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propagation (BP) decoder [33].Neural network BP decoders [34,35] and GNN decoders [36,37] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [38].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [39].BP localized statistics decoding (BP-LSD) that exploits error clustering [40].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [41].BP guided decimation (BPGD) decoder [42].SymBreak decoder, which adaptively modifies the decoding graph to break the degeneracy of the BP decoder [43].Ambiguity clustering (AC) decoder, in which measurement data is divided into clusters and decoded independently [44].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [45]. For CSS codes, syndrome extraction can be implemented in constant depth [46].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) [47].The MWPM decoder for surface codes may be generalizable to QLDPC codes [48].Extensions of the union-find decoder for qubit QLDPC codes [49–51].Sliding-window decoding [52].Closed-branch decoder [53].BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes [54].Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code [55].BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder [56].Cluster decoder [57].BP approximate degenerate OSD (BP+ADOSD) decoder [58].Decision tree decoders (DTDs), one that provably finds the minimum-weight correction, and one that is heuristic [59].AutDEC decoder for codes with large automorphism groups [60].Tesseract ML decoder [61].Relay-BP decoder [62].HyperBlossom [63].Post-selection strategies for clustering based decoders [64].Decoder switching between soft-output and higher-accuracy decoders [65].Graph augmentation and rewiring for interference (GARI) framework for circuit-level noise [66].

Fault Tolerance

Lattice surgery techniques with ancilla qubits [67–69]. 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 qubit QLDPC codes [55], e.g., quantum expander codes [70].Transversal Clifford circuits on CSS QLDPC codes, with magic-state inputs and correlated decoding, need only a constant number of syndrome extraction rounds per logical operation [71].Error-corrected GHZ state distillation for Steane error correction [72].Fault-tolerant logical measurements by gauging logical operators have worst-case qubit overhead \(O(W\log^{2}W)\) for a weight-\(W\) Pauli and improve earlier QLDPC measurement schemes [17]. This can be used for a generalization of lattice surgery for CSS QLDPC codes [18]. There are conditions on when fault-tolerant surgery can be performed with constant-time overhead [19].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [24].Fault-tolerant batched gadgets for CSS QLDPC codes with constant spacetime overhead [25].High-rate surgery, which yields parallelizable logical Pauli-product measurements [73].Fault-tolerant state preparation can be done in an overhead that is constant with the number of qubits \(n\) [15].Any qubit QLDPC code admits an exact syndrome-extraction circuit of depth \(O(\max\{1,n/\sqrt{N}\})\) on a square grid of \(N\geq n\) sites, for a data placement chosen in advance [74; Thm. 7.2]. The circuit concentrates the parity of each check onto one data qubit with CNOT gates, measures that qubit, and undoes the concentration [74].

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 [75–77]. In particular, any family of qubit QLDPC codes with superlogarithmic distance achieves a threshold [76].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [78].

Threshold

Qubit QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [55].

Notes

Links to code tables of notable QLDPC codes [79].Collection of QLDPC qubit codes based on hyperbolic tilings in the QEC-Pages software library [80].High-rate QLDPC codes can be used for Bell-pair distillation [81].Qldpc code circUIT Simulator (QUITS) Python software library for simulating QLDPC code circuits [82,83].sQetch GPU (CUDA via PyTorch) Python package for CSS-code minimum-distance estimation. It runs a randomized information-set decoder over random low-dimensional sketches of check-matrix null spaces [84,85].See [86] for a pedagogical introduction to QLDPC codes.

Cousins

  • Low-density parity-check (LDPC) code— Qubit QLDPC codes are quantum analogues of binary LDPC codes.
  • 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 [87].
  • Single-shot code— Qubit QLDPC codes satisfying linear confinement are single shot [88]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [68].
  • Low-density generator-matrix (LDGM) code— LDGM codes can yield CSS [89–92] and non-CSS [93,94] qubit QLDPC codes. Some of the LDGM-based CSS codes have \(n\)-independent minimum distance and no code capacity threshold [95; 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 [10].
  • Algebraic LDPC code— Algebraic LDPC codes made from Latin squares can be used to make qubit QLDPC codes [96; Ch. 15].
  • Asymmetric quantum code (AQC)— There are recipes to determine transversal gates for asymmetric qubit QLDPC codes [16].
  • Lattice stabilizer code— Chain complexes describing some QLDPC codes [97,98], and, more generally, CSS codes [99] can be ‘lifted’ into higher-dimensional manifolds admitting some notion of geometric locality.
  • QLDPC subsystem code— Any qubit QLDPC code with stabilizer-generator weights \(w_i\) can be mapped constructively to a sparse subsystem qubit code with the same number of logical qubits and distance, using \(n=O(\sum_i w_i)\) physical qubits and constant-weight gauge generators [100].
  • Honeycomb Floquet code— The Floquet check operators are weight-two, and each qubit participates in one check each round.
  • Dynamical code— Using ZX calculus, an \([[n,k,d]]\) qubit stabilizer code admitting stabilizer generators of weight no more than \(m\) can be Floquetified into an \([[n+\lceil m/2 \rceil+\ell,k,d^{\prime}]]\) dynamical code with single- and two-qubit operations, where \(\ell \leq \log_{2} m\) and \(d^{\prime} \geq d\) [101] (see also Ref. [102]). A more general locality-preserving spacetime concatenation procedure yields a dynamical code out of any qubit stabilizer code by structuring measurement gadgets using low-weight measurements while ensuring the preservation of logical information [103]. In particular, spacetime concatenation reformulates the notion of a dynamical code associated with a stabilizer code in terms of code concatenation for every qubit (spatial concatenation) and measurements between these codes (temporal concatenation), leading to a temporal evolution of the stabilizer state [103]. A matrix rank condition on the bond operators connecting the gadgets, called the Bond-Kernel-Rank Condition, and a strict locality preservation condition (SLPC), along with preservation of the operator algebra of the stabilizer code under the gadget action, preserves fault-tolerance and the spacetime distance of the code [103].
  • Spacetime circuit code— Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit QLDPC codes. Some spacetime circuit codes can be sparsified to yield QLDPC spacetime circuit codes [104].
  • EA QLDPC code— EA QLDPC codes utilize additional ancillary qubits in a pre-shared entangled state, but reduce to qubit QLDPC codes when said qubits are interpreted as noiseless physical qubits.
  • Mapping cone code— Gauging-based logical measurement for QLDPC codes [17] is a regular height-1 cone built from a connected graph [105; Sec. V.A]. The \(Z\)-type logical operators of this cone are those of the input code modulo the measured one [105; Sec. V]. Homological measurement is also formulated using height-1 cones [105,106].
  • Nested HA-MN CSS code— Nested HA-MN CSS codes have sparse extended parity-check matrices with hidden variables, but their stabilizer generator matrices on the visible qubits are generally dense [107; Sec. 2.4].
  • Concatenated Steane code— The combination of the concatenated Steane code and QLDPC codes with non-vanishing rate yields fault-tolerant quantum computation with constant space and polylogarithmic time overheads, even when classical computation time is taken into account [108].
  • Hierarchical code— Hierarchical codes are concatenations of constant-rate QLDPC codes with rotated surface codes, but they are not themselves QLDPC codes. Their stabilizer generators have weight of order \(O(\log n)\) because the surface-code side length grows logarithmically with the block length [109].
  • Symplectic cone code— For a QLDPC input and a bounded-degree expanding measurement graph, the symplectic-cone construction remains QLDPC [110]. It measures a single non-CSS logical operator without first converting it to a CSS representative by a local Clifford circuit [110].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Parents
Qubit stabilizer code
Quantum locally recoverable code (QLRC)QECC Quantum
Qubit QLDPC codes are stabilizer QLRCs whose locality \(r \leq w\), where \(w\) is the maximum stabilizer-generator weight [4].
QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Qubit QLDPC code
Children
Crystalline-circuit qubit code
Majorana color code
The Majorana color code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
Majorana surface code
The Majorana surface code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
Majorana checkerboard code
The Majorana checkerboard code is a 3D qubit stabilizer code with respect to the Majorana operator basis.
Majorana box qubit
The Majorana box qubit is a 1D qubit stabilizer code with respect to the Majorana operator basis.
Quantum repetition code
The codespace of the quantum repetition code is the ground-state space of a frustration-free 1D classical Ising model with nearest-neighbor interactions.
Transverse-field Ising model (TFIM) code
Quantum convolutional code
Lechner-Hauke-Zoller (LHZ) code
The LHZ code has weight-three and weight-four \(Z\)-type parity checks arranged on a square lattice, with each physical qubit participating in at most four of them [111,112].
Bosonization code
The \(D\)-dimensional bosonization code encodes fermionic modes into a \(D\)-dimensional qubit stabilizer code.
Checkerboard model code
Four Color Cube (FCC) fracton model code
Haah cubic code (CC)
Hsieh-Halasz (HH) code
Hsieh-Halasz-Balents (HHB) code
Layer code
Layer codes are QLDPC codes constructed by coupling layers of 2D surface codes according to the check-qubit incidence structure of a QLDPC input code [110,113].
X-cube model code
Camara-Ollivier-Tillich code
Raussendorf-Bravyi-Harrington (RBH) cluster-state code
Square-lattice cluster-state code
Actively orthogonal CSS code
Mirror codeBB
Every qubit of a mirror code lies in the support of at most \(|A|+|B|\) stabilizer generators [114; Prop. 3.1].
Perturbed bivariate bicycle (PBB) codeBB
PBB codes have stabilizer generators of weight at most the total number of terms of \(A\), \(B\), \(C\), and \(D\) [115].
Classical-product code
XYZ product code
XYZ product codes are non-CSS qubit QLDPC codes whenever the three input parity-check matrices are sparse [116].
Pair-partition CPM code
Both stabilizer generator matrices are sparse arrays of circulant permutation matrices with constant column weight \(J\) and constant row weight \(L\) [117]. The reported instances have Tanner girth at least six [117].
Finite-geometry (FG) qubit QLDPC code
Generalized homological-product qubit CSS codeBB Homological Kitaev surface
Homological products are a primary tool for generating qubit QLDPC codes with favorable parameters. Typically, whenever the input codes are binary LDPC or qubit QLDPC, the resulting code will be qubit QLDPC with non geometrically local stabilizer generators.
Quantum spatially coupled (SC-QLDPC) code
Tile quantum code
Tile codes are two-dimensionally local CSS QLDPC codes on a planar lattice with open boundaries, generalizing the surface code by allowing higher-weight checks confined to a fixed-size box.
Two-branch coset CSS code
The check weight \(L=2|M|\) and qubit degree \(J\) of a two-branch coset CSS code are independent of the field size and of the lift [118].
3D XYZ color code
Color code
Twist-defect color code
Matching code
3D fermionic surface code
Clifford-deformed surface code (CDSC)Kitaev surface
Twist-defect surface codeKitaev surface
Three-fermion (3F) Walker-Wang model code

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Zoo Code ID: qldpc

Cite as:
“Qubit QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qldpc, arXiv:2606.11484
BibTeX:
@incollection{eczoo_qldpc,
title={Qubit QLDPC code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/qldpc}
}
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“Qubit QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qldpc, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/qldpc.yml.