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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 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.

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 \(r = w\), 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 qubit 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].There exist distance-dependent [9; Thm. 1] (in terms of the geometric entanglement measure; cf. Ref. [10]) and rate-dependent [9; Thm. 3ii] lower bounds on the degree of entanglement of a qubit QLDPC code.Fault-tolerant state preparation can be done in an overhead that is constant with the number of qubits \(n\) [11].

Transversal Gates

There are recipes to determine transversal gates for asymmetric qubit QLDPC codes [12].

Gates

Fault-tolerant logical measurements [13] that generalize a previous construction [14] 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 [15]. There are conditions on when fault-tolerant surgery can be performed with constant-time overhead [16].Repetition-code adapter for logical Pauli measurements and logical CNOT gates via Dehn twists [17].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [18].Fault-tolerant batched gadgets for CSS QLDPC codes with constant spacetime overhead [19].

Decoding

Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms [20].Quantum belief propagation (BP) decoder [21–23] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [24]. Various post-processing algorithms have been proposed (see below and also Refs. [25,26]).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 [27].Neural network BP decoders [28,29] and GNN decoders [30,31] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [32].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [33].BP localized statistics decoding (BP-LSD) that exploits error clustering [34].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [35].BP guided decimation (BPGD) decoder [36].SymBreak decoder, which adaptively modifies the decoding graph to break the degeneracy of the BP decoder [37].Ambiguity clustering (AC) decoder, in which measurement data is divided into clusters and decoded independently [38].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [39]. For CSS codes, syndrome extraction can be implemented in constant depth [40].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) [41].The MWPM decoder for surface codes may be generalizable to QLDPC codes [42].Extensions of the union-find decoder for qubit QLDPC codes [43–45].Sliding-window decoding [46].Closed-branch decoder [47].BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes [48].Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code [49].BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder [50].Cluster decoder [51].BP approximate degenerate OSD (BP+ADOSD) decoder [52].Decision tree decoders (DTDs), one that provably finds the minimum-weight correction, and one that is heuristic [53].AutDEC decoder for codes with large automorphism groups [54].Tesseract ML decoder [55].Relay-BP decoder [56].HyperBlossom [57].Post-selection strategies for clustering based decoders [58].Decoder switching between soft-output and higher-accuracy decoders [59].Graph augmentation and rewiring for interference (GARI) framework for circuit-level noise [60].

Fault Tolerance

Lattice surgery techniques with ancilla qubits [14,61,62]. 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 [49], e.g., quantum expander codes [63].Error-corrected GHZ state distillation for Steane error correction [64].Fault-tolerant logical measurements [13] that generalize a previous construction [14] 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 [15]. There are conditions on when fault-tolerant surgery can be performed with constant-time overhead [16].Fault-tolerant logical measurements based on an extractor system and allowing for universal computation [18].Fault-tolerant batched gadgets for CSS QLDPC codes with constant spacetime overhead [19].High-rate surgery, which yields parallelizable logical Pauli-product measurements [65].Fault-tolerant state preparation can be done in an overhead that is constant with the number of qubits \(n\) [11].

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 [66–68]. In particular, any family of qubit QLDPC codes with superlogarithmic distance achieves a threshold [67].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [69].

Threshold

Qubit QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [49].

Notes

Links to code tables of notable QLDPC codes [70].Collection of QLDPC qubit codes based on hyperbolic tilings in the QEC-Pages software library [71].High-rate QLDPC codes can be used for Bell-pair distillation [72].Qldpc code circUIT Simulator (QUITS) Python software library for simulating QLDPC code circuits [73,74].

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 [75].
  • Single-shot code— Qubit QLDPC codes satisfying linear confinement are single shot [76]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [61].
  • Low-density generator-matrix (LDGM) code— LDGM codes can yield CSS [77–80] and non-CSS [81,82] qubit QLDPC codes. Some of the LDGM-based CSS codes have \(n\)-independent minimum distance and no code capacity threshold [83; 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 qubit QLDPC codes [84; Ch. 15].
  • Asymmetric quantum code— There are recipes to determine transversal gates for asymmetric qubit QLDPC codes [12].
  • 2D lattice stabilizer code— Chain complexes describing qubit QLDPC codes can be converted to 2D lattice stabilizer codes [85].
  • Lattice stabilizer code— Chain complexes describing some QLDPC codes [86,87], and, more generally, CSS codes [88] 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 [85].
  • 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 \(d^{\prime} \geq d\) [89] (see also Ref. [90]). 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 [91]. 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 [91]. 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 [91].
  • 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.
  • Bicycle code— Bicycle codes are the first QLDPC codes [92].
  • 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].
  • \(D\)-dimensional twisted toric code— It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with sublinear distance scaling of \(N^{1-\epsilon}\) for any \(\epsilon>0\) and logarithmic-weight stabilizer generators [94].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Qubit stabilizer code
QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Qubit QLDPC code
Children
Crystalline-circuit qubit code
Spacetime circuit code
Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit QLDPC codes. General spacetime circuit codes can be sparsified to yield QLDPC spacetime circuit codes [95].
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
2D bosonization code
The 2D bosonization code encodes fermionic modes into a 2D qubit stabilizer code.
3D bosonization code
The 3D bosonization code encodes fermionic modes into a 3D qubit stabilizer code.
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 constructed by coupling layers of 2D surface codes according to the Tanner graph of a QLDPC code.
X-cube model code
Raussendorf-Bravyi-Harrington (RBH) cluster-state code
Square-lattice cluster-state code
Hierarchical code
Classical-product code
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
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.), 2025. https://errorcorrectionzoo.org/c/qldpc
BibTeX:
@incollection{eczoo_qldpc, title={Qubit QLDPC code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qldpc} }
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“Qubit QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/qldpc

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