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

Also known as Sparse quantum code.

Description

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.

\(k\)

\(d\)

Code

\(2\)

\(\sqrt{n}\)

Kitaev toric

\(2\)

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

Freedman-Meyer-Luo

\(\Theta(n)\)

\(\sqrt{n}\)

hypergraph product

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

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

high-dimensional expander (HDX)

\(\sqrt{n}\)

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

tensor-product HDX

\(n^{3/5}/\text{polylog}(n)\)

\(n^{3/5}/\text{polylog}(n)\)

fiber-bundle

\(\log n\)

\(n/\log n\)

lifted-product (LP)

\(\Theta(n)\)

\(\Theta(n)\)

expander LP

\(\Theta(n)\)

\(\Theta(n)\)

quantum Tanner

\(\Theta(n)\)

\(\Theta(n)\)

Dinur-Hsieh-Lin-Vidick

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.

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. 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

Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction. Bounds generalizing the BPT bound to 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 \(\Omega(d)\) edges of length \(\Omega(d/n^{1/D})\) [4].

Gates

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 [5].

Decoding

Belief-propagation (BP) decoder [6] is a quantum version of the classical BP decoder, but performance suffers due to degeneracy [7]. Various post-processing algorithms have been proposed (see below and also Ref. [8]).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 [9].Neural BP decoder [10] for qubit codes.Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise [11].Message-passing decoder utilizing stabilizer inactivation (MP-SI a.k.a. BP-SI) for CSS-type QLDPC qubit codes [12].Syndrome-based linear programming (SB-LP) algorithm can be applied as a post-processing step after syndrome-based min-sum (SM-MS) decoding [13].BP guided decimation (BPGD) decoder [14].Non-binary decoding algorithm for CSS-type QLDPC codes [15].2D geometrically local syndrome extraction circuits with bounded depth using order \(O(n^2)\) ancilla qubits [16].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) [17].Extension of the union-find decoder for qubit QLDPC codes, as well as a related heuristic decoder [18].Sliding-window decoding [19].Closed-branch decoder [20].

Fault Tolerance

Lattice surgery techniques with ancilla qubits [21,22].Fault-tolerance with constant overhead can be performed on certain QLDPC codes [23], e.g., quantum expander codes [24].GHZ state distillation for Steane error correction [25].

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 [2628].Bounds on code capacity thresholds for various noise models exist in terms of stabilizer generator weights [29].

Threshold

QLDPC codes with a constant encoding rate can reduce the overhead of fault-tolerant quantum computation to be constant [23].

Notes

Links to code tables of notable QLDPC codes [30].Reviews of QLDPC codes provided in Refs. [15,30].

Parents

Children

Cousins

  • 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 [31]. 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 [32].
  • Hamiltonian-based code — QLDPC code Hamiltonians can be simulated by two-dimensional Hamiltonians with non-commuting terms whose interactions scale with \(n\) [33].
  • Single-shot code — Qubit QLDPC codes satisfying linear confinement are single shot [34]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [22].
  • Finite-geometry LDPC (FG-LDPC) code — Quantum versions of EG LDPC codes can be constructed via the CSS construction [35].
  • 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.
  • Spacetime circuit code — General spacetime circuit codes can be sparsified to yield QLDPC spacetime circuit codes [36].
  • 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 non-zero 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 [37].
  • Two-block quantum code — When matrices \(A\) and \(B\) have row and column weights bounded by \(W\), a two-block quantum code is a quantum LDPC code with stabilizer generators bounded by \(2W\).

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) arXiv:quant-ph/0304161 DOI
[2]
N. Baspin and A. Krishna, “Connectivity constrains quantum codes”, Quantum 6, 711 (2022) arXiv:2106.00765 DOI
[3]
N. Baspin et al., “Improved rate-distance trade-offs for quantum codes with restricted connectivity”, (2023) arXiv:2307.03283
[4]
N. Baspin and A. Krishna, “Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive”, Physical Review Letters 129, (2022) arXiv:2109.10982 DOI
[5]
S. Bravyi and R. König, “Classification of Topologically Protected Gates for Local Stabilizer Codes”, Physical Review Letters 110, (2013) arXiv:1206.1609 DOI
[6]
D. Poulin and Y. Chung, “On the iterative decoding of sparse quantum codes”, (2008) arXiv:0801.1241
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N. Raveendran and B. Vasić, “Trapping Sets of Quantum LDPC Codes”, Quantum 5, 562 (2021) arXiv:2012.15297 DOI
[8]
J. Kim, H. Jung, and J. Ha, “Low-Complexity Decoding Algorithm Utilizing Degeneracy for Quantum LDPC Codes”, MILCOM 2023 - 2023 IEEE Military Communications Conference (MILCOM) (2023) DOI
[9]
P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
[10]
S. Miao et al., “Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices”, (2023) arXiv:2212.10245
[11]
Z. Yi et al., “Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes”, (2023) arXiv:2305.17505
[12]
J. du Crest, M. Mhalla, and V. Savin, “Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes”, (2023) arXiv:2205.06125
[13]
S. Javed et al., “Low-Complexity Linear Programming Based Decoding of Quantum LDPC codes”, (2024) arXiv:2311.18488
[14]
H. Yao et al., “Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation”, (2023) arXiv:2312.10950
[15]
Z. Babar et al., “Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies”, IEEE Access 3, 2492 (2015) DOI
[16]
N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
[17]
N. Raveendran et al., “Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors”, (2022) arXiv:2205.02341
[18]
L. Berent, L. Burgholzer, and R. Wille, “Software Tools for Decoding Quantum Low-Density Parity-Check Codes”, Proceedings of the 28th Asia and South Pacific Design Automation Conference (2023) arXiv:2209.01180 DOI
[19]
S. Huang and S. Puri, “Improved Noisy Syndrome Decoding of Quantum LDPC Codes with Sliding Window”, (2023) arXiv:2311.03307
[20]
A. deMarti iOlius and J. E. Martinez, “The closed-branch decoder for quantum LDPC codes”, (2024) arXiv:2402.01532
[21]
L. Z. Cohen et al., “Low-overhead fault-tolerant quantum computing using long-range connectivity”, Science Advances 8, (2022) arXiv:2110.10794 DOI
[22]
Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
[23]
D. Gottesman, “Fault-Tolerant Quantum Computation with Constant Overhead”, (2014) arXiv:1310.2984
[24]
O. Fawzi, A. Grospellier, and A. Leverrier, “Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes”, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (2018) arXiv:1808.03821 DOI
[25]
N. Rengaswamy et al., “Entanglement Purification with Quantum LDPC Codes and Iterative Decoding”, (2024) arXiv:2210.14143
[26]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[27]
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) arXiv:1208.2317 DOI
[28]
A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
[29]
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) arXiv:1412.6172 DOI
[30]
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[31]
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[32]
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[33]
H. Apel and N. Baspin, “Simulating LDPC code Hamiltonians on 2D lattices”, (2023) arXiv:2308.13277
[34]
A. O. Quintavalle et al., “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
[35]
S. A. Aly, “A Class of Quantum LDPC Codes Constructed From Finite Geometries”, IEEE GLOBECOM 2008 - 2008 IEEE Global Telecommunications Conference (2008) arXiv:0712.4115 DOI
[36]
N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
[37]
N. Koukoulekidis et al., “Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes”, (2024) arXiv:2401.07583
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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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@incollection{eczoo_qldpc, title={Quantum low-density parity-check (QLDPC) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qldpc} }
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