Quantum low-density parity-check (QLDPC) code[1]
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 |
---|---|---|
\(2\) | \(\sqrt{n}\) | |
\(2\) | \(\sqrt{n\sqrt{\log n}}\) | |
\(\Theta(n)\) | \(\sqrt{n}\) | |
\(\sqrt{n}/\log n\) | \(\sqrt{n} \log n\) | |
\(\sqrt{n}\) | \(\sqrt{n} \log^c n\) | |
\(n^{3/5}/\text{polylog}(n)\) | \(n^{3/5}/\text{polylog}(n)\) | |
\(\log n\) | \(n/\log n\) | |
\(\Theta(n)\) | \(\Theta(n)\) | |
\(\Theta(n)\) | \(\Theta(n)\) | |
\(\Theta(n)\) | \(\Theta(n)\) |
Strictly speaking, the term parity check describes only bitwise qubit error syndromes. Nevertheless, qudit stabilizer codes satisfying the above criteria are also called QLDPC codes.
Protection
Rate
Decoding
Fault Tolerance
Code Capacity Threshold
Threshold
Notes
Parents
Children
- Generalized homological-product code — 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.
- Good QLDPC code
- Lattice stabilizer code — Lattice stabilizer codes are QLDPC codes on Euclidean geometries.
- Bivariate bicycle code
- Classical-product code
- Hierarchical code
- Yoked surface code
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 [30]. 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 [31].
- Hamiltonian-based code — QLDPC code Hamiltonians can be simulated by two-dimensional Hamiltonians with non-commuting terms whose interactions scale with \(n\) [32].
- Single-shot code — Qubit QLDPC codes satisfying linear confinement are single shot [33]. Any code that admits a local greedy decoder also satisfies linear confinement, and so is single shot [21].
- Finite-geometry LDPC (FG-LDPC) code — Quantum versions of EG LDPC codes can be constructed via the CSS construction [34].
- 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.
- 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 [35].
- 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 \(2W\).
- 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 [36].
- Distance-balanced code — Lattice surgery techniques for QLDPC codes [20,21] utilize weight reduction.
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Page edit log
- Victor V. Albert (2022-10-02) — most recent
- 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 low-density parity-check (QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qldpc