Research FAQ

The CSS condition (H_XH_Z^T=0) is essential, but enforcing it directly at the level of sparse regular binary matrices strongly restricts edge placement. This often introduces short cycles and limits the freedom normally available in classical LDPC design. Kasai-code constructions control which matrix components must commute so that the active graph can retain more classical-LDPC-like design freedom.

Kasai-code overview

The construction separates algebraic commutativity constraints from the combinatorial choice of graph edges. Candidate bases and lifts are screened for short cycles, and explicit instances are accompanied by verification data. The exact mechanism depends on the construction family.

Other QLDPC constructions

They provide controlled rate and sparsity while making degree-distribution benchmarks and finite-length comparisons transparent. The choice is a design point, not a claim that one regular degree pair is universally optimal.

Degree-distribution benchmark

Classical design principles remain valuable, but quantum commutativity, degeneracy, logical equivalence, and syndrome-measurement constraints must be handled explicitly. The goal is to retain useful classical structure while adapting it to the quantum setting.

No. In the constructions discussed here, matrices over a larger finite field are expanded into binary matrices, and the physical system remains qubits. The non-binary representation supplies algebraic structure and a useful decoding domain.

Non-binary codes

It denotes a CSS code with blocklength 9216 and 4612 logical qubits, together with an upper bound (d\le48) on the minimum distance. An upper-bound witness does not by itself provide a matching lower bound.

Upper-bound definitions

No. It is a typical-relative-distance benchmark derived from an average weight enumerator. Exact or certified statements about a finite code must come from explicit witnesses, exhaustive screening, or other code-specific arguments.

Minimum-distance benchmark

Ruling out every low-weight logical operator is a combinatorial search problem whose cost grows quickly. This is why the catalog distinguishes exact distances, certified lower bounds, and method-specific upper bounds instead of presenting a single undifferentiated distance column.

Live code catalog

The waterfall is the steep improvement in error rate near a decoder's transition region. The error floor is the slower-decaying low-error-rate regime often governed by specific trapping or low-weight structures. Good waterfall behavior does not by itself establish a low error floor.

Density evolution follows message distributions under a locally tree-like, infinite-length approximation. Its threshold is an ensemble-level analytical benchmark for the stated decoder and channel model, not a finite-length guarantee.

The baseline is a joint sum-product BP decoder. The X- and Z-side factor graphs are coupled by the local Pauli prior so that correlation between the two binary error components is retained. When used, lightweight post-processing is reported separately.

Post-processing algorithm note

Separate decoding discards the local correlation induced by Y errors. Joint sum-product BP keeps that correlation through a shared local prior. It can also be viewed as a CSS-explicit representation of a four-state Pauli-message calculation.

Matching the syndrome is not enough. Decoding succeeds only when the residual between the true and estimated errors is a stabilizer, so that it acts trivially on the logical state. A syndrome-consistent residual that represents a nontrivial logical operator is counted as a failure.

Most coding-theoretic comparisons on these pages use ideal syndrome information unless a linked work explicitly states a circuit-level model. Results with measurement and circuit noise are labeled separately.