Abstract We present a new propagation rule for CSS codes. Starting with a CSS code
[[n,k,d]]q [ [ n , k , d ] ] q , we construct a CSS code with parameters
[[n−2,k,d−1]]q [ [ n - 2 , k , d - 1 ] ] q . In general, one would only obtain a code with parameters
[[n−2,k,d−2]]q [ [ n - 2 , k , d - 2 ] ] q . The construction applies to asymmetric quantum codes from the CSS construction as well.
Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure
X and pure
Z stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For
m possibly redundant generator rows on
n qubits, the algorithm uses
O(mn2+n4) binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.
Quantum low-density parity-check codes can be decoded using a syndrome based GF(4) belief propagation decoder (where GF denotes Galois field). However, the performance of this decoder is limited both by unavoidable 4-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of 4-cycles can be reduced by breaking an error into an
X and
Z component and decoding each with an individual
GF(2) based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a
GF(2) based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a
GF(4) decoder for any code, a
GF(2) decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a
GF(2) based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.