Stabilizer-Code Channel Transforms Beyond Repetition Codes for Improved Hashing Bounds
For quantum communication theorists, this provides a systematic method to improve achievable rates for asymmetric Pauli channels, though the improvements are incremental and limited to small transforms.
The paper generalizes the use of stabilizer codes as channel transforms to improve achievable rates for asymmetric Pauli channels beyond the standard hashing bound. It reports specific small transforms that beat the baseline hashing bound for a family of skewed independent Pauli channels.
The quantum hashing bound guarantees that rates up to $1-H(p_I, p_X, p_Y, p_Z)$ are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work.