Bivariate Bicycle Codes and Metachecks: Syndrome Repair, Measurement-Fault Ambiguity, and Logical Obstructions

Abstract:Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so their measured syndromes obey parity constraints that can be used as metachecks. We study when this built-in redundancy actually supports syndrome repair and how it interacts with the logical structure of the code. Annihilator quotients identify single-block logical classes and show when a mixed-block search is unavoidable. For syndrome repair, the metasyndrome is equivalent to reduction modulo the valid-syndrome ideal. The translation group therefore maps into the unit group of a quotient algebra of dimension $k/2$, with kernel $K_M$. This gives a distance-two characterization, a unit-group bound on distinguishable single faults, and a family-level obstruction: for bounded-$k$ BB families with growing block length, metasyndrome-only exact repair fails with probability tending to one at any fixed measurement-error rate. With no intervening data fault, the same orbit count also gives the minimum partial second measurement needed to remove every single-fault ambiguity. Exact calculations separate the standard examples. The $[[72,12,6]]$ code has syndrome distance three and minimum-weight repair corrects every single measurement fault, whereas Gross $[[144,12,12]]$ has 36 indistinguishable single-fault pairs. Sustained phenomenological experiments show the same qualitative contrast and favor joint data--measurement decoding over a separated repair stage on the codes with stronger ambiguity. The analysis extends our earlier coprime-period treatment to general two-block BB codes and separates measurement ambiguity from the data error it can induce.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2609.39121 [quant-ph]

(or arXiv:2609.39121v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.39121 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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