Lifting an $[n,k,d]$ Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer Code

Abstract:The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary $[n,k,d]$ classical binary linear code can be lifted to an $[[n,k,d]]$ quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given $[n,k,d]$ binary linear code $\mathcal{C}$ and an Erdős-Rényi random graph $\mathrm{G}(n, \frac12)$. We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance $d$ of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the $n,k,d$ parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same $n,k,d$ parameters.

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

arXiv:2610.14192 [quant-ph]

(or arXiv:2610.14192v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.14192 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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