Quantum stabilizer codes from classical binary codes and random graphs
Lifting an $[n,k,d]$ Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer Code
Information Theory
Summary
Quantum error correction is important for protecting information in quantum computers. This paper studies how to turn classical binary error-correcting codes into quantum codes using a method involving graphs. The authors show that for many random choices of graphs, the resulting quantum codes keep the error-correcting strength of the original classical codes. They also give conditions on the classical codes which guarantee the existence of such quantum codes with matching parameters.
What this means in practice
- •For quantum hardware engineers: Design quantum error correction modules using classical binary codes combined with random graphs to achieve desired quantum code parameters.
- •For quantum software developers: Implement quantum stabilizer codes derived from known classical codes to enhance fault tolerance in quantum computing simulations and software.
Authors
Thamadathil Aswanth, Navin Kashyap
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.