Efficient Computations of Encodings for Quantum Error Correction

TL;DR

The paper presents a universal method to efficiently encode any quantum stabilizer code by constructing a gate array of size $O(n d)$ that maps $k=n-d$ data qubits to $n$ code qubits using only in-place one- and two-qubit gates. It achieves this by transforming the stabilizer generator matrices into a standard form via Gaussian elimination, yielding a structured division into seed, secondary, and primary generators and then realizing encoding as a two-stage pre- and post-mapping circuit that can be reversed for decoding. The authors provide concrete counts and illustrate the construction with example codes (e.g., $n=8$, $d=5$) and conjecture asymptotic optimality for general stabilizer codes. This work enables practical, scalable quantum encoding and decoding within stabilizer-code frameworks, supporting efficient quantum error correction on real devices.

Abstract

We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the codewords can be constructed. For an n-qubit code whose stabilizer has d generators, the resulting gate array consists of O(n d) operations, and converts k-qubit data (where k = n - d) into n-qubit codewords.