Robust Syndrome Extraction via BCH Encoding

TL;DR

This paper introduces a new method for quantum error correction using BCH codes for syndrome measurement, significantly reducing the number of extra measurements needed compared to previous methods, thus improving efficiency.

Contribution

The paper proposes using primitive narrow-sense BCH codes as syndrome measurement codes in quantum data-syndrome codes, reducing measurement overhead from O(t^3 log l) to O(t log l).

Findings

BCH codes asymptotically require O(t log l) extra measurements.

The new method reduces time overhead for syndrome error protection.

Significantly fewer measurements needed compared to previous approaches.

Abstract

Quantum data-syndrome (QDS) codes are a class of quantum error-correcting codes that protect against errors both on the data qubits and on the syndrome itself via redundant measurement of stabilizer group elements. One way to define a QDS code is to choose a syndrome measurement code, a classical block code that encodes the syndrome of the underlying quantum code by defining additional stabilizer measurements. We propose the use of primitive narrow-sense BCH codes as syndrome measurement codes. We show that these codes asymptotically require $O(t\log\ell)$ extra measurements, where $\ell$ is the number of stabilizer generators of the quantum code and $t$ is the number of errors corrected by the BCH code. Previously, the best known general method of constructing QDS codes out of quantum codes requires $O(t^3\log\ell)$ extra measurements. As the number of additional syndrome measurements is a reasonable metric for the amount of additional time a general QDS code requires, we conclude that our construction protects against the same number of syndrome errors with significantly less time overhead.