Minimal degenerate CSS quantum code with low cost circuit

TL;DR

This paper introduces a minimal 6-qubit degenerate CSS quantum error correcting code capable of correcting single bit flip errors and multiple phase flip errors, with significantly reduced gate count for fault-tolerant implementation.

Contribution

It presents the smallest binary CSS code for a single qubit that corrects specific errors and demonstrates a substantial reduction in gate count compared to existing codes.

Findings

Corrects single bit flip errors and multiple phase flip errors.

Achieves approximately 98% reduction in gate count compared to the 5-qubit code.

Offers around 50% gate count savings over 7- and 9-qubit codes.

Abstract

Error correction is of utmost necessity for large-scale quantum computing. Quantum error correcting codes can be degenerate, if more than one type of error can map the input state to the same error state. In this paper, we propose a 6-qubit degenerate CSS code. We show that our proposed code can correct a single bit flip error, but arbitrarily many phase flip errors occurring on the codeword. Furthermore, it follows from a bound on CSS codes that the proposed code is the minimal binary CSS code which can encode a single qubit of information. We also perform a comparative study of the gate count required for a fault tolerant decomposition of the error correction circuit for the proposed code, with that for the existing 9-qubit, 7-qubit and 5-qubit binary QEC codes. Our results show, on an average over the different technologies a savings of 98.14% in the gate count for the proposed code over that for the 5-qubit code. Significant improvement of around 50% in gate count is also obtained when compared to those for the 9-qubit and 7-qubit codes.