Optimal Decoder for the Error Correcting Parity Code

TL;DR

This paper introduces a two-step decoder for the parity code that achieves near-optimal performance in noiseless conditions and high fault-tolerance thresholds with unreliable measurements, making it promising for quantum computing.

Contribution

The paper proposes a novel two-step decoding strategy for the parity code that improves fault-tolerance and efficiency, especially in noisy measurement environments.

Findings

Near-optimal decoding for intermediate code sizes in noiseless measurements.

Fault-tolerance thresholds above 5% in unreliable measurement regimes.

Efficient long-range logical gates and suitability for planar implementation.

Abstract

We present a two-step decoder for the parity code and evaluate its performance in code-capacity and faulty-measurement settings. For noiseless measurements, we find that the decoding problem can be reduced to a series of repetition codes while yielding near-optimal decoding for intermediate code sizes and achieving optimality in the limit of large codes. In the regime of unreliable measurements, the decoder demonstrates fault-tolerant thresholds above 5% at the cost of decoding a series of independent repetition codes in (1 + 1) dimensions. Such high thresholds, in conjunction with a practical decoder, efficient long-range logical gates, and suitability for planar implementation, position the parity architecture as a promising candidate for demonstrating quantum advantage on qubit platforms with strong noise bias.