Fault-Tolerant Quantum Error Correction for Constant-Excitation Stabilizer Codes under Coherent Noise

TL;DR

This paper develops a comprehensive fault-tolerant quantum error correction framework for constant excitation codes that effectively handles collective coherent noise, demonstrating their robustness and practical viability.

Contribution

It introduces a complete fault-tolerant architecture for CE CSS codes, including new logical gates and syndrome extraction methods compatible with coherent noise models.

Findings

Identifies minimal CE codes like [[12,1,3]] and [[14,3,3]]

Demonstrates the [[12,1,3]] code's strong performance under coherent noise

Provides an extended stabilizer simulation algorithm for mixed noise models

Abstract

Collective coherent noise poses challenges for fault-tolerant quantum error correction (FTQEC), as it falls outside the usual stochastic noise models. While constant excitation (CE) codes can naturally avoid coherent noise, a complete fault-tolerant framework for the use of these codes under realistic noise models has been elusive. Here, we introduce a complete fault-tolerant architecture for CE CSS codes based on dual-rail concatenation. After showing that transversal CNOT gates violate CE code constraints, we introduce CE-preserving logical CNOT gates and modified Shor- and Steane-type syndrome extraction schemes using zero-controlled NOT gates and CE-compatible ancilla. This enables fault-tolerant syndrome-extraction circuits fully compatible with CE constraints. We also present an extended stabilizer simulation algorithm that efficiently tracks both stochastic and collective coherent noise. Using our framework, we identify minimal CE codes, including the $[[12,1,3]]$ and $[[14,3,3]]$ codes, and demonstrate that the $[[12,1,3]]$ code achieves strong performance under coherent noise. Our results establish the first complete FTQEC framework for CE codes, demonstrating their robustness to coherent noise. This highlights the potential of CE codes as a possible solution for quantum processors dominated by collective coherent noise.