Systematic Algebraic Method to Identify Clifford Operations for Quantum Error-Correction Codes

In the domain of quantum error correction, a critical task involves identifying logical operations on logical qubits for various quantum codes. However, owing to the inherent complexity of many quantum codes, devising an efficient method to implement the desired logical operations utilizing the structure of these codes presents a significant challenge. In previous studies, several methods were used to realize specific logical operations for certain quantum codes; however, they usually do not work for other quantum codes. In this study, we propose a systematic method to find solutions to any Clifford logical operation for Calderbank-Shor-Steane codes, which represent a broad category of many quantum codes. In contrast to previous geometrical methods, our method is based on an algebraic approach. The solved logical operation can be further decomposed into a sequence of single-qubit and two-qubit gates that can be directly realized using quantum hardware. By applying our method, we can find many geometrically difficult logical operations for two different types of quantum code, that is, toric and hyperbolic surface codes. We believe that the present study will find wide applications in the design of logical operations for many quantum low-density parity-check codes, which can play an important role in future fault-tolerant quantum computing.