A systematical algebraic method to find Clifford operations for quantum error-correction codes

In the area of quantum error correction, one important task is to find logical operations on logical qubits for various quantum codes. However, due to the complexity of many quantum codes, it is very challenging to find an efficient way to realize the target logical operations with the structure of these codes kept. In the previous studies, several methods have been used to realize some specific logical operations for some particular quantum codes, but usually they will not work for other quantum codes. In this work, we propose a systematical method to find the solutions to any Clifford logical operations for Calderbank-Shor-Steane (CSS) codes, which represent a broad category of many quantum codes. In contract to previous geometrical ways, our method is based on an algebraic view. The solved logical operation can be further decomposed into a sequence of single-qubit and two-qubit gates, which can be directly realized in the quantum hardware. By applying our method, we can find many geometrically difficult logical operations for two different kinds of quantum codes, i.e., toric code and hyperbolic surface code. We believe the present contribution will find wide applications in designing of logical operations for many quantum low-density parity-check (LDPC) codes, which can play an important role in future fault-tolerant quantum computing.