Efficient Quantum Circuit for Encoding and Decoding of the [[8,3,5]] Stabilizer Code

A general protocol for constructing a complete efficient encoding and decoding quantum circuit of the [[8,3,5]] stabilizer code is proposed. The [[8,3,5]] stabilizer code is an eight-qubit code that protects a three-qubit state with up to one error, which is very important for quantum information processing. Single-qubit operations, two-qubit controlled gates and Toffoli gates are required in the proposed circuit. The current protocol can be generalized to all quantum stabilizer codes satisfying quantum Hamming bound, and implemented in some quantum systems.