Efficient fault‐tolerant logical Hadamard gates implementation in Reed–Muller quantum codes

We investigate the implementation of fault‐tolerant logical Hadamard gates in Reed–Muller quantum codes (RMQCs). During the realization, we consider the influences of random single‐qubit errors and some error‐detecting stabilizers are simplified by the existing syndromes. We first identify the errors and modify the gauge‐fixing syndromes, then refer to the modified syndromes to select the fix operations, and finally perform the error‐correcting and fix operations together. Furthermore, we establish a graph model for the RMQCs and exhibit a progress of finding the fix operations for the unsatisfied stabilizers. For the circuit design, we optimize the choice of gauge operators, the positions of the ancillary qubits and design a parallel circuit for implementing a fault‐tolerant logical Hadamard gate in 15‐qubit RMQC. We simulate the progress of finding corresponding fix operations for 31‐qubit and 63‐qubit RMQCs and the whole process of realizing logical Hadamard gate with random single‐qubit errors for 15‐qubit and 31‐qubit RMQCs. Results show that correct fix operations can be obtained and fault‐tolerant logical Hadamard gates can be realized as expected. The performance comparisons are also given and the results show that our method can achieve a higher success rate with a reasonable higher cost. With the implementation of the logical Hadamard gate, a universal fault‐tolerant gate set is achieved in single RMQC.