Fault-Tolerant Logical Hadamard Gates Implementation in Reed-Muller Quantum Codes

We investigate how to implement fault-tolerant logical Hadamard gates in Reed-Muller quantum codes(RMQCs) using the gauge-fixing method. During the realization, we consider the influence of random single-qubit errors by performing the error-detecting measurements. Moreover, some error-detecting stabilizers are simplified by the existing syndromes. Then we identify the errors and modify the syndromes, and refer to the modified syndromes to select the fix operations, and finally perform the error-correcting and fix operations together. Further, we establish a graph model for the RMQCs and exhibit a progress of how to find the fix operations for the unsatisfied stabilizers. We simulate the progress of finding corresponding fix operations for 31-quibt and 63-qubit RMQCs and the whole process of realizing logical Hadamard gate with random single-qubit errors for 15-qubit and 31-quibt RMQCs. Results show that correct fix operations can be obtained and fault-tolerant logical Hadamard gates can be realized as expected. With the implementation of the logical Hadamard gate, a universal fault-tolerant gate set is achieved in single RMQC.