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.