Fault-tolerant error correction for quantum Hamming codes with only two ancillary qudits

High-dimensional quantum error correction codes (QECCs) are widely used to enhance the security and coding capabilities of the entire quantum systems. While these codes show promise in achieving reliable quantum computing by executing a set of universal quantum gates, quantum gate faults present a challenge for implementing universal quantum computing. The ability to perform universal quantum gates in a fault-tolerant manner is required. In this paper, we propose a fault-tolerant error correction scheme which allows the entire correction process to proceed smoothly, even in the presence of quantum gate faults. Firstly, by equipping the syndrome-extracted circuit with a flag qudit, it is possible to parallel the error detection of coding blocks with the fault detection of quantum gates to detect noise and correlated errors. Then, by giving a feasible permutation of quantum gates, the syndromes can identify the error type and location. We interpret the permutation with an example of Hamming code [[13,7,3]]_3 , a three-dimensional QECC, using only two ancillary qudits. Finally, the results show that our scheme is valid for any quantum Hamming code. The overhead of qudits required in our scheme is independent of the weight and quantity of stabilizer generators, and only two ancillary qudits are involved. Consequently, our scheme is very promising for achieving a low logical error rate in a fault-tolerant manner.