A Method to Reduce Resources for Quantum Error Correction

In a quantum logic circuit, the minimum number of qubits required in a quantum error-correcting code (QECC) to correct a single error was shown by Laflamme to be five. Due to the presence of multi-control gates in the circuit block for a 5-qubit QECC, this block cannot be readily implemented with present day technology. Further, the fault-tolerant decomposition of the QECC circuit block requires a large number of quantum logic gates (resources). In this paper, we (i) propose a smaller 5-qubit error detection circuit which can also correct a single error in 2 of the 5 qubits, and (ii) establish how to use a 3-qubit error correction circuit to correct the single errors when detected in the other 3 qubits. This approach to quantum error-correction circuit design, functionally equivalent to a 5-qubit QECC, yields a significant reduction in the number of quantum logic gates. For a given quantum logic circuit, we also provide a scheme to decide the locations where these error detection and error correction blocks are to be placed in attaining reduction in gate requirement compared to the case where the original 5-qubit QECC block is used. A comparative study of the resource requirement for the benchmark circuits shows that the proposed method outperforms even Shor and Steane codes in terms of resources. Thus, our proposed method provides quantum error correction with minimum qubit requirement and reduced resource requirement on the average.