Resource Optimal Realization of Fault-Tolerant Quantum Circuit

Encoding of quantum information and carrying out computation on encoded state is an essential requirement for improving the reliability of a quantum computer. Resource limitation in today's noisy intermediate scale quantum (NISQ) processors further restricts carrying out fault-tolerant quantum gate operations on such systems. Recent experiments conducted on physical qubits of superconducting transmon type and trapped atomic ions using the fault-tolerant scheme based on [[4, 2, 2]] code have shown a systematic improvement in the fidelity of all logical quantum gate operations except the logical controlled-NOT (CNOT) operation that requires 3 physical SWAP operations for fault-tolerant realization. In this present work we propose an optimal realization of logical CNOT operations on a single or two separate [[4, 2, 2]] code-words using 4 physical CNOT operations and an additional qubit. We further introduce logical two-qubit positive and negative controlled-phase operations with varying rotation angle, and also propose the fault-tolerant realization of logical 2-controlled-phase (C2Z) and 2-controlled-NOT (C2NOT) operations that are required for universal computation using [[4, 2, 2]] encoding. The implementation requires less number of encoded operations and one additional qubit. Through experiments conducted on the 15-qubit IBM Quantum Experience processor and QASM simulator the fidelity and validity of all these proposed gate operations have been verified.