Improving the Circuit Realization of Grover's Quantum Search Algorithm by Replacing Hadamard with √X Gates

Jozsa, Bernstein-Vazirani, and Grover, utilize Hadamard gates to create uniform superposition states for the input qubits of an oracle. However, Hadamard gates are non-native (non-supported) gates in all real quantum computers. For this reason, Hadamard gates are considered cost-expensive gates when realizing (transpiling) such algorithms into a real quantum computer. This paper introduces a new methodology for cost-effective transpilation of Grover’s algorithm into real quantum computers, by replacing all Hadamard gates with √ X gates. In quantum computing, the Hadamard and √ X gates create uniform superposition states of a qubit on the Xaxis and Y-axis of the Bloch sphere, respectively. Hence, we utilize a different axis of the Bloch sphere to construct a cost-effective realization for the final transpiled circuit of Grover’s algorithm. Theoretically, the final transpiled circuit of Grover’s algorithm using √ X gates has been minimized to approximately 64% than the circuit of Grover’s algorithm using Hadamard gates. Experimentally, because of the √ X is an IBM native gate, it has been proven that the final transpiled circuit of Grover’s algorithm using √ X gates always has a lower quantum cost of 6% than the circuit of Grover’s algorithm using Hadamard gates, due to the limited layout (architecture) connectivity of the physical neighboring qubits for IBM quantum computers.