Conventionally, the quantum oracular algorithms, such as Deutsch-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 X-axis 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.
We present an optical scheme to conditionally generate even or odd squeezed superpositions of coherent states (SSCSs). The optical setup consists of an unbalanced beam splitter whose transmittance tends to unity, and additional balanced beam splitters and photodetectors in auxiliary modes. Squeezed coherent states with different amplitudes are the input states in the optical scheme. The single-qubit operations are probabilistic and employ two- and three-photon subtractions from initial beams as the driving force. Generation of the even or odd SSCSs is observed in a wide diapason of values of used parameters. We consider a possibility to realize a one-way Hadamard gate for the squeezed coherent states when the base states are transformed into superposition states. States approximating the output states of a Hadamard gate with high fidelity can be realized by imposing restrictions on the values of used parameters. Higher-order subtractions from input beams are necessary to generate the SSCSs with larger amplitudes and higher fidelities. The problem is resolved in a Wigner representation to take into account imperfections of the optical devices.