Automatic Generation of an Efficient Less-Than Oracle for Quantum Amplitude Amplification

Grover's algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence faster than any classical algorithm. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. A generalization of it is the oracle for Amplitude Amplification, that marks multiple desired states. In this work we present a classical algorithm that builds a phase-marking oracle for Amplitude Amplification. This oracle performs a less-than operation, marking states representing natural numbers smaller than a given one. Results of both simulations and experiments are shown to prove its functionality. This less-than oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with the one generated by Qiskit automatic method, UnitaryGate. We show that the depth of our less-than oracle implementation is always lower. This difference is significant enough for our method to outperform UnitaryGate on real quantum hardware.