Efficient Implementation of a Quantum Search Algorithm for Arbitrary N

TL;DR

This paper improves Grover's quantum search algorithm for arbitrary N by reducing oracle calls through an efficient superposition state preparation method, especially when N is near a power of two, achieving up to 29.33% reduction.

Contribution

It introduces a new method for preparing superposition states that reduces oracle calls in Grover's algorithm for arbitrary N without using ancilla qubits.

Findings

Achieves up to 29.33% reduction in oracle calls for certain N

Requires only O(log N) gate complexity and no ancilla qubits

Significantly improves efficiency over traditional Grover's algorithm for non-power-of-two N

Abstract

This paper presents an enhancement to Grover's search algorithm for instances where the number of items (or the size of the search problem) $N$ is not a power of 2. By employing an efficient algorithm for the preparation of uniform quantum superposition states over a subset of the computational basis states, we demonstrate that a considerable reduction in the number of oracle calls (and Grover's iterations) can be achieved in many cases. For special cases (i.e., when $N$ is of the form such that it is slightly greater than an integer power of 2), the reduction in the number of oracle calls (and Grover's iterations) asymptotically approaches 29.33\%. This improvement is significant compared to the traditional Grover's algorithm, which handles such cases by rounding $N$ up to the nearest power of 2. The key to this improvement is our algorithm for the preparation of uniform quantum superposition states over a subset of the computational basis states, which requires gate complexity and circuit depth of only $ O (\log_2 (N)) $, without using any ancilla qubits.