An optimized quantum minimum searching algorithm with sure-success probability and its experiment simulation with Cirq

TL;DR

The paper addresses minimum-finding in unsorted databases using quantum speedups and proposes OQMSA, a sure-success quantum minimum search based on Grover-Long exact search with a dynamic iteration strategy and oracle simplifications. It analyzes success probability and circuit complexity, showing improved performance over the DHA approach and providing a complexity bound $R \approx \dfrac{\pi}{2}(\sqrt{2}+1)\left(\sqrt{2N}-\sqrt{N/M_0}\right)+ (\log_2 N)^2$. Theoretical results indicate a higher success rate and reduced gate counts, while Cirq-based simulations on a 6-qubit register validate feasibility with a high empirical success rate (982/1000) in a 12-qubit circuit. The work suggests that OQMSA can serve as a practical subroutine for broader quantum optimization tasks on near-term hardware.

Abstract

Finding a minimum is an essential part of mathematical models, and it plays an important role in some optimization problems. Durr and Hoyer proposed a quantum searching algorithm (DHA), with a certain probability of success, to achieve quadratic speed than classical ones. In this paper, we propose an optimized quantum minimum searching algorithm with sure-success probability, which utilizes Grover-Long searching to implement the optimal exact searching, and the dynamic strategy to reduce the iterations of our algorithm. Besides, we optimize the oracle circuit to reduce the number of gates by the simplified rules. The performance evaluation including the theoretical success rate and computational complexity shows that our algorithm has higher accuracy and efficiency than DHA algorithm. Finally, a simulation experiment based on Cirq is performed to verify its feasibility.

Figures (8)

• Figure 1: The schematic diagram of the first simplified rule. (a) The circuit for marking two continuous states. (b) The circuit for marking ${2^r}$ continuous states. (c) The circuit for marking $2^m$ continuous states.
• Figure 2: A schematic diagram of the second equivalent simplified rule. (a) The original circuit. (b) The simplified circuit.
• Figure 3: The optimized circuit for the oracle in OQMSA.
• Figure 4: The general circuit for $I_{0}$ operation.
• Figure 5: Complexity comparison between DHA and our algorithms.