[Paper Review] An Optimized Quantum Maximum or Minimum Searching Algorithm and its Circuits

Yanhu Chen, Shijie Wei+4 more|arXiv (Cornell University)|Aug 21, 2019

Quantum Computing Algorithms and Architecture31 references4 citations

TL;DR

This paper proposes an optimized quantum algorithm for finding the maximum or minimum in a dataset, improving upon Durr and Hoyer's quantum algorithm by integrating the optimal quantum exact search algorithm. It achieves near-100% success probability when the ratio of solutions to search space is known, reduces gate complexity, and demonstrates superior performance over DHA in both simulation and IBM quantum processor experiments with a two-qubit system.

ABSTRACT

Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential searching algorithm (QESA), which is based on Grover algorithm, is not a sure-success algorithm. Meanwhile, quantum circuits encounter the gate decomposition problem due to variation of the scale of data. In this paper, we propose an optimized quantum algorithm for searching maximum and minimum, based on DHA and the optimal quantum exact search algorithm. Furthermore, we provide the corresponding quantum circuits, together with three equivalent simplifications. In circumstances when we can exactly estimate the ratio of the number of solutions M and the searched space N, our method can improve the successful probability close to 100%. Furthermore, compared with DHA, our algorithm shows an advantage in complexity with large databases and in the gate complexity of constructing oracles. Experiments have been executed on an IBM superconducting processor with two qubits, and a practical problem of finding the minimum from Titanic passengers' age was numerically simulated. Both showed that our optimized maximum or minimum performs more efficiently compared with DHA. Our algorithm can serve as an important subroutine in various quantum algorithms which involves searching maximum or minimum.

Motivation & Objective

• To address the low success probability of the quantum exponential searching algorithm (QESA) in Durr and Hoyer's original maximum/minimum search algorithm.
• To overcome the gate decomposition problem in quantum circuits due to varying data scale.
• To enhance the success probability of quantum maximum/minimum searching to near 100% when the ratio M/N of solutions to search space is known.
• To reduce the quantum circuit complexity, especially in oracle construction, compared to Durr and Hoyer's approach.
• To provide practical quantum circuits with three equivalent simplifications for implementation on NISQ devices.

Proposed method

• The algorithm combines Durr and Hoyer's framework with the optimal quantum exact search algorithm to ensure high success probability.
• It leverages prior knowledge of the ratio M/N (number of solutions to search space size) to tailor the amplitude amplification process.
• The method introduces three equivalent circuit simplifications to reduce gate count and improve circuit efficiency.
• The quantum circuits are designed with optimized oracle constructions that reduce gate complexity compared to standard Grover-based oracles.
• The algorithm uses amplitude amplification with precise iteration counts derived from the known M/N ratio to achieve exact success.
• The implementation is validated via numerical simulation of the Titanic age-minimum problem and executed on a two-qubit IBM superconducting processor.

Experimental results

Research questions

• RQ1 Can the success probability of quantum maximum/minimum search be improved to near 100% when the ratio M/N is known?
• RQ2 How can gate complexity in quantum circuits for max/min search be reduced, especially in oracle construction?
• RQ3 Can the proposed algorithm outperform Durr and Hoyer’s original quantum algorithm in terms of circuit depth and success rate?
• RQ4 What are effective circuit simplifications that preserve functionality while reducing gate count in max/min quantum circuits?
• RQ5 Can the algorithm be practically implemented and validated on real quantum hardware, such as IBM’s superconducting processors?

Key findings

• The proposed algorithm achieves a success probability approaching 100% when the ratio M/N is exactly known, significantly improving upon the non-sure-success nature of Durr and Hoyer’s QESA.
• The gate complexity of the oracle construction is reduced compared to the original Durr and Hoyer algorithm, enhancing scalability for large databases.
• Numerical simulation of the Titanic age-minimum problem confirmed higher efficiency and faster convergence compared to DHA.
• Experimental execution on a two-qubit IBM superconducting processor demonstrated the feasibility and practicality of the proposed quantum circuits.
• The three equivalent circuit simplifications effectively reduce gate count without altering functionality, improving circuit fidelity and reducing error rates.
• The algorithm shows a clear advantage in complexity over DHA for large-scale databases due to optimized amplitude amplification and reduced circuit depth.