[Paper Review] Solving mathematical problems with quantum search algorithm

Rubens Viana Ramos, Paulo Benicio de Sousa+1 more|ArXiv.org|Apr 29, 2006

Quantum Computing Algorithms and Architecture18 references3 citations

TL;DR

This paper presents a quantum algorithm based on Grover's search to solve mathematical problems such as finding zeros of logical functions, extremal values in databases, and primality testing of odd numbers. It introduces a novel quantum bit string comparator as an oracle to enable conditional logic and efficiently identify target values in unstructured data, demonstrating the feasibility of implementing complex logical operations in quantum computation.

ABSTRACT

Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.

Motivation & Objective

• To develop a quantum algorithm that leverages Grover's search for solving mathematical problems beyond simple database search.
• To design and implement a quantum bit string comparator as a fundamental building block for conditional logic in quantum algorithms.
• To enable the detection of specific values such as zeros of logical functions, maximum/minimum values, and primality of odd numbers using quantum search.
• To demonstrate the practical utility of the comparator in constructing oracles for real mathematical problem-solving tasks.
• To extend the applicability of Grover's algorithm beyond unstructured search by embedding problem-specific logic through the comparator.

Proposed method

• The paper proposes a quantum bit string comparator circuit that evaluates the lexicographical order of two quantum states to determine equality or relative magnitude.
• This comparator is used as an oracle in Grover's search algorithm to identify specific target states in an unstructured database.
• The oracle is constructed using controlled quantum operations that compare qubit strings qubit-by-qubit in a reversible manner.
• The algorithm applies amplitude amplification via Grover iteration to increase the probability of measuring the desired solution state.
• The comparator enables conditional logic by allowing the algorithm to branch on comparison outcomes, essential for implementing decision-making in quantum programs.
• The approach is applied to three mathematical problems: finding function zeros, locating extremal values, and testing odd number primality.

Experimental results

Research questions

• RQ1 Can Grover's quantum search algorithm be effectively extended to solve mathematical problems such as finding zeros of logical functions?
• RQ2 How can a quantum bit string comparator be designed to serve as a reliable oracle for complex search tasks?
• RQ3 To what extent can the comparator enable conditional logic in quantum algorithms, facilitating problem-specific decision-making?
• RQ4 Can the proposed method efficiently determine the primality of odd numbers using quantum search and comparison circuits?
• RQ5 What is the role of the comparator in enabling the detection of maximum or minimum values in an unstructured quantum database?

Key findings

• The quantum bit string comparator successfully compares two quantum states and determines equality or lexicographical order, forming a reliable oracle for Grover's algorithm.
• The comparator enables the implementation of conditional statements in quantum computation, a crucial capability for algorithmic design.
• The method allows for the detection of zeros of logical functions by encoding the condition f(x) = 0 into the oracle function.
• The algorithm can identify the maximum or minimum value in an unstructured database by using the comparator to compare candidate values iteratively.
• Primality testing for odd numbers is achieved by searching for a non-trivial divisor; if none is found, the number is declared prime.
• The proposed approach demonstrates that Grover's algorithm can be adapted to solve non-trivial mathematical problems beyond simple search, using a custom comparator oracle.