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The International Arab Journal of Information Technology

Quantum Computing for Solving a System of Nonlinear Equations over GF(q).

Essam Al-Daoud

Published Jan 1, 2007• 4 citations
OpenAlex

Abstract

Abstract: Grover’s quantum search algorithm is one of the most widely studied and has produced results in some search applications faster than their classical counterpart by a square-root. This paper modifies Grover’s algorithm to solve nonlinear equations over Galois Finite field GF(q) in O( mn 2) iteration, while the best classical general solution takes O(2 nm) iteration. The modification is done by using a register for each variable and represent it by n qubits. The paper also introduces the implementation of the suggested algorithm by using the simulator QCL 5.1.

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Abstract: Grover’s quantum search algorithm is one of the most widely studied and has produced results in some search applications faster than their classical counterpart by a square-root. This paper modifies Grover’s algorithm to solve nonlinear equations over Galois Finite field GF(q) in O( mn 2) iteration, while the best classical general solution takes O(2 nm) iteration. The modification is done by using a register for each variable and represent it by n qubits. The paper also introduces the implementation of the suggested algorithm by using the simulator QCL 5.1.
The solution of nonlinear Boolean equations in a binary field plays a crucial part in cryptanalysis and computational mathematics. To speed up the process of solving Boolean equations is an urgent task that needs to be addressed. In this paper, we propose a method for solving Boolean equations based on the Grover algorithm combined with preprocessing using classical algorithms, optimizing the quantum circuit for solving the equations, and implementing the automatic generation of quantum circuits. The method first converted Boolean equations into Boolean expressions to construct the oracle in the Grover algorithm. The quantum circuit was emulated based on the IBM Qiskit framework and then simulated the Grover algorithm on this basis. Finally, the solution of the Boolean equation was implemented. The experimental results proved the feasibility of using the Grover algorithm to solve nonlinear Boolean equations in a binary field, and the correct answer was successfully found under the conditions that the search space was 221 and three G iterations were used. The method in this paper increases the solving scale and solving speed of Boolean equations and enlarges the application area of the Grover algorithm.
Quantum computation is a new computing model based on fundamental quantum mechanical principle. Grover's algorithm finds the solution for a searching problem in the square root time of exhaustive search. Brassard, Hoyer, Tapp's algorithm counts the number of solutions for a searching problem. Through exploiting the two quantum algorithms, we propose a quantum algorithm for solving a new cryptography problem -- polynomial root finding problem, which could be used to design a cryptosystem. The algorithm will take O(rootM/t) steps for finding one of the t solutions to the problem, where M is the modular of the equation. The success rate of the algorithm is a constant and the cost of the algorithm depends on the calculations of modular exponentiation and the number of iterations.
his study explores the application of quantum computing in solving complex mathematical problems through the utilization of Grover's Algorithm. Grover's Algorithm, renowned for its efficiency in unstructured search, has been adapted for various mathematical applications requiring rapid solution searches within large solution spaces. In this study, we implement Grover's Algorithm on a quantum computing platform to address a range of complex mathematical problems, including nonlinear equations and combinatorial optimization. Experimental results demonstrate that Grover's Algorithm significantly reduces computational time compared to classical methods, highlighting the immense potential of quantum computing in advanced mathematics. These findings pave the way for the development of more sophisticated quantum algorithms and their applications across science, engineering, and technology domains.
The article proposes a method for solving a system of equations based on a quantum Grover search algorithm. Finding a solution to a system of equations is a computationally complex process and can be considered as an algorithmic primitive for solving various problems. The computational complexity of finding a solution to a system of equations has led to attempts to implement this problem using quantum computing. So, the concept of Quantum Linear System Problem (QLSP) is well known – the solution of systems of linear equations using a quantum computer. The method proposed in the article is considered within the framework of solving a system of algebraic equations. A feature of this method is a modification of the Grover algorithm, which consists in placing the condition of each equation in a separate Grover iteration, which differs from the usual use of Grover iterations – repeats of the oracle and the diffusion operator, in which the oracle does not change. Thus, the construction of its own oracle function of the Grover algorithm for each equation of the system is implemented within the framework of the implementation of the general scheme. A feature of the proposed method is the approximation of the problem of solving a system of equations to a problem resembling the pRSCIiple of training generative-adversarial neural networks (GAN) using Grover’s algorithm, since Grover’s algorithm allows analyzing all possible values of variables. Thanks to the use of the modified Grover algorithm, the proposed method is not limited by the mandatory condition that the number of equations is equal to the number of unknowns, since solutions to incomplete systems of equations can be found within the limits imposed by the size of the allocated quantum registers. A quantum circuit optimization method is also proposed, which consists in implementing some calculations directly in the body of the Grover algorithm. The claimed efficiency of the proposed method is O(2n/m). The method proposed in the article allows us to obtain a quantum primitive for solving a wide range of practical problems.
Nonlinear boolean equation systems play an important role in a wide range of applications. Grover's algorithm is one of the best-known quantum search algorithms in solving the nonlinear boolean equation system on quantum computers. In this paper, we propose three novel techniques to improve the efficiency under Grover's algorithm framework. A W-cycle circuit construction introduces a recursive idea to increase the solvable number of boolean equations given a fixed number of qubits. Then, a greedy compression technique is proposed to reduce the oracle circuit depth. Finally, a randomized Grover's algorithm randomly chooses a subset of equations to form a random oracle every iteration, which further reduces the circuit depth and the number of ancilla qubits. Numerical results on boolean quadratic equations demonstrate the efficiency of the proposed techniques.