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Electronics

Optimizing the Quantum Circuit for Solving Boolean Equations Based on Grover Search Algorithm

Hui Liu, Fukun Li, Yilin Fan

Published Aug 8, 2022• 4 citations
Publisher

Abstract

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.

DOI
10.3390/electronics11152467

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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.
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.
Quantum computing is an exciting new field in the intersection of computer science, physics and mathematics. It refines the central concepts from Quantum mechanics into its least difficult structures, peeling away the complications from the physical world. Any combinational circuit that has only one stuck at fault can be tested by applying a set of inputs that drive the circuit to verify the output response. The outputs of that circuit will be different from the one desired if the faults exist. This project describes a method of generating test patterns using the Boolean satisfaction method. First, the Boolean formula is constructed to express the Boolean difference between a fault-free circuit and a faulty circuit. Second, the Boolean satisfaction algorithm is applied to the formula in the previous step. The Grover algorithm is used to solve the Boolean satisfaction problem. The Boolean Satisfiability problem for Automatic Test Pattern Generation(ATPG) is implemented on IBM Quantum Experience. The Python program initially generates the boolean expression from the file and converts it into Conjunctive Normal Form(CNF) which is passed on to Grover Oracle and runs on IBM simulator and produces excellent results on combinational circuits for test pattern generation with a quadratic speedup. Grover’s Algorithm on this problem has a run time of O(√N).
This paper introduces a comprehensive quantum calculator using Qiskit that integrates quantum arithmetic and logic functions into a single calculator. The calculator relies on quantum Fourier transforms (QFT), controlled-phase gates, and quantum modular arithmetic to perform basic arithmetic operations such as addition, subtraction, and multiplication. These arithmetic techniques used quantum parallelism and interference to perform calculations more efficiently, particularly for tasks involving large numerical inputs. Numerical values are stored in qubit registers and processed using QFT-based circuits, which enable efficient and reversible mathematical computations. In addition to arithmetic operations, the calculator also implements quantum logic algorithms, including Grover’s Search and the Deutsch-Jozsa algorithm. Grover’s algorithm is an unstructured search algorithm that operates by constructing an oracle and iteratively amplifying the probability amplitude of the target solution. This amplitude amplification significantly increases the likelihood of measuring the desired outcome. Due to this property, Grover’s algorithm has practical relevance in applications such as database searching and solving combinatorial optimization problems. The Deutsch-Jozsa algorithm determines whether or not a certain Boolean function is constant or balanced and offers an exponential speed up compared to classical methods for this class of problem. A key fundamental function of the calculator is the ability to generate hybrid quantum circuits that support both arithmetic and logical functions. That allows the exploration of simple as well as complex problem-solving strategies that take advantage of both quantum arithmetic and decision-making algorithms. For instance, combining Grover’s amplification with function characterizations from the Deutsch-Jozsa algorithm provides not only interesting circuit configurations but also adds value to the possibilities of the algorithms. This research offers a modular idea and allows exploring the fundamental principles of quantum computing generally speaking and will be a valuable resource for educational as well as research purposes in the rapidly growing field of quantum algorithm development.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.