A Method for Solving a System of Equations Based on the PRSCIiple of Training Generative-adversarial Neural Networks (GAN) Using a Modified Grover Algorithm
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.
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.
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.
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.
This paper highlights a practical research of the possibility of forming quantum circuits for training neural networks. The demonstrated quantum circuits were based on the principles of Grover's Search Algorithm. The perceptron was chosen as the architecture for the example neural network. The multilayer perceptron is a popular neural network architecture due to its scalability and applicability for solving a wide range of problems. Large scale neural networks can require significant computing resources for training purposes, therefore Grover’s quantum search algorithm could be used as a way to speed up this process, thanks to its ability of simultaneously processing multiple input values of a given function.
Abstract The article proposes a modification of Grover’s quantum oracle quantum search algorithm, which makes it easier to search the database. The algorithm is implemented in the Python programming language using the Reggeti Forest cloud quantum service. The authors of the article use the mean flipping method, which solves the search problem during the iterative order √ (2 ^ n). Development offers great potential for the practical application of the Grover quantum algorithm, as it is characterized by higher performance and speed when performing research. Theoretically, the algorithm provides quadratic acceleration compared to conventional computers. It is not an exponential acceleration, but it remains important for large data carriers. The quantum parallelism of the Grover search algorithm is based on a simultaneous change in the amplitudes of all the inputs. This is done through a superposition of states, which is a purely quantum concept. In addition, the research is carried out globally, which indicates a significant improvement in optimization procedures. Grover’s algorithm, on the other hand, is sensitive to the number of iterations. The more iterations, the smaller the amplitude of the correct answer, so the wrong choice of this parameter can digest the solution. In addition, the operation of the algorithm is limited in the case of the introduction of noise into a quantum system, which is real in modern quantum computers.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
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.