This paper presents a quantum optimization problem and solid-state quantum computing architectures. Quantum approach to global optimization and NP-complete problems are considered. Our approach to global optimization based on quantum mechanical entanglement, quantum resonant tunneling, cellular automaton and geometric control methods. A quantum optimization algorithm combines the properties of classical simulated annealing with the possibility of quantum tunneling between the minima. Quantum computation exploits the property of quantum states to implement quantum parallelism for global nonconvex optimization problem. This paper considers new mathematical models of classical (CL) and quantum-mechanical lattices (QML). System-theoretic results on the observability, controllability and minimal realizability theorems are formulated for CL. The cellular dynamaton (CD) based on quantum oscillators is presented. We investigate the conditions when stochastic resonance can occur through the interaction of dynamical neurons with intrinsic deterministic noise and an external periodic control.
This paper presents a quantum optimization problem and solid-state quantum computing architectures. Quantum approach to global optimization and NP-complete problems are considered. Our approach to global optimization based on quantum mechanical entanglement, quantum resonant tunneling, cellular automaton and geometric control methods. A quantum optimization algorithm combines the properties of classical simulated annealing with the possibility of quantum tunneling between the minima. Quantum computation exploits the property of quantum states to implement quantum parallelism for global nonconvex optimization problem. This paper considers new mathematical models of classical (CL) and quantum-mechanical lattices (QML). System-theoretic results on the observability, controllability and minimal realizability theorems are formulated for CL. The cellular dynamaton (CD) based on quantum oscillators is presented. We investigate the conditions when stochastic resonance can occur through the interaction of dynamical neurons with intrinsic deterministic noise and an external periodic control.
This paper deals with the progress made in applications of quantum computing in control and optimization. It concentrates on applying the geometric technique in order to investigate a finite control problem of a two-level quantum system, resonance control of a three-level system, simulation of bilinear quantum control systems, and optimal control using the Bellman principle. We show that a quantum object described by a Schroedinger equation can be controlled in an optimal way by electromagnetic modes. We also demonstrate an application of these techniques and an algebra-geometric approach to the study of dynamic processes in nonlinear systems. The information processing by means of controlled quantum lattices is discussed: we present new mathematical models of classical (CL) and quantum-mechanical lattices (QML) and their application to information processing. system-theoretical results on the observability, controllability and minimal realizability theorems are formulated for cl. The cellular dynamaton (CD) based on quantum oscillators is presented. Cellular's quantum computational search procedure can provide the basis for implementing adaptive global optimization algorithms. A brief overview of the procedure is given and a framework called lattice adaptive search is set up. A method of Yatsenko and one introduced by the authors fit into this framework and are compared.
Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.
Quantum resonance would be exploited in a proposed quantum-computing approach to the solution of combinatorial optimization problems. In quantum computing in general, one takes advantage of the fact that an algorithm cannot be decoupled from the physical effects available to implement it. Prior approaches to quantum computing have involved exploitation of only a subset of known quantum physical effects, notably including parallelism and entanglement, but not including resonance. In the proposed approach, one would utilize the combinatorial properties of tensor-product decomposability of unitary evolution of many-particle quantum systems for physically simulating solutions to NP-complete problems (a class of problems that are intractable with respect to classical methods of computation). In this approach, reinforcement and selection of a desired solution would be executed by means of quantum resonance. Classes of NP-complete problems that are important in practice and could be solved by the proposed approach include planning, scheduling, search, and optimal design.
In the past few decades, there has been an acute problem of creating a quantum computer that uses quantum mechanical effects such as quantum parallelism and quantum entanglement for computations. Using these mechanisms, the quantum computer is able to solve some of the NP-class problems in polynomial time. A hardware approach to modeling quantum computing is considered, and a general mathematical model of the operation of a quantum computer is described, and a technique for mathematical modeling of quantum computing is presented. At the stage of consideration of the method of mathematical modeling, the most resource-intensive parts of the model are indicated. Also, issues related to data parallelization on a hardware accelerator, which are planned to be used to simulate quantum computing, were considered, and algorithms for the operation of this type of accelerator were given. A more compact data format is proposed, which is recommended to be used when implementing the accelerator. Using the proposed format, it is possible to reduce the resources spent on elementary arithmetic operations. The possibility of introducing an optimization algorithm to minimize the time spent in computations, as well as to speed up the execution of quantum algorithms in simulators of a quantum computer at the stage of the effect of quantum gates on the quantum register model, is proposed. The results of the optimization algorithm and its comparison with the classical mathematical approach using a software model are presented.
A primary challenge of computational science is solving NP-hard optimization problems since they are too complicated and cannot be effectively solved deterministically in large instances. Even though quantum computing can solve these problems with the help of its key properties quantum parallelism and tunneling, the modern hardware is yet not sufficiently developed. In this case, QICAs are considered as a new method in which the key aspects of quantum mechanics, such as superposition, entanglement, interference and tunneling are replicated in classical, non-quantum computers. It aims to attain rapid optimization of NP-hard problems through a hybrid approach that combines cartography-based exploration, variable state reduction with tensor networks as well as an inclusion of a variant-based approach inspired by QAOA. A number of standard problems are used to test the model, e.g., the Traveling Salesman Problem (TSP), the 0/1 Knapsack Problem and Max-Cut Problem. Genetic Algorithms, Simulated Annealing and Ant Colony Optimization trials all concur that QICA converges to superior solutions much faster and it can nonetheless handle bigger-sized problems. More precisely, the algorithm provides better approximation results as well as lesser computing power requirements, thus demonstrating the effectiveness of quantum-inspired methods in the case of classical systems. They establish the fact that QICAs have the potential to solve challenging optimization problems, particularly in the cases where quantum machines are absent or inaccessible. The way of solving the problem represented in the study is quite innovative and serves as a bridge between the theory of quantum and practical implementation in the optimization field.
This paper presents a novel approach to optimization using quantum computation, specifically focusing on geometric optimization. The core concept revolves around transforming optimization problems into quantum states, leveraging quantum entanglement and quantum computing's unique properties to achieve enhanced efficiency. We explore a method for mapping complex geometric structures to a quantum state space, utilizing quantum entanglement as a key enabling mechanism. The potential benefits include the ability to tackle optimization problems previously intractable with classical algorithms, offering a fundamentally new perspective on geometric problem-solving. The research investigates the feasibility of this approach through theoretical analysis and initial simulations, highlighting the advantages and challenges associated with implementing this quantum optimization strategy.