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Quantum optimization methods in water flow control

Dilnoz Muhamediyeva, Nilufar A’loxanovna Niyozmatova, Dilfuza Yusupova, Boymirzo Samijonov

Published Jan 1, 2024• 2 citations
Publisher

Abstract

This paper examines the problem of optimizing water flow control in order to minimize costs, represented as the square of the water flow. This takes into account restrictions on this flow, such as the maximum flow value. To solve this problem, two optimization methods are used: the classical optimization method Sequential Least SQuares Programming (SLSQP) and the quantum optimization method Variational Quantum Eigensolver (VQE). First, the classical SLSQP method finds the optimal control (water flow) according to the given cost function and constraints. Then the obtained result is refined using the quantum VQE method. The quantum method uses an ansatz to represent the quantum circuit and a Hamiltonian to describe the system. The problem comes down to finding the minimum eigenvalue of the Hamiltonian, which makes it possible to determine the optimal parameters of the quantum circuit that minimize the cost of controlling the water flow. Thus, the proposed approach combines the strengths of classical and quantum optimization to effectively solve the water flow control optimization problem.

DOI
10.1051/e3sconf/202459002003

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This paper examines the problem of optimizing water flow control in order to minimize costs, represented as the square of the water flow. This takes into account restrictions on this flow, such as the maximum flow value. To solve this problem, two optimization methods are used: the classical optimization method Sequential Least SQuares Programming (SLSQP) and the quantum optimization method Variational Quantum Eigensolver (VQE). First, the classical SLSQP method finds the optimal control (water flow) according to the given cost function and constraints. Then the obtained result is refined using the quantum VQE method. The quantum method uses an ansatz to represent the quantum circuit and a Hamiltonian to describe the system. The problem comes down to finding the minimum eigenvalue of the Hamiltonian, which makes it possible to determine the optimal parameters of the quantum circuit that minimize the cost of controlling the water flow. Thus, the proposed approach combines the strengths of classical and quantum optimization to effectively solve the water flow control optimization problem.
The variational quantum eigensolver (VQE) is a hybrid algorithm to find the minimum eigenvalue/vector of the given Hamiltonian by optimizing a parameterized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parameterized gates of PQCs. In this paper, we focus on the case where the components to be optimized are single-qubit gates, in which the analytic optimization of a single-qubit gate is sequentially performed. The analytical solution is given by diagonalization of a matrix whose elements are computed from the expectation values of observables specified by a set of predetermined parameters, which we refer to as the parameter configurations. In this paper, we first show that the optimization accuracy significantly depends on the choice of parameter configurations owing to the statistical errors in the expectation values. We then identify a metric that quantifies the optimization accuracy of a parameter configuration for all possible statistical errors, named configuration overhead/cost or C-cost. We theoretically provide the lower bound of C-cost and show that, for the minimum size of parameter configurations, the lower bound is achieved if and only if the parameter configuration satisfies the so-called equiangular line condition. Finally, we provide numerical experiments demonstrating that the optimal parameter configuration exhibits the best result in several VQE problems. We hope that this general statistical methodology will enhance the efficacy of sequential optimization of PQCs for solving practical problems with near-term quantum devices.
In this chapter, we will take a look at the various optimization techniques that use quantum computing in their formulation. A couple of such algorithms that we are going to work through in great detail are the variational quantum eigensolver, popularly known as VQE, and the quantum approximate optimization algorithm, also known as QAOA. The central idea in both methods is to define cost objectives as an expectation of appropriate Hamiltonians pertaining to quantum systems. Based on the maximization or minimization problem, we look to derive the maximum or minimum eigenvalue state through these optimization techniques. Both of these methods are variational in that they combine both quantum and classical methods for the optimization. Since the topic is centered around Hamiltonians, we will study the popular Isling Hamiltonian model as well in this chapter. Similarly, the QAOA technique is based on the adiabatic evolution of a quantum system, and hence we will study its underlying math in great detail in this chapter.
Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson's equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains.
Efficient management of water resource systems requires solving complex optimization problems involving nonlinear dynamics, large datasets, and multiple operational constraints. Traditional optimization techniques such as linear programming, dynamic programming, and evolutionary algorithms have been widely applied in water resource management; however, these methods often face computational challenges when applied to large-scale hydrological systems. This paper proposes a hybrid quantum-classical optimization framework for water resource management using topological qubit models. The water allocation problem is formulated as a network-based optimization model and transformed into a quadratic unconstrained binary optimization (QUBO) problem suitable for quantum optimization algorithms. A hybrid computational approach combining classical hydrological simulation with quantum optimization is developed to improve computational efficiency. Simulation results demonstrate that the proposed model achieves faster convergence and lower operational cost compared with classical optimization methods. The study shows that quantum computing provides a promising computational approach for solving complex water resource optimization problems. Future work will focus on implementing the proposed model on real quantum hardware and integrating real-world hydrological datasets.
Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks, including solving optimization problems. Nonetheless, optimization problems with constraints may not have been handled in a disciplined fashion thus far. To address this gap, this work proposes a hybrid quantum-classical algorithmic paradigm termed the variational quantum eigensolver with constraints (VQEC) that extends the celebrated VQE to handle optimization with constraints. As with the standard VQE, the vector of optimization variables is captured by the state of a variational quantum circuit (VQC). To deal with constraints, VQEC optimizes a Lagrangian function classically over both the VQC parameters as well as the dual variables associated with constraints. To comply with the quantum setup, variables are updated via a perturbed primal-dual method leveraging the parameter shift rule. Among a wide gamut of potential applications, we showcase how VQEC can approximately solve quadratically constrained binary optimization problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs over the probability simplex. Under an assumption on the error for the VQC to approximate an arbitrary probability mass function, we provide bounds on the optimality gap attained by a VQC. Numerical tests on a quantum simulator investigate the effect of various parameters and corroborate that VQEC can generate high-quality solutions.
The possibility of adopting quantum computing in the field of financial services can bring a new breakthrough in portfolio management since the optimization of this method requires large numbers for the solution that exceed the capabilities of classical computing algorithms. This study focuses on using a hybrid quantum-classical computational method known as the Variational Quantum Eigensolver (VQE) in optimizing portfolio selection. The VQE aims at finding the best allocation of the portfolio; the best weights that correspond to minimum risk and maximum return as predicted by the modern portfolio theory. The study assesses the suitability and speed of VQE in solving matters involving portfolio management tasks such as handling vast data and nonlinear constraints which are hard to solve through other optimization methods. Thus, under the theoretical framework of quantum computing, the study seeks to address some of the following challenges: computational time and scalability in portfolio optimization. The VQE algorithm work on different portfolio configurations; their performance in comparison with classical techniques; special attention is paid to the rate of convergence and the achievement of the optimal solution. Based on the results of this work we can conclude that there is a considerable literature on quantum finance and that this field can benefit from utilization of quantum algorithms as a tool to improve decision-making in the financial market. That is why the study proves that quantum optimization techniques, to wit VQE, can provide better solutions when it comes to portfolio management within the sphere of finance.