The numerical solution of partial differential equations by discretization techniques is ubiquitous in computational physics. In this work we investigate this approach in the quantum realm by solving the heat equation for a square plate subject to fixed temperatures at the edges and random heat sources and sinks within the domain. The hybrid classical-quantum approach consists of the solution on a quantum computer of the coupled linear system of equations that results from the discretization step. Owing to the limitations on the number of qubits and their connectivity, we use the Gauss-Seidel method to divide the full system of linear equations into subsystems, which are solved iteratively in block fashion. Each of the linear subsystems was solved using 2000Q and Advantage quantum computers developed by D-Wave Systems Inc. By comparing classical numerical and quantum solutions, we observe that the errors and chain break fraction are, on average, greater on the 2000Q system. Unlike in the classical Gauss-Seidel method, the errors of the quantum solutions level off after a few iterations of our algorithm. This is partly a result of the span of the real-number line available from the mapping of the chosen size of the set of qubit states. We verified this by using techniques to progressively shrink the range mapped by the set of qubit states at each iteration (increasing floating-point accuracy). As a result, no leveling off is observed. However, an increase in qubits does not translate to an overall lower error. This is believed to be indicative of the increasing length of chains required for the mapping to real numbers and the ensuing limitations of hardware.
The numerical solution of partial differential equations by discretization techniques is ubiquitous in computational physics. In this work we investigate this approach in the quantum realm by solving the heat equation for a square plate subject to fixed temperatures at the edges and random heat sources and sinks within the domain. The hybrid classical-quantum approach consists of the solution on a quantum computer of the coupled linear system of equations that results from the discretization step. Owing to the limitations on the number of qubits and their connectivity, we use the Gauss-Seidel method to divide the full system of linear equations into subsystems, which are solved iteratively in block fashion. Each of the linear subsystems was solved using 2000Q and Advantage quantum computers developed by D-Wave Systems Inc. By comparing classical numerical and quantum solutions, we observe that the errors and chain break fraction are, on average, greater on the 2000Q system. Unlike in the classical Gauss-Seidel method, the errors of the quantum solutions level off after a few iterations of our algorithm. This is partly a result of the span of the real-number line available from the mapping of the chosen size of the set of qubit states. We verified this by using techniques to progressively shrink the range mapped by the set of qubit states at each iteration (increasing floating-point accuracy). As a result, no leveling off is observed. However, an increase in qubits does not translate to an overall lower error. This is believed to be indicative of the increasing length of chains required for the mapping to real numbers and the ensuing limitations of hardware.
The numerical solution of partial differential equations (PDEs) is essential in computational physics. Over the past few decades, various quantum-based methods have been developed to formulate and solve PDEs. Solving PDEs incur high time complexity for real-world problems with high dimensions, and using traditional methods becomes practically inefficient. This paper presents a fast hybrid classical-quantum paradigm based on successive over-relaxation (SOR) to accelerate solving PDEs. Using the discretization method, this approach reduces the PDE solution to solving a system of linear equations, which is then addressed using the block SOR method. Due to limitations in the number of qubits, the block SOR method is employed, where the entire system of linear equations is decomposed into smaller subsystems. These subsystems are iteratively solved block-wise using Advantage quantum computers developed by D-Wave Systems, and the solutions are subsequently combined to obtain the overall solution. The performance of the proposed method is evaluated by solving the heat equation for a square plate with fixed boundary temperatures and comparing the results with the best existing method. Experimental results show that the proposed method can accelerate the solution of high-dimensional PDEs by using a limited number of qubits up to 2 times the existing method.
The numerical solution of partial differential equations (PDEs) is essential in computational physics. Over the past few decades, various quantum-based methods have been developed to formulate and solve PDEs. Solving PDEs incurs high-time complexity for high-dimensional real-world problems, and using traditional methods becomes practically inefficient. This paper presents a fast hybrid classical-quantum paradigm based on successive over-relaxation (SOR) to accelerate solving PDEs. Using the discretization method, this approach reduces the PDE solution to solving a system of linear equations, which is then addressed using the block SOR method. The block SOR method is employed to address qubit limitations, where the entire system of linear equations is decomposed into smaller subsystems. These subsystems are iteratively solved block-wise using Advantage quantum computers developed by D-Wave Systems, and the solutions are subsequently combined to obtain the overall solution. The performance of the proposed method is evaluated by solving the heat equation for a square plate with fixed boundary temperatures and comparing the results with the best existing method. Experimental results show that the proposed method can accelerate the solution of high-dimensional PDEs by using a limited number of qubits up to 2 times the existing method.
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.
This paper presents a quantum algorithm for solving the one-dimensional heat equation with Dirichlet boundary conditions. The algorithm utilizes discretization techniques and employs quantum gates to emulate the heat propagation operator. Central to the algorithm is the Trotter–Suzuki decomposition, enabling the simulation of the time evolution of the temperature distribution. The initial temperature distribution is encoded into quantum states, and the evolution of these states is driven by quantum gates tailored to mimic the heat propagation process. As per the literature, quantum algorithms exhibit an exponential computational speedup with increasing qubit counts, albeit facing challenges such as exponential growth in relative error and cost functions. This study addresses these challenges by assessing the potential impact of quantum simulations on heat conduction modeling. Simulation outcomes across various quantum devices, including simulators and real quantum computers, demonstrate a decrease in the relative error with an increasing number of qubits. Notably, simulators like the simulator_statevector exhibit lower relative errors compared to the ibmq_qasm_simulator and ibm_osaka. The proposed approach underscores the broader applicability of quantum computing in physical systems modeling, particularly in advancing heat conductivity analysis methods. Through its innovative approach, this study contributes to enhancing modeling accuracy and efficiency in heat conduction simulations across diverse domains.
Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.