Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs.
Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs.
Quantum computing has the potential to transform computational problem-solving by leveraging quantum mechanical principles of superposition and entanglement. This capability is particularly important for the numerical solution of complex and/or multidimensional partial differential equations (PDEs). There are currently many quantum techniques available for solving PDEs, which are mainly based on variational quantum algorithms (VQAs). However, the existing quantum PDE solvers, particularly those based on VQAs suffer from limitations such as low accuracy, high execution times, and low scalability on both simulators as well as on real quantum devices. In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated our proposed algorithm by solving several practically useful PDEs such as Poisson, heat, Black-Scholes, and Navier-Stokes equations. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQA-based solvers on noise-free and noisy quantum simulators from IBM and achieved promising results on real quantum hardware.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
Quantum computing has the potential to revolutionize computational problem-solving by leveraging the quantum mechanical phenomena such as superposition and entanglement, enabling fundamentally different computational strategies compared to classical computing. Classical methods for solving complex and high dimensional partial differential equations (PDEs), crucial for modeling diverse physical phenomena, suffer from an exponential growth in computational complexity known as the curse of dimensionality. This can limit their effectiveness for certain high-resolution simulations and real-time computational applications. In contrast, quantum computing inherently exploits superposition and entanglement to efficiently represent and simultaneously manipulate large-scale discretized PDEs, using exponentially fewer resources compared to classical systems. There are currently many quantum techniques available for solving partial differential equations (PDEs), which are mainly based on variational quantum circuits. However, the existing quantum PDE solvers, particularly those based on variational quantum eigensolver (VQE) techniques, suffer from several limitations. These include low accuracy, high execution times, and low scalability on quantum simulators as well as on noisy intermediate-scale quantum (NISQ) devices, especially for multidimensional PDEs. In this work, we propose an efficient and scalable algorithm for solving multidimensional PDEs. We present two variants of our algorithm: the first leverages finite-difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, while the second employs FDM, C2Q, and column-by-column decomposition (CCD). Both variants are designed to enhance accuracy and scalability while reducing execution times. We have validated and evaluated our proposed concepts using a number of case studies including multidimensional Poisson equation, multidimensional heat equation, Black Scholes equation for financial option pricing, and Navier-Stokes equation for computational fluid dynamics (CFD), achieving promising results. Our results demonstrate higher accuracy, higher scalability, and faster execution times compared to VQE-based solvers on noise-free and noisy quantum simulators from IBM. Additionally, we validated our approach on hardware emulators and actual quantum hardware, employing noise mitigation techniques. This work establishes a practical and effective approach for solving PDEs using quantum computing for engineering and scientific applications.
Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate-scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE---the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices---especially when considering large system sizes and more complicated nonlinear PDEs that are required in order to be competitive with classical high-end solvers.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
This paper proposes a variational quantum algorithm to address the large-scale and low computational efficiency issues of high-dimensional partial differential equations (PDEs) in climate science. In this method, parameterized quantum circuits encode the discrete solutions of PDEs into quantum states. Then, through repeated quantum measurements, we can obtain the expectation values of the operators. By iteratively adjusting the circuit parameters, a hybrid quantum-classical optimization loop can reduce the error in the solution. The Schrödinger equation and the Poisson equation are benchmark partial differential equations for numerical experiments, and they are four-dimensional and five-dimensional, respectively. According to the above results, the convergence speed is relatively fast; after 100 iterations, the median relative errors of all test cases are less than 1.6% and 3.1%, respectively, with the final target loss ranging between 0.014 and 0.045. The above comparison indicates that in high-dimensional and ill-posed problems, quantum solvers can achieve the same or better accuracy and robustness compared to traditional methods. Resource analysis indicates that quantum algorithms use less memory and fewer sub-exponential growth gates and runtime in larger systems. According to the above findings, variational quantum solvers can be used for complex scientific computing problems. In addition, they hope that future climate models will be more accurate and efficient.