Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large computational models is essential for improving product performance and reducing development costs. However, solving large-scale PDEs, particularly for systems with spatially varying material properties, poses significant computational challenges. In this paper, we propose a quantum algorithm for solving second-order linear PDEs of nonconservative systems with spatially varying parameters, using the linear combination of the Hamiltonian simulation (LCHS) method. Our approach transforms those PDEs into ordinary differential equations represented by qubit operators, through spatial discretization using the finite-difference method. Then, we provide an algorithm that efficiently constructs the operator corresponding to the spatially varying parameters of PDEs via a logic minimization technique, which reduces the number of terms and subsequently the circuit depth. We also develop a scalable method for realizing a quantum circuit for LCHS, using a tensor-network-based technique, specifically a matrix product state (MPS). We validate our method with applications to the acoustic equation with spatially varying parameters and the dissipative heat equation. Our approach includes a detailed recipe for constructing quantum circuits for PDEs, leveraging efficient encoding of spatially varying parameters of PDEs and scalable implementation of LCHS, which we believe marks a significant step towards advancing quantum computing's role in solving practical engineering problems.
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large computational models is essential for improving product performance and reducing development costs. However, solving large-scale PDEs, particularly for systems with spatially varying material properties, poses significant computational challenges. In this paper, we propose a quantum algorithm for solving second-order linear PDEs of nonconservative systems with spatially varying parameters, using the linear combination of the Hamiltonian simulation (LCHS) method. Our approach transforms those PDEs into ordinary differential equations represented by qubit operators, through spatial discretization using the finite-difference method. Then, we provide an algorithm that efficiently constructs the operator corresponding to the spatially varying parameters of PDEs via a logic minimization technique, which reduces the number of terms and subsequently the circuit depth. We also develop a scalable method for realizing a quantum circuit for LCHS, using a tensor-network-based technique, specifically a matrix product state (MPS). We validate our method with applications to the acoustic equation with spatially varying parameters and the dissipative heat equation. Our approach includes a detailed recipe for constructing quantum circuits for PDEs, leveraging efficient encoding of spatially varying parameters of PDEs and scalable implementation of LCHS, which we believe marks a significant step towards advancing quantum computing's role in solving practical engineering problems.
Partial differential equations (PDEs) play an important role in engineering since various physical phenomena can be well described by PDEs. The key problems in solving PDEs are memory usage and computational time when the size of the system of interest grows. Given the potential of quantum computing, we propose a quantum algorithm for solving PDEs based on the Hamiltonian simulation, which is well known as the possible application of quantum computing for physical simulation. First, we review the method of linear combination of Hamiltonian simulation and propose its implementation via a tensor network technique. Second, we discuss how to deal with the spatially varying physical constants via a logic minimizer technique. Finally, we provide numerical experiments to demonstrate our proposed method.
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 methods for solving partial differential equations (PDEs) generally fall into a few general categories, each leveraging different quantum mechanisms to achieve potential speedups. One well-known approach is the quantum linear system solver, epitomized by the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses phase estimation and eigenvalue inversion to solve large, sparse linear systems arising from discretized linear PDEs [1]. This method can yield exponential improvements over classical solvers under certain conditions (mostly when the matrix is well-conditioned and sparse). More recent developments, such as Quantum Singular Value Transformation (QSVT), provide a framework for matrix inversion in a more flexible way, often improving on the cost and error dependence of earlier solvers [2]. Time-dependent PDEs can be tackled via Hamiltonian simulations. The Trotter-Suzuki method decomposes the exponential of a Hamiltonian into products of exponentials of simpler sub-Hamiltonians, allowing a piecewise approximation of continuous dynamics [3]. Continuous-time quantum walks, which treat a PDE operator as a Hamiltonian that governs the walk, similarly simulate processes like diffusion or wave propagation on discretized grids or graphs [4]. Beyond direct simulation, variational quantum algorithms have emerged as a promising route in the noisy intermediate-scale quantum era. These approaches encode the PDE solution as a parameterized quantum state and optimize the parameters via classical feedback, thereby minimizing a cost function tied to the PDE residual or energy functional [5]. Quantum annealing or adiabatic methods, meanwhile, can solve optimization problems derived from PDE constraints by adiabatically evolving a simple initial Hamiltonian to the final problem Hamiltonian and staying in the system's ground state [3]. In our study, we implement a new parameterized quantum model, specifically designed to solve large linear systems on quantum hardware. This model is less deep and computationally faster than the proposed one in [5], reducing the complexity and speeding up convergence times. Such improvements are particularly valuable in cases where classical algorithms may fail due to the problem's size or complexity. Ongoing research is focusing on generalized quantum models that can be adapted to practical PDE problems for solving large electromagnetic problems where classical computers can fail.
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.
Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a potential and promising approach to achieve this purpose. Actually, there are several oracle-based Hamiltonian simulations with potential quantum speedup, but their detailed implementations and accordingly the detailed computational complexities are all unclear. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation; the key technique is the explicit gate construction of differential operators contained in the target partial differential equation discretized by the finite difference method. Moreover, we show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms. We also provide numerical experiments and an experiment on a real device for the wave equation to demonstrate the validity of our proposed method. Published by the American Physical Society 2024
One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for solving PDEs. Furthermore, the algorithm is extended to the multidimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.