Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d+L (for dissipative equations) or d+2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M, while also showing potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.
Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d+L (for dissipative equations) or d+2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M, while also showing potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.
Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.
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.
Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction.
Abstract Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular—Schrödinger’s equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs—both classical and quantum—are digital (they must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrödingerisation, we show how to directly map D-dimensional linear PDEs onto a ( D + 1 ) -qumode quantum system where analog or continuous-variable (CV) Hamiltonian simulation on D + 1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker–Planck equation, Black–Scholes equations, wave equation and Maxwell’s equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or CV framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.
Abstract. We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the (constant and variable coefficient) backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation, which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit-based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost [Formula: see text] where [Formula: see text] is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any nonunitary dynamical system introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603].
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.