Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, Fernando G. S. L. Brandão
This chapter covers quantum algorithms for numerically solving differential equations and the areas of application where such capabilities might be useful, such as computational fluid dynamics, semiconductor chip design, and many engineering workflows. We focus mainly on algorithms for linear differential equations (covering both partial and ordinary linear differential equations), but we also mention the additional nuances that arise for nonlinear differential equations. We discuss important caveats related to both the data input and output aspects of an end-to-end differential equation solver, and we place these quantum methods in the context of existing classical methods currently in use for these problems.
This chapter covers quantum algorithms for numerically solving differential equations and the areas of application where such capabilities might be useful, such as computational fluid dynamics, semiconductor chip design, and many engineering workflows. We focus mainly on algorithms for linear differential equations (covering both partial and ordinary linear differential equations), but we also mention the additional nuances that arise for nonlinear differential equations. We discuss important caveats related to both the data input and output aspects of an end-to-end differential equation solver, and we place these quantum methods in the context of existing classical methods currently in use for these problems.
In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.
For quantum computers to become useful tools to physicists, engineers and computational scientists, quantum algorithms for solving nonlinear differential equations need to be developed. Despite recent advances, the quest for a solver that can integrate nonlinear dynamical systems with a quantum advantage, whilst being realisable on available (or near-term) quantum hardware, is an open challenge. In this paper, we propose to transform a nonlinear dynamical system into a linear system, which we integrate with quantum algorithms. Key to the method is the Fokker-Planck equation, which is a non-normal partial differential equation. Three integration strategies are proposed: (i) Forward-Euler stepping by unitary block encoding; (ii) Schroedingerisation, and (iii) Forward-Euler stepping by linear addition of unitaries. We emulate the integration of prototypical nonlinear systems with the proposed quantum solvers, and compare the output with the benchmark solutions of classical integrators. We find that classical and quantum outputs are in good agreement. This paper opens opportunities for solving nonlinear differential equations with quantum algorithms.
Quantum computers have the potential to solve some complex problems much faster than classical equivalents. Significant research efforts are aimed at determining which algorithms give which advantages for different use-cases. An application where quantum computers can possibly bring great advantages is solving (partial) differential equations, which has a broad set of applications, for instance in wave-propagation models. Few differential equations admit an analytical solution. For most heuristic methods are required to approximate a solution. Well-known heuristic techniques include the finite element and finite difference method, where the considered space is partitioned and systems of linear equations resulting from this partitioning need to be solved. Quantum computing puts forward new methods to solve these systems of linear equations and hence these differential equations. First, the HHL algorithm gives an efficient way to solve a linear system of equations. The HHL algorithm comes with drawbacks, but in this specific use-case some of these objections might be circumvented. Quantum computers furthermore offer the variational approach: an optimization-based path to solving differential equations. With this approach, the devices might even ‘learn’ the noise patterns emerging in present day quantum computers and compensate for it. In this work we revisit quantum methods for solving partial differential equations and consider how well they work in solving differential equations in practice. We also discuss possible caveats and bottlenecks of the approaches.
Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on classical, that is, silicon-based computer hardware, becomes prohibitive. Quantum computing, though still in its infancy, holds the promise of enabling a new generation of algorithms that can execute the most cost-demanding parts of PDE solvers up to exponentially faster than classical methods, at least theoretically. Also, increasing research and availability of quantum computing hardware spurs the hope of scientists and engineers to start using quantum computers for solving PDE problems much faster than classically possible. This work reviews the contributions that deal with the application of quantum algorithms to solve PDEs in structural mechanics. The aim is not only to discuss the theoretical possibility and extent of advantage for a given PDE, boundary conditions and input/output to the solver, but also to examine the hardware requirements of the methods proposed in literature.
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.
A large spectrum of problems in classical physics and engineering, such as turbulence, is governed by nonlinear differential equations, which typically require high-performance computing to be solved. Over the past decade, however, the growth of classical computing power has slowed down because the miniaturisation of chips has been approaching the atomic scale. This is marking an end to Moore's law, which calls for a new computing paradigm: Quantum computing is a prime candidate. In this paper, we offer a perspective on the current challenges that need to be overcome in order to use quantum computing for the simulation of nonlinear dynamics. We review and discuss progress in the development of both quantum algorithms for nonlinear equations and quantum hardware. We propose pairings between quantum algorithms for nonlinear equations and quantum hardware concepts. These avenues open new opportunities for the simulation of nonlinear systems and turbulence.