Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
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.
We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the d-dimensional fractional equation is reformulated as a local partial differential equation in d+1 dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schrödingerization approach from \cite{JLY22SchrShort, JLY22SchrLong, analogPDE}. The Schrödingerization technique transforms general linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage -- with respect to the inverse of the mesh size in high dimensions -- compared to its classical counterpart. Specifically, while the classical method requires O(d1/233d/2h−d−2) operations, the quantum counterpart requires O(d33d/2h−2.5) queries to the block-encoding input models, with the quantum complexity being independent of the dimension d in terms of the inverse mesh size h−1. Numerical experiments are conducted to verify the validity of our formulation.
We focus on the electromagnetic fluid dynamics governing plasmas and develop a quantum algorithm for nonlinear partial differential equations. Specifically, we apply Koopman-von Neumann (KvN) linearization to transform the electromagnetic fluid dynamics into a Schrödinger equation. The time evolution of this system is then computed using Hamiltonian simulation via Quantum Singular Value Transformation (QSVT). The proposed quantum algorithm has the potential to significantly reduce computational complexity compared to an ideal classical high-precision scheme, which scales asO(Nx4)whereNxis the number of grid points per direction. By leveraging quantum computation, this complexity is reduced toO(NxlogNx)offering a potential advantage in large-scale simulations. To assess the feasibility and performance of quantum computing in plasma simulations, we implement the proposed quantum algorithm and conduct numerical validations. Specifically, we present results for two test cases: one with a linear solution and the other with a nonlinear solution. As a result, the behavior of composite numerical errors arising from discretization, KvN, and QSVT is elucidated, demonstrating the effectiveness of the proposed quantum algorithm. The novelty of this study lies in the application of a promising quantum algorithm for nonlinear partial differential equations to plasma simulations, along with a concrete numerical evaluation of its computational performance. Furthermore, this work presents a novel direction by identifying a nonlinear, high- degree-of-freedom physical system that can be efficiently solved by a quantum algorithm under theoretical guarantees.
We present a quantum preconditioning framework for solving linear systems arising from a finite difference discretization of the Poisson equation. It is based on the combination of the Schrödingerization technique \cite{JLY22b,JLYPRL24} and the BPX multilevel preconditioner in order to achieve near-optimal complexity. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems with unitary evolution in one higher dimension, making them suitable for quantum simulation. A key contribution is a structure-aware construction of the block-encoding for the symmetrically preconditioned matrix AS=S⊤AS, where A is the stiffness matrix and S encodes the BPX preconditioner in factored form. By establishing a novel commuting identity, we avoid the unfavorable normalization scaling that would otherwise arise from naive multiplication of block-encodings. This yields an exact block-encoding of AS with normalization O(d2(L+1)), where d is the spatial dimension and L is the number of levels. Combined with the Schrödingerization-based Hamiltonian simulation, the overall quantum algorithm achieves a query complexity of O(poly(d)ε−1polylog(ε−1)) for estimating linear functionals of the solution to a given tolerance ε.
Abstract Quantum methods are proposed for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, the Jacobi iteration is implemented on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, quantum methods are extended to Gauss‐Seidel iterative methods. Additionally, a quantum‐suitable resolvent decomposition is proposed based on the Woodbury identity. From a technical perspective, tools are developed and utilized for the block encoding of specific matrices as well as their multiplication. The approach is benchmarked on paradigmatic fluid dynamics problems. These results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid‐type computations and leverage corresponding advances in scientific computing.
Abstract Solving computational fluid dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion (Gilyén et al 2019 Proc. 51st Annual ACM SIGACT Symp. on Theory of Computing 193–204). However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing. In this work we propose a scheme for measuring the output of a quantum singular value transform (QSVT) matrix inversion algorithm specifically for the CFD use case. We use a quantum signal processing based amplitude estimation algorithm (Rall and Fuller 2023 Quantum 7 937) and show how it can be combined with the QSVT matrix inversion algorithm. We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction. We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focussing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.