Multigrid has become a popular method for solving some of the most challenging real-world computational problems, such as computational fluid dynamics (CFD). The reason for this is the very good scaling properties of multigrid, which is often linear, or close to linear, with respect to problem size. In this paper a method is presented, which can be used to implement a quantum version of the multigrid algorithm. The method relies upon a quantum state that is maintained in a equal superposition throughout the calculation, and where information is encoded digitally in the qubits in a way more similar to a classical computer. This differs from many existing quantum algorithms where information is encoded in the amplitudes of the quantum states in the superposition. At the core of the method is an algorithm for sharing information between the states in the superposition. An exponential speedup is provided for classes of problems where the solution vector can be compressed efficiently, and where a quantum compiler can reduce the quantum circuit depth efficiently.
Multigrid has become a popular method for solving some of the most challenging real-world computational problems, such as computational fluid dynamics (CFD). The reason for this is the very good scaling properties of multigrid, which is often linear, or close to linear, with respect to problem size. In this paper a method is presented, which can be used to implement a quantum version of the multigrid algorithm. The method relies upon a quantum state that is maintained in a equal superposition throughout the calculation, and where information is encoded digitally in the qubits in a way more similar to a classical computer. This differs from many existing quantum algorithms where information is encoded in the amplitudes of the quantum states in the superposition. At the core of the method is an algorithm for sharing information between the states in the superposition. An exponential speedup is provided for classes of problems where the solution vector can be compressed efficiently, and where a quantum compiler can reduce the quantum circuit depth efficiently.
Quantum linear system algorithms (QLSAs) can provide exponential speedups for the solution of linear systems, but the growth of the condition number for finite element problems can eliminate the exponential speedup. QLSAs are also incapable of using an initial guess of a solution to improve upon it. To circumvent these issues, we present a Quantum Multigrid Algorithm (qMG) for the iterative solution of linear systems by applying the sequence of multigrid operations on a quantum state. Given an initial guess with error e_0, qMG can produce a vector encoding the entire sequence of multigrid iterates with the final iterate having a relative error e'=e/e_0, as a subspace of the final quantum state, with exponential advantage in O( poly log (N/e') ) time using O( poly log (N/e') ) qubits. Although extracting the final iterate from the sequence is efficient, extracting the sequence of iterates from the final quantum state can be inefficient. We provide an analysis of the complexity of the method along with numerical analysis.
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.
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 document is meant to review and discuss the possible applications of Quantum computing to computational fluid dynamics (CFD) problems.A review of the current state-of-the-art of quantum computing applied to computational fluid dynamics has been carried out, highlining how the technology is still in an early stage of development but very promising.Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations.The devices that perform quantum computations are known as quantum computers.Quantum Computing is currently a very active field of study and development, with expected applications in various fields, including engineering problems.Indeed, several approaches have been proposed for the solutions of CFD problems with the use of quantum algorithms and / or quantum computers.Possible approaches, applicable to compact heat exchangers used in aeronautics, include the development of quantum algorithms, e.g., based on the solutions of Navier Stokes equations or the Lattice Boltzmann Method., the design of an "analog" quantum computer able to mimic the thermal fluid system under investigation, and, finally, the implementation of a scientific machine learning approach.
Recent advances in quantum hardware and quantum computing algorithms promise significant breakthroughs in computational capabilities. Quantum computers can achieve exponential improvements in speed vs classical computers by employing principles of quantum mechanics like superposition and entanglement. However, designing quantum algorithms to solve the nonlinear partial differential equations governing fluid dynamics is challenging due to the inherent linearity of quantum mechanics, which requires unitary transformation. In this study, we first address in detail several challenges that arise when trying to deal with nonlinearity using quantum algorithms and then propose a novel pure quantum algorithm for solving a nonlinear Burgers' equation. We employed multiple copies of the state vector to calculate the nonlinear term, which is necessary due to the no-cloning theorem. By reusing qubits from the previous time steps, we significantly reduced the number of qubits required for multi-step simulations, from exponential/quadratic scaling in earlier studies to linear scaling in time in the current study. We also employed various advanced quantum techniques, including block-encoding, quantum Hadamard product, and the linear combination of unitaries, to design a quantum circuit for the proposed quantum algorithm. The quantum circuit was executed on quantum simulators, and the obtained results demonstrated excellent agreement with those from classical simulations.
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.