Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.
Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.
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.
Boundary value problems (BVPs) arise in various scientific and engineering disciplines, including physics, finance, and biology. Classical computers may not be able to solve these problems quickly or accurately due to their computational complexity. Quantum Fourier transform (QFT) may efficiently solve differential equations including the Schrödinger equation and Helmholtz equation, according to recent quantum computing research. Quantum annealing determines a Hamiltonian's ground state using quantum fluctuation. The quantum computing BVP approach is novel to numerical methods. Quantum methods like quantum phase estimation have not been employed to solve BVPs in heat transfer, groundwater modelling, or advection diffusion response. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm solves steady-state heat and radon-diffusion equations in this research study. The BVP linear operator eigenvalues are computed using the quantum phase estimation approach. We used IBM's 'Qiskit' package to compute in Python. IBM Qiskit's 'Linear Solver' program provided the quantum computing numerical solution. Tridiagonal Toeplitz matrix role is shown. Comparisons with traditional results and future applications were made.
Near-term quantum algorithms are a promising route to solving partial differential equations, but gauging their true potential requires separating algorithmic performance from sampling and hardware noise. We benchmark a ground-state variational quantum eigensolver (VQE), cast as a variational quantum linear solver, against the Trotterization, variational quantum imaginary time evolution, and adaptive variational quantum dynamics simulation methods applied to the one-dimensional advection-diffusion equation in the recent quantum-dynamics study by Alipanah et al. [Phys. Rev. Res. 7, 043318 (2025)2643-156410.1103/ndc3-bdwt] at matched grid and problem size. On a noiseless state-vector simulator the N=4 VQE drives the final-time infidelity to a numerical floor (∼10^{-14}) once the depth reaches L≈5, an algorithmic ceiling set by exact expectation values. Evaluating the same solver with a finite number S of measurement shots, still without hardware noise, makes the infidelity sampling limited, following 1-f≈c/S (a best-case readout-sampling estimate, with the solution's signs assumed known), providing a regime-matched comparison with the shot-based emulator of Alipanah et al. and explaining the gap to their noisy hardware runs (>10^{-1}). The benchmark thus decomposes the near-term error budget into algorithmic, sampling, and hardware contributions, with a matched-depth resource comparison. The formulation applies without modification across N=4,5,6 qubits and to a two-dimensional (eight-qubit, 16×16) problem evolved to t=1, where the state-vector VQE holds a ∼10^{-7} algorithmic-ceiling infidelity against the sampling-limited ∼10^{-5} of the corresponding shot-based simulation, a difference of measurement regime rather than algorithmic superiority.
A fully quantum algorithm for solving the one-dimensional linear advection-diffusion equation using the Lattice Boltzmann method as a numerical procedure is presented in this work. We start by presenting a state of the art of the current usage of quantum algorithms for solving ordinary and partial differential equations. We then describe two algorithms for the one-dimensional Lattice Boltzmann method with two degrees of freedom. The first one is an existing hybrid quantum-classical algorithm with measurements at each time step, and the second one is our improved version, viz. a fully quantum algorithm where only one measurement is needed at the end of the algorithm. The fully quantum algorithm is first executed on a quantum simulator and then compared with a classical approach. Subsequently, the fully quantum algorithm is run on a quantum system with 133 qubits to investigate the effect of noise and the depth of the circuit on the output state. We find fluctuations in the final result due to the decoherence noise of the qubits.
Time-fractional diffusion equations have emerged as powerful models for describing anomalous transport phenomena in physics, biology and engineering. To address the computational challenges arising from their non-local operators, we employ the WEB-spline finite element method, which provides a flexible and accurate discretization framework. The resulting linear system of equations are then explored in the context of quantum computing. Specifically, we investigate three prominent quantum linear solvers: the variational quantum linear solver (VQLS), the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum annealing (QA). VQLS leverages hybrid variational techniques and shallow circuits, making it well-suited for noisy intermediate-scale quantum (NISQ) devices, while HHL offers a theoretically exponential speedup for sparse systems but requires deep fault-tolerant circuits. QA, in contrast, reformulates the problem into a quadratic unconstrained binary optimization (QUBO) instance, enabling approximate solutions through energy minimization on specialized hardware. We present a comparative analysis in terms of circuit depth, noise resilience, scalability and solution extraction, including the role of quantum state tomography in reconstructing classical information. Numerical experiments on a time-fractional diffusion problem highlight the complementary strengths and limitations of each method. This study bridges advanced numerical discretization with emerging quantum algorithms, providing insights into the feasibility and future potential of quantum-enhanced solvers for fractional partial differential equations.
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it restricts the mesh size since the solution of the linear system becomes computationally intractable when the mesh resolution increases. The reliance on the ability of High-Performance Computers (HPC) to scale up and meet these requirements is myopic; such very high-fidelity simulations require a paradigm shift in computing. This paper presents an exploration of quantum methodologies aimed at achieving high accuracy in solving such a large system of equations. Leveraging recent works in Quantum Linear Solver Algorithms (QLSA) and variational algorithms suitable for Quantum Simulation in HPC, we aspire to push the boundaries of CFD-relevant problems that can be solved on hybrid quantum-classical framework. To this end, we consider the 2D, transient, incompressible, viscous, non-linear coupled Burgers equation as a test problem and investigate the accuracy of our approach by comparing results with a classical linear system of equation solvers, such as the Generalized Minimal RESidual method (GMRES). Through rigorous testing, our findings demonstrate that our quantum methods yield results comparable in accuracy to traditional approaches. Additionally, we demonstrate the accuracy, scalability, and consistency of our quantum method. Lastly, we present an insightful estimation of the resources our quantum algorithm needs to solve systems with nearly 2 billion mesh points.