For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.
For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.
Quantum computing presents a possible paradigm shift in computing, given its advantages in memory and speed. However, there is a growing need to demonstrate its utility in solving practical problems that are nonlinear, such as in fluid dynamics, which is the subject of this work. To facilitate this objective, it is essential to have a dedicated toolkit that enables the development, testing, and simulation of new quantum algorithms and flow problems, taken together. To this end, we present here a high performance, quantum computational simulation package called Quantum Flow Simulator (QFlowS), designed for computational fluid dynamics simulations. QFlowS is a versatile tool that can create and simulate quantum circuits using an in-built library of fundamental quantum gates and operations. We outline here all its functionalities with illustrations. Algorithms to solve flow problems can be built using the expanding list of the core functionalities of QFlowS with its hybrid quantum–classical type workflow. This is demonstrated here by solving an example, one-dimensional, diffusion flow problem. These simulations serve as a check on the algorithm's correctness as well as an ideal test-bed for making them more efficient and better suited for near-term quantum computers for addressing flow problems.
Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, nonlinear PDEs. We embed the PDEs in a truncated, high-dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods with a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of the time-marching system size, simulation time, accuracy parameters, and post-selection parameters. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts, and query complexity. A physically motivated measure of nonlinearity is connected to a parameter similar to the flow Reynolds number ReH, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of Burgers, Fisher--KPP, damped Kuramoto--Sivashinsky, and real Ginzburg--Landau/Allen--Cahn equations. Together, these examples encompass cubic nonlinearity and saturation, quadratic transport, and fourth-order dissipative stiffness. This work shows the potential of hybrid quantum algorithms for simulating nonlinear problems on near-term and fault-tolerant devices.
Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension D at Kolmogorov microscale resolution with O(Re43(1+2D)×qM), where qM is a multiplicative overhead for data extraction with qM=O(Re83) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re83D). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM) for D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
Recent advancements in quantum computing provide opportunities to address the challenges of computational resource limitations in computational fluid dynamics (CFD). This work presents a hybrid quantum-classical CFD framework that leverages quantum linear algorithms to practical flow simulations. By transforming nonlinear fluid dynamics problems into linear systems, the framework employs quantum linear algorithms to compute solutions, effectively integrating quantum and classical computing approaches. Furthermore, this framework employs subspace methods to map the original large-scale linear systems to small ones solved by quantum linear algorithms, thereby enabling the solution of large-scale problems using the currently limited quantum resources. We applied the famous Harrow–Hassidim–Lloyd (HHL) algorithm and variational quantum linear solver to simulate complex flows, including unsteady flows around a cylinder, simulation of an aircraft with turbulence model, and combustion flows. The performance and quantum resource consumption of these algorithms were evaluated in practical flow scenarios, demonstrating their effectiveness with an average relative error lower than 0.001%. We conducted quantum resource and convergence analyses to adapt this framework for use with near-term quantum computers. Our framework supports simulations of up to 9.4 × 106 grid cells, marking a significant advancement from previous quantum approaches limited to simple flows. This paper not only offers a way for employing quantum linear algorithms in solving complex flow problems but also provides insights into optimizing quantum algorithms for CFD applications, pushing forward the practical utilization of quantum CFD.
As the energy sector transitions towards more sustainable forms of energy production, classical Computational Fluid Dynamics (CFD) increasingly struggles with the bottle- necks of simulating high complexity turbulent flows. While quantum computing (QC) theoretically offers an exponential speedup for large-scale linear systems, current Noisy Intermediate-Scale Quantum (NISQ) hardware lacks the error correction required for deep and complex circuits that mathematically ideal algorithms require. Furthermore, the inherent linearity of quantum mechanics seriously impedes the solving non-linear partial differential equations (PDEs), which are critical for fluid modeling. To bridge thisgap, thisthesisevaluatestheVariationalQuantumLinearSolver(VQLS),a shallow- circuit, hybrid algorithm, applied to the 1D viscous Burgers’ equation while utilizing the Cole-Hopf transformation to analytically linearize the equation. In both simulated and actual hardware, the VQLS algorithm is implemented across increasingly scaled topographies (2, 4, and 8 qubits). This work intends to identify the limitations of implementing quantum fluid dynamics in near-term quantum hardware.
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.