Fluid flow simulations marshal our most powerful computational resources. In many cases, even this is not enough. Quantum computers provide an opportunity to speed up traditional algorithms for flow simulations. We show that lattice-based mesoscale numerical methods can be executed as efficient quantum algorithms due to their statistical features. This approach revises a quantum algorithm for lattice gas automata to reduce classical computations and state preparation at every time step. For this, the algorithm approximates the qubit relative phases and subtracts them at the end of each time step. Phases are evaluated using the iterative phase estimation algorithm and subtracted using single-qubit rotation phase gates. This method optimizes the quantum resource required and makes it more appropriate for near-term quantum hardware. We also demonstrate how the checkerboard deficiency that the D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is validated by simulating two canonical partial differential equations: the diffusion and Burgers' equations on different quantum simulators. We find good agreement between quantum simulations and classical solutions for the presented algorithm.
Fluid flow simulations marshal our most powerful computational resources. In many cases, even this is not enough. Quantum computers provide an opportunity to speed up traditional algorithms for flow simulations. We show that lattice-based mesoscale numerical methods can be executed as efficient quantum algorithms due to their statistical features. This approach revises a quantum algorithm for lattice gas automata to reduce classical computations and state preparation at every time step. For this, the algorithm approximates the qubit relative phases and subtracts them at the end of each time step. Phases are evaluated using the iterative phase estimation algorithm and subtracted using single-qubit rotation phase gates. This method optimizes the quantum resource required and makes it more appropriate for near-term quantum hardware. We also demonstrate how the checkerboard deficiency that the D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is validated by simulating two canonical partial differential equations: the diffusion and Burgers' equations on different quantum simulators. We find good agreement between quantum simulations and classical solutions for the presented algorithm.
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.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.
The application of Quantum Computing (QC) to fluid dynamics simulation has developed into a dynamic research topic in recent years. With many flow problems of scientific and engineering interest requiring large computational resources, the potential of QC to speed-up simulations and facilitate more detailed modeling forms the main motivation for this growing research interest. Despite notable progress, many important challenges to creating quantum algorithms for fluid modeling remain. The key challenge of non-linearity of the governing equations in fluid modeling is investigated here in the context of lattice-based modeling of fluids. Quantum circuits for the D1Q3 (one-dimensional, three discrete velocities) Lattice Boltzmann model are detailed along with design trade-offs involving circuit width and depth. Then, the design is extended to a one-dimensional lattice model for the non-linear Burgers equation. To facilitate the evaluation of non-linear terms, the presented quantum circuits employ quantum computational basis encoding. The second part of this work introduces a novel, modular quantum-circuit implementation for non-linear terms in multi-dimensional lattice models. In particular, the evaluation of kinetic energy in two-dimensional models is detailed as the first step toward quantum circuits for the collision term of two- and three-dimensional Lattice Boltzmann methods. The quantum circuit analysis shows that with O(100) fault-tolerant qubits, meaningful proof-of-concept experiments could be performed in the near future.
The computational power to resolve large complex aerospace flow problems has increased over the past decades but is still yet not able to simulate these problems sufficiently accurate to a scale resolved level. At the same time, the interest in quantum computing has recently increased rapidly due to the massive improvements of the quantum computer hardware technology. Many fields of research explore the possible advantages that quantum computing can offer in their respective fields. For fluid flow problems, the exponentially increasing storage capacity of flow properties by quantum amplitude encoding offers an approach to efficiently deal with highly resolved flow fields. Within the DLR-QCI project ToQuaFlics, we adjust and extend the quantum lattice Boltzmann method (QLBM) from the literature for single time steps and present our extension to multiple time steps. The structure of our algorithm is shown in Fig. 1. Our extended QLBM can perform all time steps of a simulation without having to extract the quantum state or parts of it at any time. Therefore, with our algorithm there is no need to measure any qubit during or at the end of the simulation. This saves the computationally expensive read out and reinitialization after every time step and enables an efficient time stepping scheme. We verify our algorithm on 1D and 2D test cases simulating a linear advection-diffusion process with periodic boundary conditions. The algorithm is simulated using the IBM simulator package Qiskit with the shot method. For verification reasons, we measure the entire flow field at the end of the simulation to demonstrate the agreement with the reference solution given by the classic lattice Boltzmann solution. The results show excellent agreement and arbitrary convergence towards the reference. We further discuss the inherent amplitude decay of a QLBM time step and how that effects our solution in terms of signal loss of the quantum state. An outlook to include a body with a surface interaction model and to capture nonlinearities is given and briefly sketched.
An algorithm is presented to simulate fluid dynamics on a three qubit type II quantum computer: a lattice of small quantum computers that communicate classical information. The algorithm presented is called a three qubit factorized quantum lattice gas algorithm. It is modeled after classical lattice gas algorithms which move virtual particles along an imaginary lattice and change the particles' momentums using collision rules when they meet at a lattice node. Instead of moving particles, the quantum algorithm presented here moves probabilities, which interact via a unitary collision operator. Probabilities are determined using ensemble measurement and are moved with classical communications channels. The lattice node spacing is defined to be a microscopic scale length. A mesoscopic governing equation for the lattice is derived for the most general three qubit collision operator which preserves particle number. In the continuum limit of the lattice, a governing macroscopic partial differential equation-the diffusion equation-is derived for a particular collision operator using a Chapman-Enskog expansion. A numerical simulation of the algorithm is carried out on a conventional desktop computer and compared to the analytic solution of the diffusion equation. The simulation agrees very well with the known solution.