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Volume 33, Issue 2
Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics

Bofeng Liu, Lixing Zhu, Zixuan Yang & Guowei He

Commun. Comput. Phys., 33 (2023), pp. 425-451.

Published online: 2023-03

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  • Abstract

We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.

  • AMS Subject Headings

68Q12, 76M20

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COPYRIGHT: © Global Science Press

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@Article{CiCP-33-425, author = {Liu , BofengZhu , LixingYang , Zixuan and He , Guowei}, title = {Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {2}, pages = {425--451}, abstract = {

We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0081}, url = {http://global-sci.org/intro/article_detail/cicp/21494.html} }
TY - JOUR T1 - Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics AU - Liu , Bofeng AU - Zhu , Lixing AU - Yang , Zixuan AU - He , Guowei JO - Communications in Computational Physics VL - 2 SP - 425 EP - 451 PY - 2023 DA - 2023/03 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0081 UR - https://global-sci.org/intro/article_detail/cicp/21494.html KW - Quantum computing, partial differential equations, computational fluid dynamics, finite difference, finite element. AB -

We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.

Bofeng Liu, Lixing Zhu, Zixuan Yang & Guowei He. (2023). Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics. Communications in Computational Physics. 33 (2). 425-451. doi:10.4208/cicp.OA-2022-0081
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