Abstract
We explore the Carleman linearization of the collision term of the lattice Boltzmann formulation, as a first step towards formulating a quantum lattice Boltzmann algorithm. Specifically, we deal with the case of a single, incompressible fluid with the Bhatnagar Gross and Krook equilibrium function. Under this assumption, the error in the velocities is proportional to the square of the Mach number. Then, we showcase the Carleman linearization technique for the system under study. We compute an upper bound to the number of variables as a function of the order of the Carleman linearization. We study both collision and streaming steps of the lattice Boltzmann formulation under Carleman linearization. We analytically show why linearizing the collision step sacrifices the exactness of streaming in lattice Boltzmann, while also contributing to the blow up in the number of Carleman variables in the classical algorithm. The error arising from Carleman linearization has been shown analytically and numerically to improve exponentially with the Carleman linearization order. This bodes well for the development of a corresponding quantum computing algorithm based on the lattice Boltzmann equation.
Highlights
Computational Fluid Dynamics (CFD) has accompanied computers as an application since their infancy, starting with von Neumann’s program to simulate the weather on the ENIAC machine around the 1950s
We explore the Carleman linearization of the collision term of the lattice Boltzmann formulation, as a first step towards formulating a quantum lattice Boltzmann algorithm
We present a brief survey of current ongoing research work in this direction and a preparatory technique, known as Carleman linearization, aimed at the development of a quantum computing algorithm for the lattice Boltzmann method for fluid flows
Summary
Computational Fluid Dynamics (CFD) has accompanied computers as an application since their infancy, starting with von Neumann’s program to simulate the weather on the ENIAC machine around the 1950s. This has spanned sixteen orders of magnitude in 70 years, close to a sustained Moore’s law rate, doubling every 1.5 years! CFD has been consistently on the forefront of the journey, and it continues to be to the present day. When it comes to quantum computing, CFD is yet to capture the limelight it deserves. We present a brief survey of current ongoing research work in this direction and a preparatory technique, known as Carleman linearization, aimed at the development of a quantum computing algorithm for the lattice Boltzmann method for fluid flows
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