Quantum Algorithm for the Lattice Boltzmann Method of Fluid Simulation
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We propose a quantum algorithm for solving physical problems represented by the lattice Boltzmann formulation. Specifically, we deal with the case of a single, incompressible fluid with the Bhatnagar Gross and Krook (BGK) equilibrium function. We use the framework introduced by Kowalski that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the bosonic Fock space of the bosons. We write a quantized version of the Hamiltonian describing the collision step of LB, with a corresponding non-unitary dynamics, and implement the non-unitary evolution by a linear combination of unitary operators. We use the compact mapping of the bosonic modes to qubits that uses a number of qubits which scales logarithmically with the size of truncated bosonic Fock space. We use the results of Todorova & Steijl for the streaming step. The error and complexity scaling are discussing. The resulting algorithm could be readily adjusted to account for the different physics which could be encompassed by lattice methods.
