Extensions of the Linear Embedding Quantum Algorithm for Lattice Boltzmann Fluid Simulation

We advance our prior work on a quantum algorithm for the lattice Boltzmann scheme of an incompressible, single-phase fluid. The work is based on the linearization approach due to Kowalski by noting the equivalence between classical orthogonal polynomials and bosonic operators in Hilbert space for describing the evolution of a nonlinear system. We compare embedding variables in a bosonic Fock space with those in a generic binary representation. We discuss the favorable aspects of the lattice Boltzmann formulation, as well as the issue of normalization, and present numerical results of the linearized matrix-embedding of the collision operator, acting on normalized and unnormalized state vectors. We introduce an approach to achieving non-unitary evolution, utilizing additional bosonic modes in the Fock space, benefitting from the nature of nonlinearity in the lattice Boltzmann equations, and present it as an alternative to the linear combination of unitaries that utilize additional ancilla qubits. We discuss the limitations of such an approach.