  In the present contribution, we propose a quantum computational algorithm for
the Lattice Boltzmann Method (LBM) to solve fluid flow equations in the low
Reynolds number ($Re$) regime. Firstly, we express the LBM collision and
streaming operators in matrix form. Since quantum logic gates are typically
expressed as unitary matrices, we first decompose LBM operations as a product
of unitaries. The particle distribution functions (PDFs) of LBM are encoded as
probability amplitudes of the quantum state. We have observed that the
amplitudes in the state vector (SV) can be affected: (i) by the choice of
encoding the PDFs during the quantum state preparation or (ii) when collision
is followed by streaming, as in classical LBM implementation. In the first
case, we show that the ancilla qubit must be in superposition with the compute
qubits during the quantum state preparation. The superposition allows the SV to
utilize the increased Hilbert space offered by the ancilla qubit rather than
placing the ancilla in a separate register, which restricts the space of
possible outcomes. Next, we show that the second issue can be resolved by
having an intermediate Hadamard gate before the streaming operation. The
proposed algorithm has been tested through typical benchmark problems like
advection-diffusion of a Gaussian hill, Poiseuille flow, Couette flow, and the
lid-driven cavity problem. The results are validated with the respective
analytic or reference solutions. Translating the unitaries into quantum gates
(circuit synthesis) presents a primary challenge, as a unitary matrix can be
decomposed in multiple ways. We report on the CNOT and U gate counts obtained
for the test cases with the range of qubits from 9 to 12. Although the gate
count closely agrees with the theoretical limit, the number of two qubit gates
is in the $O(10^7)$ prompts special attention to circuit synthesis.
