  We propose a quantum algorithm to tackle the quadratic nonlinearity in the
Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum
gates based on the particle distribution functions (PDF) within the coherence
time for qubits. Thus, both the operator and a state vector are linear
functions of PDFs, and upon quantum state evolution, the resulting PDFs will
have quadraticity. To this end, we decompose the collision operator for a
$DmQn$ lattice model into a product of $2(n+1)$ operators, where $n$ is the
number of lattice velocity directions. After decomposition, the $(n+1)$
operators with constant entries remain unchanged throughout the simulation,
whereas the remaining $(n+1)$ will be built based on the statevector of the
previous time step. Also, we show that such a decomposition is not unique.
Compared to the second-order Carleman-linearized LB, the present approach
reduces the circuit width by half and circuit depth by exponential order. The
proposed algorithm has been verified through the one-dimensional flow
discontinuity and two-dimensional Kolmogrov-like flow test cases.
