Decomposition of Nonlinear Collision Operator in Quantum Lattice Boltzmann Algorithm

Abstract: 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.