Quantum collision circuit, quantum invariants and quantum phase estimation procedure for fluid dynamic lattice gas automata

Lattice Gas Cellular Automata (LGCA) is a classical numerical method widely known and applied to simulate several physical phenomena. In this paper, we study the translation of LGCA on quantum computers (QC) using computational basis encoding (CBE), developing methods for different purposes. In particular, we clarify and discuss some fundamental limitations and advantages in using CBE and quantum walk as streaming procedure. Using quantum walks affect the possible encoding of classical states in quantum orthogonal states, feature linked to the unitarity of collision and to the possibility of getting a quantum advantage. Then, we give efficient procedures for optimizing collisional quantum circuits, based on the classical features of the model. This is applied specifically to fluid dynamic LGCA. Alongside, a new collision circuit for a 1-dimensional model is proposed. We address the important point of invariants in LGCA providing a method for finding how many invariants appear in their QC formulation. Quantum invariants outnumber the classical expectations, proving the necessity of further research. Lastly, we prove the validity of a method for retrieving any quantity of interest based on quantum phase estimation (QPE).