A Fully Quantum Algorithm for Hydrodynamic Lattice Gas Cellular Automata

Abstract

Lattice Gas Cellular Automata (LGCA) are a computational model widely known and applied for the simulation of many physical phenomena. Their implementation requires an amount of resources and operations which scale linearly versus the system size and number of time steps. We propose a quantum-pointers-based quantum algorithm able to simulate LGCA while exhibiting an exponential advantage in space complexity and a number of quantum operations independent from the system size. We propose a collision circuit for the FHP lattice-gas automata considering the 2-, 3-, and 4-body collisions. These are implemented with two methodologies that suggest the procedure for finding quantum circuits for LGCA with more collisions. We also propose a phase estimation algorithm to retrieve information about a single cell, whose application can be expanded for implementing other collisions. A general methodology to identify the invariants associated to quantum LGCA is also proposed.