On Quantum Extensions of Hydrodynamic Lattice Gas Automata

We consider quantum extensions of classical hydrodynamic lattice gas models. We find that the existence of local conserved quantities strongly constrains such extensions. We find the only extensions that retain local conserved quantities correspond to changing the local encoding of a subset of the bits. These models maintain separability of the state throughout the evolution and are thus efficiently classically simulable. We then consider evolution of these models in the case where any of the bits can be encoded and measured in one of two local bases. In the case that either encoding is allowed, the models are efficiently classically simulable. In the case that both encoding and measurement is allowed in either basis, we argue that efficient classical simulation is unlikely. In particular, for classical models that are computationally universal such quantum extensions can encode Simon’s algorithm, thus presenting an obstacle to efficient classical simulation.