Abstract
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.
Highlights
Lattice gas cellular automata are some of the simplest models of physical phenomena
In spite of the fact that these stabilizer states are elements of a Hilbert space whose dimensions are exponential in the number of sites of the quantum cellular automataon (QCA), such models still admit an efficient classical simulation
We show that quantum extensions in which some lattice vector occupancies are encoded in different bases are possible
Summary
Lattice gas cellular automata are some of the simplest models of physical phenomena. The microscopic state of a lattice gas is given by particles (possibly of more than one type) occupying a set of lattice vectors at each site of a lattice. Classical cellular automata and lattice-gas models are natural models for implementation on parallel supercomputers [36] They led to the construction of special-purpose hardware for their simulation [37,38,39]. In spite of the fact that these stabilizer states are elements of a Hilbert space whose dimensions are exponential in the number of sites of the quantum cellular automataon (QCA), such models still admit an efficient classical simulation. They are directly equivalent to some classical cellular automata [51,52,53]. For the HPP and FHP models, which are capable of universal classical computation [58,59], we argue that such extensions are unlikely to be amenable to efficient classical simulation
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