N - dimensional Quantum Cellular Automata — Extended abstract —

Quantum cellular automata (QCA) consist in an array of identical finite-dimensional quantum systems. The whole array evolves in discrete time steps by iterating a unitary operator G. Moreover this global evolution G is shift-invariant (acts everywhere the same), and local (information can only be transmitted at a bounded speed). Whilst this axiomatic definition is clearly the natural ‘quantization’ of the classical definition, when left as such QCA remain an excessively abstract/hardto-grasp mathematical object. In this paper we show that every QCA can be put into the form of an infinite tiling of more elementary, finite-dimensional unitary evolutions, i.e. that they can be thought of as a quantum circuit infinitely repeating across space. More precisely we represent an n-dimensional QCA of cell dimension d by an n-dimensional QCA of cell dimension d, which admits a n + 1-layered block representation, thereby generalizing the same result for one-dimensional QCA [2, 12]. Hence we now have a clear, robust definition of n-dimensional QCA, and phrased in the traditional setting of Hilbert spaces.