Intrinsically Universal One-dimensional Quantum Cellular Automata in Two Flavours

Abstract

We give a one-dimensional quantum cellular automaton (QCA) capable of simulating all others. By this we mean that the initial configuration and the local transition rule of any onedimensional QCA can be encoded within the initial configuration of the universal QCA. Several steps of the universal QCA will then correspond to one step of the simulated QCA. The simulation preserves the topology in the sense that each cell of the simulated QCA is encoded as a group of adjacent cells in the universal QCA. The encoding is linear and hence does not carry any of the cost of the computation. We do this in two flavours: a weak one which requires an infinite but periodic initial configuration and a strong one which needs only a finite initial configuration.