Vincent Nesme

One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations

One-dimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a causal, shift-invariant unitary evolution. By causal we mean that no instantaneous long-range communication can occur. In order to define these over a Hilbert space we must restrict to a base of finite, yet unbounded configurations. We show that QCA always admit a two-layered block representation, and hence the inverse QCA is again a QCA. This is a striking result since the property does not hold for classical one-dimensional cellular automata as defined over such finite configurations. As an example we discuss a bijective cellular automata which becomes non-causal as a QCA, in a rare case of reversible computation which does not admit a straightforward quantization. We argue that a whole class of bijective cellular automata should no longer be considered to be reversible in a physical sense. Note that the same two-layered block representation result applies also over infinite configurations, as was previously shown for one-dimensional systems in the more elaborate formalism of operators algebras [13]. Here the proof is simpler and self-contained, moreover we discuss a counterexample QCA in higher dimensions.

Applying causality principles to the axiomatization of probabilistic cellular automata

Cellular automata (CAs) consist of an bi-infinite array of identical cells, each of which may take one of a finite number of possible sstates. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmitted faster than some fixed number of cells per time step). At least in the classical [7], reversible [11] and quantum cases [1], these two top-down axiomatic conditions are sufficient to entail more bottom-up, operational descriptions of G. We investigate whether the same is true in the probabilistic case.

Bounds on the speedup in quantum signaling

Given a discrete reversible dynamics, we can define a quantum dynamics, which acts on basis states like the classical one, but also allows for superpositions of them. It is a curious fact that in the quantum version, local changes in the initial state, after a single dynamical step, can sometimes can be detected much farther away than classically. Here we show that this effect is no use for generating faster signals. In a run of many steps the quantum propagation neighborhood can only increase by a constant fringe, so there is no asymptotic increase in speed.