Selfsimilarity, Simulation and Spacetime Symmetries

We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse

A simple block representation of reversible cellular automata with time-symmetry

Reversible Cellular Automata (RCA) are a physics-like model of computation consisting of an array of identical cells, evolving in discrete time steps by iterating a global evolution G. Further, G is required to be shift-invariant (it acts the same everywhere), causal (information cannot be transmitted faster than some fixed number of cells per time step), and reversible (it has an inverse which verifies the same requirements). An important, though only recently studied special case is that of Time-symmetric Cellular Automata (TSCA), for which G and its inverse are related via a local operation. In this note we revisit the question of the Block representation of RCA, i.e. we provide a very simple proof of the existence of a reversible circuit description implementing G. This operational, bottom-up description of G turns out to be time-symmetric, suggesting interesting connections with TSCA. Indeed we prove, using a similar technique, that a wide class of them admit an Exact block representation (EBR), i.e. one which does not increase the state space.Comment: 6 pages, 3 figures, Automata 201

On the Probabilistic Query Complexity of Transitively Symmetric Problems

We obtain optimal lower bounds on the nonadaptive probabilistic query complexity of a class of problems defined by a rather weak symmetry condition. In fact, for each problem in this class, given a number T of queries we compute exactly the performance (i.e., the probability of success on the worst instance) of the best nonadaptive probabilistic algorithm that makes T queries. We show that this optimal performance is given by a minimax formula involving certain probability distributions. Moreover, we identify two classes of problems for which adaptivity does not help. We illustrate these results on a few natural examples, including unordered search, Simon's problem, distinguishing one-to-one functions from two-to-one functions, and hidden translation. For these last three examples, which are of particular interest in quantum computing, the recent theorems of Aaronson, of Laplante and Magniez, and of Bar-Yossef, Kumar and Sivakumar on the probabilistic complexity of black-box problems do not yield any nonconstant lower bound

Unitarity plus causality implies localizability

We consider a graph with a single quantum system at each node. The entire compound system evolves in discrete time steps by iterating a global evolution $U$. We require that this global evolution $U$ be unitary, in accordance with quantum theory, and that this global evolution $U$ be causal, in accordance with special relativity. By causal we mean that information can only ever be transmitted at a bounded speed, the speed bound being quite naturally that of one edge of the underlying graph per iteration of $U$. We show that under these conditions the operator $U$ can be implemented locally; i.e. it can be put into the form of a quantum circuit made up with more elementary operators -- each acting solely upon neighbouring nodes. We take quantum cellular automata as an example application of this representation theorem: this analysis bridges the gap between the axiomatic and the constructive approaches to defining QCA. KEYWORDS: Quantum cellular automata, Unitary causal operators, Quantum walks, Quantum computation, Axiomatic quantum field theory, Algebraic quantum field theory, Discrete space-time.Comment: V1: 5 pages, revtex. V2: Generalizes V1. V3: More precisions and reference