Results 1  10
of
23
A brief history of cellular automata
, 2000
"... Cellular automata are simple models of computation which exhibit fascinatingly complex behavior. They have captured the attention of several generations of researchers, leading to an extensive body of work. Here we trace a history of cellular automata from their beginnings with von Neumann to the pr ..."
Abstract

Cited by 66 (2 self)
 Add to MetaCart
Cellular automata are simple models of computation which exhibit fascinatingly complex behavior. They have captured the attention of several generations of researchers, leading to an extensive body of work. Here we trace a history of cellular automata from their beginnings with von Neumann to the present day. The emphasis is mainly on topics closer to computer science and mathematics rather than physics, biology or other applications. The work should be of interest to both new entrants into the field as well as researchers working on particular aspects of cellular automata.
Onedimensional quantum cellular automata over finite, unbounded configurations
"... Onedimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a local, shiftinvariant unitary evolution. By local we mean that no instantaneous longrange communication can occur. In order to def ..."
Abstract

Cited by 18 (9 self)
 Add to MetaCart
(Show Context)
Onedimensional quantum cellular automata (QCA) consist in a line of identical, finite dimensional quantum systems. These evolve in discrete time steps according to a local, shiftinvariant unitary evolution. By local we mean that no instantaneous longrange 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 twolayered block representation, and hence the inverse QCA is again a QCA. This is a striking result since the property does not hold for classical onedimensional cellular automata as defined over such finite configurations. As an example we discuss a bijective cellular automata which becomes nonlocal 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 twolayered block representation result applies also over infinite configurations,
Quantum Walks
, 2008
"... Abstract. Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discretetime (or coined) and the continuoustime quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, re ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discretetime (or coined) and the continuoustime quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, respectively. Most of the contents are based on our results. Furthermore, papers on quantum walks are listed in References. Studies of discretetime walks appeared from the late 1980s from Gudder (1988), for example. Meyer (1996) investigated the model as a quantum lattice gas automaton. Nayak and Vishwanath (2000) and Ambainis et al. (2001) studied intensively the behaviour of discretetime walks, in particular, the Hadamard walk. In contrast with the central limit theorem for the classical random walks, Konno (2002a, 2005a) showed a new type of weak limit theorems for the onedimensional lattice. Grimmett, Janson, and Scudo (2004) extended the limit theorem to a wider range of the walks. On the other hand, the continuoustime quantum walk was introduced and studied by Childs, Farhi, and Gutmann (2002). Excellent reviews on quantum walks are
On the structure of clifford quantum cellular automata
 J. Math. Phys
"... We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by sy ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the one dimensional case we also find that every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton timestep, thereby characterizing these class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions. Contents Section
Locality and information transfer in quantum operations
, 2008
"... We investigate the situation in which no information can be transferred from a quantum system B to a quantum system A, even though both interact with a common system C. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We investigate the situation in which no information can be transferred from a quantum system B to a quantum system A, even though both interact with a common system C.
Algebraic Characterizations of Unitary Linear Quantum Cellular Automata
, 2006
"... Abstract. We provide algebraic criteria for the unitarity of linear quantum cellular automata, i.e. one dimensional quantum cellular automata. We derive these both by direct combinatorial arguments, and by adding constraints into the model which do not change the quantum cellular automata’s computat ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We provide algebraic criteria for the unitarity of linear quantum cellular automata, i.e. one dimensional quantum cellular automata. We derive these both by direct combinatorial arguments, and by adding constraints into the model which do not change the quantum cellular automata’s computational power. The configurations we consider have finite but unbounded size. 1