Definition and evolution of quantum cellular automata with two qubits per cell

Studies of quantum computer implementations suggest cellular quantum computer architectures. These architectures can simulate the evolution of quantum cellular automata, which can possibly simulate both quantum and classical physical systems and processes. It is however known that except for the trivial case, unitary evolution of one-dimensional homogeneous quantum cellular automata with one qubit per cell is not possible. Quantum cellular automata that comprise two qubits per cell are defined and their evolution is studied using a quantum computer simulator. The evolution is unitary and its linearity manifests itself as a periodic structure in the probability distribution patterns.Comment: 13 pages, 4 figure

A universally programmable Quantum Cellular Automaton

We discuss the role of classical control in the context of reversible quantum cellular automata. Employing the structure theorem for quantum cellular automata, we give a general construction scheme to turn an arbitrary cellular automaton with external classical control into an autonomous one, thereby proving the computational equivalence of these two models. We use this technique to construct a universally programmable cellular automaton on a one-dimensional lattice with single cell dimension 12.Comment: 4 pages, 4 figures, minor changes in introduction, fixed typos, accepted for publication in Physical Review Letter

On the structure of Clifford quantum cellular automata

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.Comment: 28 pages, 2 figures, LaTe

Quantum Cellular Automata

Quantum cellular automata (QCA) are reviewed, including early and more recent proposals. QCA are a generalization of (classical) cellular automata (CA) and in particular of reversible CA. The latter are reviewed shortly. An overview is given over early attempts by various authors to define one-dimensional QCA. These turned out to have serious shortcomings which are discussed as well. Various proposals subsequently put forward by a number of authors for a general definition of one- and higher-dimensional QCA are reviewed and their properties such as universality and reversibility are discussed.Comment: 12 pages, 3 figures. To appear in the Springer Encyclopedia of Complexity and Systems Scienc

Index theory of one dimensional quantum walks and cellular automata

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the sense that there is a system S which locally acts like S_1 in one region and like S_2 in some other region, if and only if S_1 and S_2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S_1 into S_2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S -> ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.Comment: 38 pages. v2: added examples, terminology clarifie

Intrinsically universal one-dimensional quantum cellular automata in two flavours

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 one-dimensional 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. KEYWORDS: Quantum cellular automata, Intrinsic universality, Quantum computation.Comment: 27 pages, revtex, 23 figures. V3: The results of V1-V2 are better explained and formalized, and a novel result about intrinsic universality with only finite initial configurations is give