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 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 $${S \mapsto {\rm ind} S}$$ 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 shift

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

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

The effects of Professional and Pedagogical Program Development Environments on Novice Programmer Perceptions

Novice programmers generally have difficulty learning to program and one of the problems contributing towards this is the program development environment used at tertiary institutions. A number of pedagogical program development environments have been developed specifically for novice programmers, but these have not been compared experimentally with professional program development environments. A study was conducted that compared the perceptions of novice programmers using a representative professional program development environment to a pedagogical program development environment during an Introductory Programming module at a tertiary institution. It was found that the use of a pedagogical program development environment had a positive effect on the feelings of achievement and learning while learning to program, while the perceived ease of using the program development environment and the perceived difficulty of practical assignments were not affected

Distribution patterns and stable carbon isotopic composition of alkanes and alkan-1-ols from plant waxes of African rain forest and savannah C₃ species

soils, lakes and marine sediments. Relevant to the emerging use of plant wax derived biomarkers as proxies for past vegetation composition this study provides key data for C3 plants of tropical and subtropical Africa. We present analytical results for 45 savanna species and 24 rain forest plants sampled in their natural habitats. Contents and distribution patterns of long chain n-alkanes (n-C25 to n-C35) and n-alkan-1- ols (n-C24 to n-C34) as well as bulk and molecular carbon isotopic data are presented. The variations of the analysed parameters among different growth forms (herb, shrub, liana and tree) are small within the vegetation zones, whereas characteristic differences occur between the signatures of rain forest and savanna plants. Therefore, we provide averaged histogram representations for rain forest and savanna C3 plants. The findings were compared to previously published data of typical C4 grass waxes of tropical and subtropical Africa. Generally, trends to longer n-alkane chains and less negative carbon isotopic values are evident from rain forest over C3 savanna to C4 vegetation. For n-alkanols of rain forest plants the maximum of the averaged distribution pattern is between those of C3 savanna plants and C4 grasses. The averaged presentations for tropical and subtropical vegetation and their characteristics may constitute useful biomarker proxies for studies analysing the expansion and contraction of African vegetation zones