Showing papers by "Mark Hillery published in 2007"

Modifying quantum walks: a scattering theory approach

Abstract: We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.

Modifying quantum walks: A scattering theory approach

Abstract: We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.

Unambiguous identification of coherent states: Searching a quantum database

Abstract: We consider an unambiguous identification of an unknown coherent state with one of two unknown coherent reference states. Specifically, we consider two modes of an electromagnetic field prepared in unknown coherent states $\ensuremath{\mid}{\ensuremath{\alpha}}_{1}⟩$ and $\ensuremath{\mid}{\ensuremath{\alpha}}_{2}⟩$, respectively. The third mode is prepared either in the state $\ensuremath{\mid}{\ensuremath{\alpha}}_{1}⟩$ or in the state $\ensuremath{\mid}{\ensuremath{\alpha}}_{2}⟩$. The task is to identify (unambiguously) which of the two modes are in the same state. We present a scheme consisting of three beam splitters capable to perform this task. Although we do not prove the optimality, we show that the performance of the proposed setup is better than the generalization of the optimal measurement known for a finite-dimensional case. We show that a single beam splitter is capable to perform an unambiguous quantum state comparison for coherent states optimally. Finally, we propose an experimental setup consisting of $2N\ensuremath{-}1$ beam splitters for unambiguous identification among $N$ unknown coherent states. This setup can be considered as a search in a quantum database. The elements of the database are unknown coherent states encoded in different modes of an electromagnetic field. The task is to specify the two modes that are excited in the same, though unknown, coherent state.

Application of quantum algorithms to the study of permutations and group automorphisms

Abstract: We discuss three applications of efficient quantum algorithms to determining properties of permutations and group automorphisms. The first uses the Bernstein-Vazirani algorithm to determine an unknown homomorphism from ${Z}_{p\ensuremath{-}1}^{m}$ to $\mathit{Aut}({Z}_{p})$ where $p$ is prime. The remaining two make use of modifications of the Grover search algorithm. The first finds the fixed point of a permutation or an automorphism (assuming it has only one besides the identity). It can be generalized to find cycles of a specified size for permutations or orbits of a specified size for automorphisms. The second finds which of a set of permutations or automorphisms maps one particular element of a set or group onto another. This has relevance to the conjugacy problem for groups. We show how two of these algorithms can be implemented via programmable quantum processors. This approach opens new perspectives in quantum information processing when both the data and the programs are represented by states of quantum registers. In particular, quantum programs that specify control over data can be treated using methods of quantum information theory.

Phase information from a conservation law

Abstract: Using a conservation law obeyed by the degenerate parametric amplifier, we find a bound to the accuracy that can be obtained by using the signal mode to measure a phase shift. In particular, we show that there is a limit, which depends only on the initial state of the pump and signal modes, on how sharp the phase distribution of the signal mode can be. If the pump mode is initially in a coherent state and the signal mode is in the vacuum state, the phase distribution of the signal mode can be sharper than that of the initial state of the pump mode by at most a factor of two.