Detecting Topological Invariants in Nonunitary Discrete-Time Quantum Walks.

Explicit definition of PT symmetry for nonunitary quantum walks with gain and loss

Exploring topological phases with quantum walks

The quantum walk was originally proposed as a quantum-mechanical analog of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete-time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigations. In particular, we de...

Correlation Functions for Some Conformal Theories on Riemann Surfaces

We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFT’s with monodromies being the discrete subgroups of SL(2,R I ) the determination of four–point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces.

ar X iv : m at h / 07 03 33 9 v 1 [ m at h . FA ] 1 2 M ar 2 00 7 APPROXIMATION OF QUANTUM LÉVY PROCESSES BY QUANTUM RANDOM WALKS

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks. The note is concerned with investigating convergence of random walks on quantum groups to quantum Lévy processes. The theory of the latter is a natural non-commutative counterpart of the theory of classical Lévy processes on groups ([Hey]). It h...

Topology and Holonomy in Discrete-time Quantum Walks

We present a research article which formulates the milestones for the understanding and characterization of holonomy and topology of a discrete-time quantum walk architecture, consisting of a unitary step given by a sequence of two non-commuting rotations in parameter space. Unlike other similar systems recently studied in detail in the literature, this system does not present continous 1D topo...