Two-level quantum walkers on directed graphs. I. Universal quantum computing

Quantum walks on directed graphs

We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices (vi, vj), if vi is connected to vj then there is a path from vj to vi. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and discuss some implications of this condition. We presen...

Universal mixing of quantum walk on graphs

We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is universal mixing if the instantaneous or average probability distribution of the quantum walk on G ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is uniform mixing if it visits the uniform distribut...

On Universal and Fault-Tolerant Quantum Computing

Universal Quantum Computing Based on Monodromy Representations

A model of quantum computing is presented, based on properties of connections with a prescribed monodromy group on holomorphic vector bundles over bases with nontrivial topology. Such connections with required properties appear in the WZW-models, in which moreover the corresponding n-point correlation functions are sections of appropriate bundles which are holomorphic with respect to the connec...

Universal quantum computation with two-level trapped ions

Although the initial proposal for ion trap quantum computation made use of an auxiliary internal level to perform logic between ions, this resource is not necessary in principle. Instead, one may perform such operations directly using sideband laser pulses, operating with an arbitrary ~sufficiently small! Lamb-Dicke parameter. We explore the potential of this technique, showing how to perform l...