In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x, t) depends mainly on the spectrum of the superoperator L kk . We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals to 1, thenP ( ν √ t , t) converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function p t (x, t) ≡ p( √ tx, t), x ∈ Z/ √ t, converges in distribution to a continuous convex combination of normal distributions. We give an necessary and sufficient condition for a U (2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk, walks 1 under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x, t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.

In this paper, we define a new type of decoherent quantum random walks with parameter 0 ≤ p ≤ 1, which becomes a unitary quantum random walk (UQRW) when p = 0 and an open quantum random walk (OPRW) when p = 1 respectively. We call this process a partially open quantum random walk (POQRW). We study the limiting distribution of a POQRW on Z 1 subject to decoherence on coins with n degrees of freedom, which converges to a convex combination of normal distributions if the superoperator L kk satisfies the eigenvalue condition, that is, 1 is an eigenvalue of L kk with multiplicity one and all other eigenvalues have absolute values less than 1. A Perron-Frobenius type of theorem is provided in determining whether or not the superoperator satisfies the eigenvalue condition. Moreover, we compute the limiting distributions of characteristic equations of the position probability functions for n = 2 and 3.

Professor Wei-Shih Yang, Chair Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement.