Convergence of quantum random walks with decoherence

Abstract

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 ${\mathcal{L}}_{kk}$. 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, then $\hat{P}(\frac{\nu }{\sqrt{t}},t)$ converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function $p_t(x,t)\equiv p(\sqrt{t}x,t)$, $x\in Z/\sqrt{t}$, 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 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.

References in corpus (3)

• Asymptotic evolution of quantum walks with random coin
• Random Quantum Operations
• Asymptotic evolution of quantum walks on the $N$-cycle subject to decoherence on both the coin and position degrees of freedom

Cited by in corpus (8)

• Quantum walks: a comprehensive review
• Open quantum random walks with decoherence on coins with $n$ degrees of freedom
• Discrete dynamics and non-Markovianity
• Unusual scaling in a discrete quantum walk with random long range steps
• The localization of quantum random walks on sierpinski gaskets
• Limit Theorems for Decoherent Two Dimensional Quantum Walks
• Time-inhomogeneous Quantum Walks with Decoherence on Discrete Infinite Spaces
• Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces