9 research outputs found
On limiting distributions of quantum Markov chains
In a quantum Markov chain, the temporal succession of states is modeled by
the repeated action of a "bistochastic quantum operation" on the density matrix
of a quantum system. Based on this conceptual framework, we derive some new
results concerning the evolution of a quantum system, including its long-term
behavior. Among our findings is the fact that the Cesro limit of any
quantum Markov chain always exists and equals the orthogonal projection of the
initial state upon the eigenspace of the unit eigenvalue of the bistochastic
quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on
the unit circle, then the quantum Markov chain converges in the conventional
sense to the said orthogonal projection. As a corollary, we offer a new
derivation of the classic result describing limiting distributions of unitary
quantum walks on finite graphs \cite{AAKV01}
A quantization procedure based on completely positive maps and Markov operators
We describe -limit sets of completely positive (CP) maps over
finite-dimensional spaces. In such sets and in its corresponding convex hulls,
CP maps present isometric behavior and the states contained in it commute with
each other. Motivated by these facts, we describe a quantization procedure
based on CP maps which are induced by Markov (transfer) operators. Classical
dynamics are described by an action over essentially bounded functions. A
non-expansive linear map, which depends on a choice of a probability measure,
is the centerpiece connecting phenomena over function and matrix spaces
Decomposition of quantum Markov chains and its applications
© 2018 Elsevier Inc. Markov chains have been widely employed as a fundamental model in the studies of probabilistic and stochastic communicating and concurrent systems. It is well-understood that decomposition techniques play a key role in reachability analysis and model-checking of Markov chains. (Discrete-time) quantum Markov chains have been introduced as a model of quantum communicating systems [1] and also a semantic model of quantum programs [2]. The BSCC (Bottom Strongly Connected Component) and stationary coherence decompositions of quantum Markov chains were introduced in [3–5]. This paper presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. We further establish a limit theorem for them. As an application, an algorithm to find a maximum dimensional noiseless subsystem of a quantum communicating system is given using decomposition techniques of quantum Markov chains
Quantum walks: a comprehensive review
Quantum walks, the quantum mechanical counterpart of classical random walks,
is an advanced tool for building quantum algorithms that has been recently
shown to constitute a universal model of quantum computation. Quantum walks is
now a solid field of research of quantum computation full of exciting open
problems for physicists, computer scientists, mathematicians and engineers.
In this paper we review theoretical advances on the foundations of both
discrete- and continuous-time quantum walks, together with the role that
randomness plays in quantum walks, the connections between the mathematical
models of coined discrete quantum walks and continuous quantum walks, the
quantumness of quantum walks, a summary of papers published on discrete quantum
walks and entanglement as well as a succinct review of experimental proposals
and realizations of discrete-time quantum walks. Furthermore, we have reviewed
several algorithms based on both discrete- and continuous-time quantum walks as
well as a most important result: the computational universality of both
continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing
Journa