72 research outputs found
Entanglement dynamics and quasi-periodicity in discrete quantum walks
We study the entanglement dynamics of discrete time quantum walks acting on
bounded finite sized graphs. We demonstrate that, depending on system
parameters, the dynamics may be monotonic, oscillatory but highly regular, or
quasi-periodic. While the dynamics of the system are not chaotic since the
system comprises linear evolution, the dynamics often exhibit some features
similar to chaos such as high sensitivity to the system's parameters,
irregularity and infinite periodicity. Our observations are of interest for
entanglement generation, which is one primary use for the quantum walk
formalism. Furthermore, we show that the systems we model can easily be mapped
to optical beamsplitter networks, rendering experimental observation of
quasi-periodic dynamics within reach.Comment: 9 pages, 8 figure
Quantum Walks with Entangled Coins
We present a mathematical formalism for the description of unrestricted
quantum walks with entangled coins and one walker. The numerical behaviour of
such walks is examined when using a Bell state as the initial coin state, two
different coin operators, two different shift operators, and one walker. We
compare and contrast the performance of these quantum walks with that of a
classical random walk consisting of one walker and two maximally correlated
coins as well as quantum walks with coins sharing different degrees of
entanglement.
We illustrate that the behaviour of our walk with entangled coins can be very
different in comparison to the usual quantum walk with a single coin. We also
demonstrate that simply by changing the shift operator, we can generate widely
different distributions. We also compare the behaviour of quantum walks with
maximally entangled coins with that of quantum walks with non-entangled coins.
Finally, we show that the use of different shift operators on 2 and 3 qubit
coins leads to different position probability distributions in 1 and 2
dimensional graphs.Comment: Two new sections and several changes from referees' comments. 12
pages and 12 (colour) figure
Asymptotic entanglement in a two-dimensional quantum walk
The evolution operator of a discrete-time quantum walk involves a conditional
shift in position space which entangles the coin and position degrees of
freedom of the walker. After several steps, the coin-position entanglement
(CPE) converges to a well defined value which depends on the initial state. In
this work we provide an analytical method which allows for the exact
calculation of the asymptotic reduced density operator and the corresponding
CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the
von Neumann entropy of the reduced density operator as an entanglement measure.
The method is applied to the case of a Hadamard walk for which the dependence
of the resulting CPE on initial conditions is obtained. Initial states leading
to maximum or minimum CPE are identified and the relation between the coin or
position entanglement present in the initial state of the walker and the final
level of CPE is discussed. The CPE obtained from separable initial states
satisfies an additivity property in terms of CPE of the corresponding
one-dimensional cases. Non-local initial conditions are also considered and we
find that the extreme case of an initial uniform position distribution leads to
the largest CPE variation.Comment: Major revision. Improved structure. Theoretical results are now
separated from specific examples. Most figures have been replaced by new
versions. The paper is now significantly reduced in size: 11 pages, 7 figure
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
Universal Quantum Walk Control Plane for Quantum Networks
Quantum networks are complex systems formed by the interaction among quantum
processors through quantum channels. Analogous to classical computer networks,
quantum networks allow for the distribution of quantum operations among quantum
processors. In this work, we describe a Quantum Walk Control Protocol (QWCP) to
perform distributed quantum operations in a quantum network. We consider a
generalization of the discrete-time coined quantum walk model that accounts for
the interaction between quantum walks in the network graph with quantum
registers inside the network nodes. QWCP allows for the implementation of
networked quantum services, such as distributed quantum computing and
entanglement distribution, abstracting hardware implementation and the
transmission of quantum information through channels. Multiple interacting
quantum walks can be used to propagate entangled control signals across the
network in parallel. We demonstrate how to use QWCP to perform distributed
multi-qubit controlled gates, which shows the universality of the protocol for
distributed quantum computing. Furthermore, we apply the QWCP to the task of
entanglement distribution in a quantum network.Comment: 27 pages; 2 figures. A preliminary version of this work was presented
at IEEE International Conference on Quantum Computing and Engineering 2021
(QCE21). arXiv admin note: text overlap with arXiv:2106.0983
Two quantum walkers sharing coins
We consider two independent quantum walks on separate lines augmented by
partial or full swapping of coins after each step. For classical random walks,
swapping or not swapping coins makes little difference to the random walk
characteristics, but we show that quantum walks with partial swapping of coins
have complicated yet elegant inter-walker correlations. Specifically we study
the joint position distribution of the reduced two-walker state after tracing
out the coins and analyze total, classical and quantum correlations in terms of
the mutual information, the quantum mutual information, and the
measurement-induced disturbance. Our analysis shows intriguing quantum features
without classical analogues.Comment: 10 pages, 4 figure
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