88 research outputs found
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 stochastic walks: A generalization of classical random walks and quantum walks
We introduce the quantum stochastic walk (QSW), which determines the
evolution of generalized quantum mechanical walk on a graph that obeys a
quantum stochastic equation of motion. Using an axiomatic approach, we specify
the rules for all possible quantum, classical and quantum-stochastic
transitions from a vertex as defined by its connectivity. We show how the
family of possible QSWs encompasses both the classical random walk (CRW) and
the quantum walk (QW) as special cases, but also includes more general
probability distributions. As an example, we study the QSW on a line, the QW to
CRW transition and transitions to genearlized QSWs that go beyond the CRW and
QW. QSWs provide a new framework to the study of quantum algorithms as well as
of quantum walks with environmental effects.Comment: 5 pages, 2 figures, 1 table. Video Abstract: http://vimeo.com/474903
Digital image processing with quantum approaches
Digital image processing with quantum approaches
Microscopic observation of magnon bound states and their dynamics
More than eighty years ago, H. Bethe pointed out the existence of bound
states of elementary spin waves in one-dimensional quantum magnets. To date,
identifying signatures of such magnon bound states has remained a subject of
intense theoretical research while their detection has proved challenging for
experiments. Ultracold atoms offer an ideal setting to reveal such bound states
by tracking the spin dynamics after a local quantum quench with single-spin and
single-site resolution. Here we report on the direct observation of two-magnon
bound states using in-situ correlation measurements in a one-dimensional
Heisenberg spin chain realized with ultracold bosonic atoms in an optical
lattice. We observe the quantum walk of free and bound magnon states through
time-resolved measurements of the two spin impurities. The increased effective
mass of the compound magnon state results in slower spin dynamics as compared
to single magnon excitations. In our measurements, we also determine the decay
time of bound magnons, which is most likely limited by scattering on thermal
fluctuations in the system. Our results open a new pathway for studying
fundamental properties of quantum magnets and, more generally, properties of
interacting impurities in quantum many-body systems.Comment: 8 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
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