210 research outputs found
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
Fast Escape from Quantum Mazes in Integrated Photonics
Escaping from a complex maze, by exploring different paths with several
decision-making branches in order to reach the exit, has always been a very
challenging and fascinating task. Wave field and quantum objects may explore a
complex structure in parallel by interference effects, but without necessarily
leading to more efficient transport. Here, inspired by recent observations in
biological energy transport phenomena, we demonstrate how a quantum walker can
efficiently reach the output of a maze by partially suppressing the presence of
interference. In particular, we show theoretically an unprecedented improvement
in transport efficiency for increasing maze size with respect to purely quantum
and classical approaches. In addition, we investigate experimentally these
hybrid transport phenomena, by mapping the maze problem in an integrated
waveguide array, probed by coherent light, hence successfully testing our
theoretical results. These achievements may lead towards future bio-inspired
photonics technologies for more efficient transport and computation.Comment: 13 pages, 10 figure
Coined quantum walks on percolation graphs
Quantum walks, both discrete (coined) and continuous time, form the basis of
several quantum algorithms and have been used to model processes such as
transport in spin chains and quantum chemistry. The enhanced spreading and
mixing properties of quantum walks compared with their classical counterparts
have been well-studied on regular structures and also shown to be sensitive to
defects and imperfections in the lattice. As a simple example of a disordered
system, we consider percolation lattices, in which edges or sites are randomly
missing, interrupting the progress of the quantum walk. We use numerical
simulation to study the properties of coined quantum walks on these percolation
lattices in one and two dimensions. In one dimension (the line) we introduce a
simple notion of quantum tunneling and determine how this affects the
properties of the quantum walk as it spreads. On two-dimensional percolation
lattices, we show how the spreading rate varies from linear in the number of
steps down to zero, as the percolation probability decreases to the critical
point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after
referee comments, added extra figur
Discrete time quantum walks on percolation graphs
Randomly breaking connections in a graph alters its transport properties, a
model used to describe percolation. In the case of quantum walks, dynamic
percolation graphs represent a special type of imperfections, where the
connections appear and disappear randomly in each step during the time
evolution. The resulting open system dynamics is hard to treat numerically in
general. We shortly review the literature on this problem. We then present our
method to solve the evolution on finite percolation graphs in the long time
limit, applying the asymptotic methods concerning random unitary maps. We work
out the case of one dimensional chains in detail and provide a concrete, step
by step numerical example in order to give more insight into the possible
asymptotic behavior. The results about the case of the two-dimensional integer
lattice are summarized, focusing on the Grover type coin operator.Comment: 22 pages, 3 figure
Continuous-time quantum walks on one-dimension regular networks
In this paper, we consider continuous-time quantum walks (CTQWs) on
one-dimension ring lattice of N nodes in which every node is connected to its
2m nearest neighbors (m on either side). In the framework of the Bloch function
ansatz, we calculate the spacetime transition probabilities between two nodes
of the lattice. We find that the transport of CTQWs between two different nodes
is faster than that of the classical continuous-time random walk (CTRWs). The
transport speed, which is defined by the ratio of the shortest path length and
propagating time, increases with the connectivity parameter m for both the
CTQWs and CTRWs. For fixed parameter m, the transport of CTRWs gets slow with
the increase of the shortest distance while the transport (speed) of CTQWs
turns out to be a constant value. In the long time limit, depending on the
network size N and connectivity parameter m, the limiting probability
distributions of CTQWs show various paterns. When the network size N is an even
number, the probability of being at the original node differs from that of
being at the opposite node, which also depends on the precise value of
parameter m.Comment: Typos corrected and Phys. ReV. E comments considered in this versio
Symmetry in quantum walks
A discrete-time quantum walk on a graph is the repeated application of a
unitary evolution operator to a Hilbert space corresponding to the graph.
Hitting times for discrete quantum walks on graphs give an average time before
the walk reaches an ending condition. We derive an expression for hitting time
using superoperators, and numerically evaluate it for the walk on the hypercube
for various coins and decoherence models. We show that, by contrast to
classical walks, quantum walks can have infinite hitting times for some initial
states. We seek criteria to determine if a given walk on a graph will have
infinite hitting times, and find a sufficient condition for their existence.
The phenomenon of infinite hitting times is in general a consequence of the
symmetry of the graph and its automorphism group. Symmetries of a graph, given
by its automorphism group, can be inherited by the evolution operator. Using
the irreducible representations of the automorphism group, we derive conditions
such that quantum walks defined on this graph must have infinite hitting times
for some initial states. Symmetry can also cause the walk to be confined to a
subspace of the original Hilbert space for certain initial states. We show that
a quantum walk confined to the subspace corresponding to this symmetry group
can be seen as a different quantum walk on a smaller quotient graph and we give
an explicit construction of the quotient graph. We conjecture that the
existence of a small quotient graph with finite hitting times is necessary for
a walk to exhibit a quantum speed-up. Finally, we use symmetry and the theory
of decoherence-free subspaces to determine when the subspace of the quotient
graph is a decoherence-free subspace of the dynamics.Comment: 136 pages, Ph.D. thesis, University of Southern California, 200
Decoherence and classicalization of continuous-time quantum walks on graphs
We address decoherence and classicalization of continuous-time quantum walks
(CTQWs) on graphs. In particular, we investigate three different models of
decoherence, and employ the quantum-classical (QC) dynamical distance as a
figure of merit to assess whether, and to which extent, decoherence
classicalizes the CTQW, i.e. turns it into the analogue classical process. We
show that the dynamics arising from intrinsic decoherence, i.e. dephasing in
the energy basis, do not fully classicalize the walker and partially preserves
quantum features. On the other hand, dephasing in the position basis, as
described by the Haken-Strobl master equation or by the quantum stochastic walk
(QSW) model, asymptotically destroys the quantumness of the walker, making it
equivalent to a classical random walk. We also investigate the speed of the
classicalization process, and observe a faster convergence of the QC-distance
to its asymptotic value for intrinsic decoherence and the QSW models, whereas
in the Haken-Strobl scenario, larger values of the decoherence rate induce
localization of the walker.Comment: 15 pages, 4 figure
On analog quantum algorithms for the mixing of Markov chains
The problem of sampling from the stationary distribution of a Markov chain
finds widespread applications in a variety of fields. The time required for a
Markov chain to converge to its stationary distribution is known as the
classical mixing time. In this article, we deal with analog quantum algorithms
for mixing. First, we provide an analog quantum algorithm that given a Markov
chain, allows us to sample from its stationary distribution in a time that
scales as the sum of the square root of the classical mixing time and the
square root of the classical hitting time. Our algorithm makes use of the
framework of interpolated quantum walks and relies on Hamiltonian evolution in
conjunction with von Neumann measurements.
There also exists a different notion for quantum mixing: the problem of
sampling from the limiting distribution of quantum walks, defined in a
time-averaged sense. In this scenario, the quantum mixing time is defined as
the time required to sample from a distribution that is close to this limiting
distribution. Recently we provided an upper bound on the quantum mixing time
for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we
also extend and expand upon our findings therein. Namely, we provide an
intuitive understanding of the state-of-the-art random matrix theory tools used
to derive our results. In particular, for our analysis we require information
about macroscopic, mesoscopic and microscopic statistics of eigenvalues of
random matrices which we highlight here. Furthermore, we provide numerical
simulations that corroborate our analytical findings and extend this notion of
mixing from simple graphs to any ergodic, reversible, Markov chain.Comment: The section concerning time-averaged mixing (Sec VIII) has been
updated: Now contains numerical plots and an intuitive discussion on the
random matrix theory results used to derive the results of arXiv:2001.0630
Discrete-Time Quantum Walk - Dynamics and Applications
This dissertation presents investigations on dynamics of discrete-time
quantum walk and some of its applications. Quantum walks has been exploited as
an useful tool for quantum algorithms in quantum computing. Beyond quantum
computational purposes, it has been used to explain and control the dynamics in
various physical systems. In order to use the quantum walk to its fullest
potential, it is important to know and optimize the properties purely due to
quantum dynamics and in presence of noise. Various studies of its dynamics in
the absence and presence of noise have been reported. We propose new approaches
to optimize the dynamics, discuss symmetries and effect of noise on the quantum
walk. Making use of its properties, we propose the use of quantum walk as an
efficient new tool for various applications in physical systems and quantum
information processing. In the first and second part of this dissertation, we
discuss evolution process of the quantum walks, propose and demonstrate the
optimization of discrete-time quantum walk using quantum coin operation from
SU(2) group and discuss some of its properties. We investigate symmetry
operations and environmental effects on dynamics of the walk on a line and an
n-cycle highlighting the interplay between noise and topology. Using the
properties and behavior of quantum walk discussed in part two, in part three we
propose the application of quantum walk to realize quantum phase transition in
optical lattice, that is to efficiently control and redistribute ultracold
atoms in optical lattice. We also discuss the implementation scheme. Another
application we consider is creation of spatial entanglement using quantum walk
on a quantum many body system.Comment: 199 pages, 52 figures, Thesis completed during 2009 at University of
Waterloo (IQC), V2 : Index of figures has been made compac
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