72 research outputs found
Quantum walks on two-dimensional grids with multiple marked locations
The running time of a quantum walk search algorithm depends on both the
structure of the search space (graph) and the configuration of marked
locations. While the first dependence have been studied in a number of papers,
the second dependence remains mostly unstudied.
We study search by quantum walks on two-dimensional grid using the algorithm
of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two
marked location cases only. We move beyond two marked locations and study the
behaviour of the algorithm for an arbitrary configuration of marked locations.
In this paper we prove two results showing the importance of how the marked
locations are arranged. First, we present two placements of marked
locations for which the number of steps of the algorithm differs by
factor. Second, we present two configurations of and
marked locations having the same number of steps and probability to
find a marked location
Quantum walks can find a marked element on any graph
We solve an open problem by constructing quantum walks that not only detect
but also find marked vertices in a graph. In the case when the marked set
consists of a single vertex, the number of steps of the quantum walk is
quadratically smaller than the classical hitting time of any
reversible random walk on the graph. In the case of multiple marked
elements, the number of steps is given in terms of a related quantity
which we call extended hitting time.
Our approach is new, simpler and more general than previous ones. We
introduce a notion of interpolation between the random walk and the
absorbing walk , whose marked states are absorbing. Then our quantum walk
is simply the quantum analogue of this interpolation. Contrary to previous
approaches, our results remain valid when the random walk is not
state-transitive. We also provide algorithms in the cases when only
approximations or bounds on parameters (the probability of picking a
marked vertex from the stationary distribution) and are
known.Comment: 50 page
Quantum enigma machines and the locking capacity of a quantum channel
The locking effect is a phenomenon which is unique to quantum information
theory and represents one of the strongest separations between the classical
and quantum theories of information. The Fawzi-Hayden-Sen (FHS) locking
protocol harnesses this effect in a cryptographic context, whereby one party
can encode n bits into n qubits while using only a constant-size secret key.
The encoded message is then secure against any measurement that an eavesdropper
could perform in an attempt to recover the message, but the protocol does not
necessarily meet the composability requirements needed in quantum key
distribution applications. In any case, the locking effect represents an
extreme violation of Shannon's classical theorem, which states that
information-theoretic security holds in the classical case if and only if the
secret key is the same size as the message. Given this intriguing phenomenon,
it is of practical interest to study the effect in the presence of noise, which
can occur in the systems of both the legitimate receiver and the eavesdropper.
This paper formally defines the locking capacity of a quantum channel as the
maximum amount of locked information that can be reliably transmitted to a
legitimate receiver by exploiting many independent uses of a quantum channel
and an amount of secret key sublinear in the number of channel uses. We provide
general operational bounds on the locking capacity in terms of other well-known
capacities from quantum Shannon theory. We also study the important case of
bosonic channels, finding limitations on these channels' locking capacity when
coherent-state encodings are employed and particular locking protocols for
these channels that might be physically implementable.Comment: 37 page
Non-Markovian dynamics of a qubit coupled to an Ising spin bath
We study the analytically solvable Ising model of a single qubit system
coupled to a spin bath. The purpose of this study is to analyze and elucidate
the performance of Markovian and non-Markovian master equations describing the
dynamics of the system qubit, in comparison to the exact solution. We find that
the time-convolutionless master equation performs particularly well up to
fourth order in the system-bath coupling constant, in comparison to the
Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the
infinite bath correlation time in this model. A recently proposed
post-Markovian master equation performs comparably to the time-convolutionless
master equation for a properly chosen memory kernel, and outperforms all the
approximation methods considered here at long times. Our findings shed light on
the applicability of master equations to the description of reduced system
dynamics in the presence of spin-baths.Comment: 17 pages, 16 figure
Ergodicity breaking in a model showing many-body localization
We study the breaking of ergodicity measured in terms of return probability
in the evolution of a quantum state of a spin chain. In the non ergodic phase a
quantum state evolves in a much smaller fraction of the Hilbert space than
would be allowed by the conservation of extensive observables. By the anomalous
scaling of the participation ratios with system size we are led to consider the
distribution of the wave function coefficients, a standard observable in modern
studies of Anderson localization. We finally present a criterion for the
identification of the ergodicity breaking (many-body localization) transition
based on these distributions which is quite robust and well suited for
numerical investigations of a broad class of problems.Comment: 5 pages, 5 figures, final versio
Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve
We study first-order quantum phase transitions in models where the mean-field
traitment is exact, and the exponentially fast closure of the energy gap with
the system size at the transition. We consider exactly solvable ferromagnetic
models, and show that they reduce to the Grover problem in a particular limit.
We compute the coefficient in the exponential closure of the gap using an
instantonic approach, and discuss the (dire) consequences for quantum
annealing.Comment: 6 pages, 3 figure
Quantum walks on quotient graphs
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. If
this unitary evolution operator has an associated group of symmetries, then for
certain initial states the walk will be confined to a subspace of the original
Hilbert space. Symmetries of the original graph, given by its automorphism
group, can be inherited by the evolution operator. 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. We give an explicit
construction of the quotient graph for any subgroup of the automorphism group
and illustrate it with examples. The automorphisms of the quotient graph which
are inherited from the original graph are the original automorphism group
modulo the subgroup used to construct it. We then analyze the behavior of
hitting times on quotient graphs. Hitting time is the average time it takes a
walk to reach a given final vertex from a given initial vertex. It has been
shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting
time can be infinite. We give a condition which determines whether the quotient
graph has infinite hitting times given that they exist in the original graph.
We apply this condition for the examples discussed and determine which quotient
graphs have infinite hitting times. All known examples of quantum walks with
fast hitting times correspond to systems with quotient graphs much smaller than
the original 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.Comment: 18 pages, 7 figures in EPS forma
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