1,534 research outputs found
Quantum speedup of classical mixing processes
Most approximation algorithms for #P-complete problems (e.g., evaluating the
permanent of a matrix or the volume of a polytope) work by reduction to the
problem of approximate sampling from a distribution over a large set
. This problem is solved using the {\em Markov chain Monte Carlo} method: a
sparse, reversible Markov chain on with stationary distribution
is run to near equilibrium. The running time of this random walk algorithm, the
so-called {\em mixing time} of , is as shown
by Aldous, where is the spectral gap of and is the minimum
value of . A natural question is whether a speedup of this classical
method to , the diameter of the graph
underlying , is possible using {\em quantum walks}.
We provide evidence for this possibility using quantum walks that {\em
decohere} under repeated randomized measurements. We show: (a) decoherent
quantum walks always mix, just like their classical counterparts, (b) the
mixing time is a robust quantity, essentially invariant under any smooth form
of decoherence, and (c) the mixing time of the decoherent quantum walk on a
periodic lattice is , which is indeed
and is asymptotically no worse than the
diameter of (the obvious lower bound) up to at most a logarithmic
factor.Comment: 13 pages; v2 revised several part
Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups
Capacities of quantum channels and decoherence times both quantify the extent
to which quantum information can withstand degradation by interactions with its
environment. However, calculating capacities directly is known to be
intractable in general. Much recent work has focused on upper bounding certain
capacities in terms of more tractable quantities such as specific norms from
operator theory. In the meantime, there has also been substantial recent
progress on estimating decoherence times with techniques from analysis and
geometry, even though many hard questions remain open. In this article, we
introduce a class of continuous-time quantum channels that we called
transferred channels, which are built through representation theory from a
classical Markov kernel defined on a compact group. We study two subclasses of
such kernels: H\"ormander systems on compact Lie-groups and Markov chains on
finite groups. Examples of transferred channels include the depolarizing
channel, the dephasing channel, and collective decoherence channels acting on
qubits. Some of the estimates presented are new, such as those for channels
that randomly swap subsystems. We then extend tools developed in earlier work
by Gao, Junge and LaRacuente to transfer estimates of the classical Markov
kernel to the transferred channels and study in this way different
non-commutative functional inequalities. The main contribution of this article
is the application of this transference principle to the estimation of various
capacities as well as estimation of entanglement breaking times, defined as the
first time for which the channel becomes entanglement breaking. Moreover, our
estimates hold for non-ergodic channels such as the collective decoherence
channels, an important scenario that has been overlooked so far because of a
lack of techniques.Comment: 35 pages, 2 figures. Close to published versio
Perspectives on Multi-Level Dynamics
As Physics did in previous centuries, there is currently a common dream of
extracting generic laws of nature in economics, sociology, neuroscience, by
focalising the description of phenomena to a minimal set of variables and
parameters, linked together by causal equations of evolution whose structure
may reveal hidden principles. This requires a huge reduction of dimensionality
(number of degrees of freedom) and a change in the level of description. Beyond
the mere necessity of developing accurate techniques affording this reduction,
there is the question of the correspondence between the initial system and the
reduced one. In this paper, we offer a perspective towards a common framework
for discussing and understanding multi-level systems exhibiting structures at
various spatial and temporal levels. We propose a common foundation and
illustrate it with examples from different fields. We also point out the
difficulties in constructing such a general setting and its limitations
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}
Quantized recurrence time in iterated open quantum dynamics
The expected return time to the original state is a key concept
characterizing systems obeying both classical or quantum dynamics. We consider
iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a
broad class of systems that includes classical Markov chains and unitary
discrete time quantum walks on networks. Starting from a pure state, the time
evolution is induced by repeated applications of a general quantum channel, in
each timestep followed by a measurement to detect whether the system has
returned to the original state. We prove that if the superoperator is unital in
the relevant Hilbert space (the part of the Hilbert space explored by the
system), then the expectation value of the return time is an integer, equal to
the dimension of this relevant Hilbert space. We illustrate our results on
partially coherent quantum walks on finite graphs. Our work connects the
previously known quantization of the expected return time for bistochastic
Markov chains and for unitary quantum walks, and shows that these are special
cases of a more general statement. The expected return time is thus a
quantitative measure of the size of the part of the Hilbert space available to
the system when the dynamics is started from a certain state
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