27 research outputs found
The quantum to classical transition for random walks
We look at two possible routes to classical behavior for the discrete quantum
random walk on the line: decoherence in the quantum ``coin'' which drives the
walk, or the use of higher-dimensional coins to dilute the effects of
interference. We use the position variance as an indicator of classical
behavior, and find analytical expressions for this in the long-time limit; we
see that the multicoin walk retains the ``quantum'' quadratic growth of the
variance except in the limit of a new coin for every step, while the walk with
decoherence exhibits ``classical'' linear growth of the variance even for weak
decoherence.Comment: 4 pages RevTeX 4.0 + 2 figures (encapsulated Postscript). Trimmed for
length. Minor corrections + one new referenc
A Quantum Lovasz Local Lemma
The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show
the existence of combinatorial objects meeting a prescribed collection of
"weakly dependent" criteria. We show that the LLL extends to a much more
general geometric setting, where events are replaced with subspaces and
probability is replaced with relative dimension, which allows to lower bound
the dimension of the intersection of vector spaces under certain independence
conditions. Our result immediately applies to the k-QSAT problem: For instance
we show that any collection of rank 1 projectors with the property that each
qubit appears in at most of them, has a joint satisfiable
state.
We then apply our results to the recently studied model of random k-QSAT.
Recent works have shown that the satisfiable region extends up to a density of
1 in the large k limit, where the density is the ratio of projectors to qubits.
Using a hybrid approach building on work by Laumann et al. we greatly extend
the known satisfiable region for random k-QSAT to a density of
. Since our tool allows us to show the existence of joint
satisfying states without the need to construct them, we are able to penetrate
into regions where the satisfying states are conjectured to be entangled,
avoiding the need to construct them, which has limited previous approaches to
product states.Comment: 19 page
Optimizing the discrete time quantum walk using a SU(2) coin
We present a generalized version of the discrete time quantum walk, using the
SU(2) operation as the quantum coin. By varying the coin parameters, the
quantum walk can be optimized for maximum variance subject to the functional
form and the probability distribution in the position
space can be biased. We also discuss the variation in measurement entropy with
the variation of the parameters in the SU(2) coin. Exploiting this we show how
quantum walk can be optimized for improving mixing time in an -cycle and for
quantum walk search.Comment: 6 pages, 6 figure
Quantum random walks with decoherent coins
The quantum random walk has been much studied recently, largely due to its
highly nonclassical behavior. In this paper, we study one possible route to
classical behavior for the discrete quantum walk on the line: the presence of
decoherence in the quantum ``coin'' which drives the walk. We find exact
analytical expressions for the time dependence of the first two moments of
position, and show that in the long-time limit the variance grows linearly with
time, unlike the unitary walk. We compare this to the results of direct
numerical simulation, and see how the form of the position distribution changes
from the unitary to the usual classical result as we increase the strength of
the decoherence.Comment: Minor revisions, especially in introduction. Published versio
Quantum Walks driven by many coins
Quantum random walks have been much studied recently, largely due to their
highly nonclassical behavior. In this paper, we study one possible route to
classical behavior for the discrete quantum random walk on the line: the use of
multiple quantum ``coins'' in order to diminish the effects of interference
between paths. We find solutions to this system in terms of the single coin
random walk, and compare the asymptotic limit of these solutions to numerical
simulations. We find exact analytical expressions for the time-dependence of
the first two moments, and show that in the long time limit the ``quantum
mechanical'' behavior of the one-coin walk persists. We further show that this
is generic for a very broad class of possible walks, and that this behavior
disappears only in the limit of a new coin for every step of the walk.Comment: 36 pages RevTeX 4.0 + 5 figures (encapsulated Postscript). Submitted
to Physical Review
Quadratic speedup for finding marked vertices by quantum walks
A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk, resolving a question that had been open for 15 years