6,691 research outputs found
Fast Computation of Abelian Runs
Given a word and a Parikh vector , an abelian run of period
in is a maximal occurrence of a substring of having
abelian period . Our main result is an online algorithm that,
given a word of length over an alphabet of cardinality and a
Parikh vector , returns all the abelian runs of period
in in time and space , where is the
norm of , i.e., the sum of its components. We also present an
online algorithm that computes all the abelian runs with periods of norm in
in time , for any given norm . Finally, we give an -time
offline randomized algorithm for computing all the abelian runs of . Its
deterministic counterpart runs in time.Comment: To appear in Theoretical Computer Scienc
Identifying all abelian periods of a string in quadratic time and relevant problems
Abelian periodicity of strings has been studied extensively over the last
years. In 2006 Constantinescu and Ilie defined the abelian period of a string
and several algorithms for the computation of all abelian periods of a string
were given. In contrast to the classical period of a word, its abelian version
is more flexible, factors of the word are considered the same under any
internal permutation of their letters. We show two O(|y|^2) algorithms for the
computation of all abelian periods of a string y. The first one maps each
letter to a suitable number such that each factor of the string can be
identified by the unique sum of the numbers corresponding to its letters and
hence abelian periods can be identified easily. The other one maps each letter
to a prime number such that each factor of the string can be identified by the
unique product of the numbers corresponding to its letters and so abelian
periods can be identified easily. We also define weak abelian periods on
strings and give an O(|y|log(|y|)) algorithm for their computation, together
with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer
Science
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Exact quantum Fourier transforms and discrete logarithm algorithms
We show how the quantum fast Fourier transform (QFFT) can be made exact for
arbitrary orders (first for large primes). For most quantum algorithms only the
quantum Fourier transform of order is needed, and this can be done
exactly. Kitaev \cite{kitaev} showed how to approximate the Fourier transform
for any order. Here we show how his construction can be made exact by using the
technique known as ``amplitude amplification''. Although unlikely to be of any
practical use, this construction e.g. allows to make Shor's discrete logarithm
quantum algorithm exact. Thus we have the first example of an exact non black
box fast quantum algorithm, thereby giving more evidence that ``quantum'' need
not be probabilistic. We also show that in a certain sense the family of
circuits for the exact QFFT is uniform. Namely the parameters of the gates can
be calculated efficiently.Comment: 10 pages Late
Fast simulation of large-scale growth models
We give an algorithm that computes the final state of certain growth models
without computing all intermediate states. Our technique is based on a "least
action principle" which characterizes the odometer function of the growth
process. Starting from an approximation for the odometer, we successively
correct under- and overestimates and provably arrive at the correct final
state.
Internal diffusion-limited aggregation (IDLA) is one of the models amenable
to our technique. The boundary fluctuations in IDLA were recently proved to be
at most logarithmic in the size of the growth cluster, but the constant in
front of the logarithm is still not known. As an application of our method, we
calculate the size of fluctuations over two orders of magnitude beyond previous
simulations, and use the results to estimate this constant.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm
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