2,013 research outputs found
Classical and Quantum Algorithms for Constructing Text from Dictionary Problem
We study algorithms for solving the problem of constructing a text (long
string) from a dictionary (sequence of small strings). The problem has an
application in bioinformatics and has a connection with the Sequence assembly
method for reconstructing a long DNA sequence from small fragments. The problem
is constructing a string of length from strings with
possible intersections. We provide a classical algorithm with running time
where is the sum of lengths
of . We provide a quantum algorithm with running time . Additionally, we show that the lower bound for the
classical algorithm is . Thus, our classical algorithm is optimal
up to a log factor, and our quantum algorithm shows speed-up comparing to any
classical algorithm in a case of non-constant length of strings in the
dictionary
Quantum pattern matching fast on average
The -dimensional pattern matching problem is to find an occurrence of a
pattern of length within a text of length , with . This task models various problems in text and
image processing, among other application areas. This work describes a quantum
algorithm which solves the pattern matching problem for random patterns and
texts in time . For
large this is super-polynomially faster than the best possible classical
algorithm, which requires time . The
algorithm is based on the use of a quantum subroutine for finding hidden shifts
in dimensions, which is a variant of algorithms proposed by Kuperberg.Comment: 22 pages, 2 figures; v3: further minor changes, essentially published
versio
Generalized Scaling Function at Strong Coupling
We considered folded spinning string in AdS_5 x S^5 background dual to the
Tr(D^S Phi^J) operators of N=4 SYM theory. In the limit S,J-> \infty and l=pi
J/\sqrt\lambda\log S fixed we compute the string energy with the 2-loop
accuracy in the worldsheet coupling \sqrt\lambda from the asymptotical Bethe
ansatz. In the limit l-> 0 the result is finite due to the massive cancelations
with terms coming from the conjectured dressing phase. We also managed to
compute all leading logarithm terms l^{2m}\log^n l/\lambda^n/2 to an arbitrary
order in perturbation theory. In particular for m=1 we reproduced results of
Alday and Maldacena computed from a sigma model. The method developed in this
paper could be used for a systematic expansion in 1/\sqrt\lambda and also at
weak coupling
Stochastic String Motion Above and Below the World Sheet Horizon
We study the stochastic motion of a relativistic trailing string in black
hole AdS_5. The classical string solution develops a world-sheet horizon and we
determine the associated Hawking radiation spectrum. The emitted radiation
causes fluctuations on the string both above and below the world-sheet horizon.
In contrast to standard black hole physics, the fluctuations below the horizon
are causally connected with the boundary of AdS. We derive a bulk stochastic
equation of motion for the dual string and use the AdS/CFT correspondence to
determine the evolution a fast heavy quark in the strongly coupled
plasma. We find that the kinetic mass of the quark decreases by while the correlation time of world sheet
fluctuations increases by .Comment: 27 pages, 5 figures; v2 final version, small changes, references
adde
Longest Common Substring and Longest Palindromic Substring in Time
The Longest Common Substring (LCS) and Longest Palindromic Substring (LPS)
are classical problems in computer science, representing fundamental challenges
in string processing. Both problems can be solved in linear time using a
classical model of computation, by means of very similar algorithms, both
relying on the use of suffix trees. Very recently, two sublinear algorithms for
LCS and LPS in the quantum query model have been presented by Le Gall and
Seddighin~\cite{GallS23}, requiring and
queries, respectively. However, while the query
model is fascinating from a theoretical standpoint, its practical applicability
becomes limited when it comes to crafting algorithms meant for actual execution
on real hardware. In this paper we present, for the first time, a
quantum algorithm for both LCS and LPS working
in the circuit model of computation. Our solutions are simpler than previous
ones and can be easily translated into quantum procedures. We also present
actual implementations of the two algorithms as quantum circuits working in
and time,
respectively
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