Search results for "Computer and Information Science"
showing 10 items of 1335 documents
New Developments in Quantum Algorithms
2010
In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the …
A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity
2014
Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy [7], is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity and 0-sensitivity and 0-block sensitivity.
A Generalization of Girod’s Bidirectional Decoding Method to Codes with a Finite Deciphering Delay
2012
In this paper we generalize an encoding method due to Girod (cf. [6]) using prefix codes, that allows a bidirectional decoding of the encoded messages. In particular we generalize it to any finite alphabet A, to any operation defined on A, to any code with finite deciphering delay and to any key x ∈ A+ , on a length depending on the deciphering delay. We moreover define, as in [4], a deterministic transducer for such generalized method. We prove that, fixed a code X ∈ A* with finite deciphering delay and a key x ∈ A *, the transducers associated to different operations are isomorphic as unlabelled graphs. We also prove that, for a fixed code X with finite deciphering delay, transducers asso…
Implications of quantum automata for contextuality
2014
We construct zero error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded error probabilistic finite automata (PFAs). Here is a summary of our results: There is a promise problem solvable by an exact two way QFA in exponential expected time but not by any bounded error sublogarithmic space probabilistic Turing machine (PTM). There is a promise problem solvable by an exact two way QFA in quadratic expected time but not by any bounded error o(loglogn) space PTMs in polynomial expected time. The same problem can be solvable by a one way Las Vegas (or exact two way) QFA with quantum head in linear (expected) time. There is a promise problem solvable by a Las …
A note on Sturmian words
2012
International audience; We describe an algorithm which, given a factor of a Sturmian word, computes the next factor of the same length in the lexicographic order in linear time. It is based on a combinatorial property of Sturmian words which is related with the Burrows-Wheeler transformation.
Group algebras and Lie nilpotence
2013
Abstract Let ⁎ be an involution of a group algebra FG induced by an involution of the group G. For char F ≠ 2 , we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of ⁎-skew elements is nilpotent.
Quantum walks on two-dimensional grids with multiple marked locations
2015
The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration (the placement and the number) of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied.We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [3]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for several configurations of multiple marked locations.In this paper, we prove two results showing the importance of how the marked locations ar…
Symmetry-assisted adversaries for quantum state generation
2011
We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the $backslash$sc Graph Isomorphism problem. We show that for the related problem of $backslash$sc Index Erasure our method leads to a lower bound of $backslash Omega(backslash sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is …
Any AND-OR Formula of Size N Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer
2007
Consider the problem of evaluating an AND-OR formula on an $N$-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time $N^{1/2+o(1)}$. In particular, approximately balanced formulas can be evaluated in $O(\sqrt{N})$ queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.
Burrows-Wheeler transform and Run-Length Enconding
2017
In this paper we study the clustering effect of the Burrows-Wheeler Transform (BWT) from a combinatorial viewpoint. In particular, given a word w we define the BWT-clustering ratio of w as the ratio between the number of clusters produced by BWT and the number of the clusters of w. The number of clusters of a word is measured by its Run-Length Encoding. We show that the BWT-clustering ratio ranges in ]0, 2]. Moreover, given a rational number \(r\,\in \,]0,2]\), it is possible to find infinitely many words having BWT-clustering ratio equal to r. Finally, we show how the words can be classified according to their BWT-clustering ratio. The behavior of such a parameter is studied for very well-…