4,788 research outputs found
Information-theoretic lower bounds for quantum sorting
We analyze the quantum query complexity of sorting under partial information.
In this problem, we are given a partially ordered set and are asked to
identify a linear extension of using pairwise comparisons. For the standard
sorting problem, in which is empty, it is known that the quantum query
complexity is not asymptotically smaller than the classical
information-theoretic lower bound. We prove that this holds for a wide class of
partially ordered sets, thereby improving on a result from Yao (STOC'04)
Element Distinctness, Frequency Moments, and Sliding Windows
We derive new time-space tradeoff lower bounds and algorithms for exactly
computing statistics of input data, including frequency moments, element
distinctness, and order statistics, that are simple to calculate for sorted
data. We develop a randomized algorithm for the element distinctness problem
whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than
previous lower bounds for comparison-based algorithms, showing that element
distinctness is strictly easier than sorting for randomized branching programs.
This algorithm is based on a new time and space efficient algorithm for finding
all collisions of a function f from a finite set to itself that are reachable
by iterating f from a given set of starting points. We further show that our
element distinctness algorithm can be extended at only a polylogarithmic factor
cost to solve the element distinctness problem over sliding windows, where the
task is to take an input of length 2n-1 and produce an output for each window
of length n, giving n outputs in total. In contrast, we show a time-space
tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to
compute the number of distinct elements over sliding windows. The same lower
bound holds for computing the low-order bit of F_0 and computing any frequency
moment F_k, k neq 1. This shows that those frequency moments and the decision
problem F_0 mod 2 are strictly harder than element distinctness. We complement
this lower bound with a T in O(n^2/S) comparison-based deterministic RAM
algorithm for exactly computing F_k over sliding windows, nearly matching both
our lower bound for the sliding-window version and the comparison-based lower
bounds for the single-window version. We further exhibit a quantum algorithm
for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider
the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437
Quantum vs. Classical Read-once Branching Programs
The paper presents the first nontrivial upper and lower bounds for
(non-oblivious) quantum read-once branching programs. It is shown that the
computational power of quantum and classical read-once branching programs is
incomparable in the following sense: (i) A simple, explicit boolean function on
2n input bits is presented that is computable by error-free quantum read-once
branching programs of size O(n^3), while each classical randomized read-once
branching program and each quantum OBDD for this function with bounded
two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs
reading each input variable exactly once are shown to require size
2^{\Omega(n)} for computing the set-disjointness function DISJ_n from
communication complexity theory with two-sided error bounded by a constant
smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by
deterministic OBDDs of linear size. The technically most involved part is the
proof of the lower bound in (ii). For this, a new model of quantum
multi-partition communication protocols is introduced and a suitable extension
of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to
this model is presented.Comment: 35 pages. Lower bound for disjointness: Error in application of info
theory corrected and regularity of quantum read-once BPs (each variable at
least once) added as additional assumption of the theorem. Some more informal
explanations adde
Dynamic quantum clustering: a method for visual exploration of structures in data
A given set of data-points in some feature space may be associated with a
Schrodinger equation whose potential is determined by the data. This is known
to lead to good clustering solutions. Here we extend this approach into a
full-fledged dynamical scheme using a time-dependent Schrodinger equation.
Moreover, we approximate this Hamiltonian formalism by a truncated calculation
within a set of Gaussian wave functions (coherent states) centered around the
original points. This allows for analytic evaluation of the time evolution of
all such states, opening up the possibility of exploration of relationships
among data-points through observation of varying dynamical-distances among
points and convergence of points into clusters. This formalism may be further
supplemented by preprocessing, such as dimensional reduction through singular
value decomposition or feature filtering.Comment: 15 pages, 9 figure
07411 Abstracts Collection -- Algebraic Methods in Computational Complexity
From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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