113,307 research outputs found
Computational Difficulty of Computing the Density of States
We study the computational difficulty of computing the ground state
degeneracy and the density of states for local Hamiltonians. We show that the
difficulty of both problems is exactly captured by a class which we call #BQP,
which is the counting version of the quantum complexity class QMA. We show that
#BQP is not harder than its classical counting counterpart #P, which in turn
implies that computing the ground state degeneracy or the density of states for
classical Hamiltonians is just as hard as it is for quantum Hamiltonians.Comment: v2: Accepted version. 9 pages, 1 figur
Apparent Fractality Emerging from Models of Random Distributions
The fractal properties of models of randomly placed -dimensional spheres
(=1,2,3) are studied using standard techniques for calculating fractal
dimensions in empirical data (the box counting and Minkowski-sausage
techniques). Using analytical and numerical calculations it is shown that in
the regime of low volume fraction occupied by the spheres, apparent fractal
behavior is observed for a range of scales between physically relevant
cut-offs. The width of this range, typically spanning between one and two
orders of magnitude, is in very good agreement with the typical range observed
in experimental measurements of fractals. The dimensions are not universal and
depend on density. These observations are applicable to spatial, temporal and
spectral random structures. Polydispersivity in sphere radii and
impenetrability of the spheres (resulting in short range correlations) are also
introduced and are found to have little effect on the scaling properties. We
thus propose that apparent fractal behavior observed experimentally over a
limited range may often have its origin in underlying randomness.Comment: 19 pages, 12 figures. More info available at
http://www.fh.huji.ac.il/~dani
Counting points on curves using a map to P^1, II
We introduce a new algorithm to compute the zeta function of a curve over a
finite field. This method extends previous work of ours to all curves for which
a good lift to characteristic zero is known. We develop all the necessary
bounds, analyse the complexity of the algorithm and provide a complete
implementation
Alternation-Trading Proofs, Linear Programming, and Lower Bounds
A fertile area of recent research has demonstrated concrete polynomial time
lower bounds for solving natural hard problems on restricted computational
models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path,
Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs
of these lower bounds follow a certain proof-by-contradiction strategy that we
call alternation-trading. An important open problem is to determine how
powerful such proofs can possibly be.
We propose a methodology for studying these proofs that makes them amenable
to both formal analysis and automated theorem proving. We prove that the search
for better lower bounds can often be turned into a problem of solving a large
series of linear programming instances. Implementing a small-scale theorem
prover based on this result, we extract new human-readable time lower bounds
for several problems. This framework can also be used to prove concrete
limitations on the current techniques.Comment: To appear in STACS 2010, 12 page
Direct Counting Analysis on Network Generated by Discrete Dynamics
A detail study on the In-degree Distribution (ID) of Cellular Automata is
obtained by exact enumeration. The results indicate large deviation from
multiscaling and classification according to ID are discussed. We further
augment the transfer matrix as such the distributions for more complicated
rules are obtained. Dependence of In-degree Distribution on the lattice size
have also been found for some rules including R50 and R77.Comment: 8 pages, 11 figure
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