5,389 research outputs found
Topological lower bounds for arithmetic networks
We prove a complexity lower bound on deciding membership in a semialgebraic
set for arithmetic networks in terms of the sum of Betti numbers with respect
to "ordinary" (singular) homology. This result complements a similar lower
bound by Montana, Morais and Pardo for locally close semialgebraic sets in
terms of the sum of Borel-Moore Betti numbers. We also prove a lower bound in
terms of the sum of Betti numbers of the projection of a semialgebraic set to a
coordinate subspace.Comment: 19 pages, 10 figure
On the Effect of Quantum Interaction Distance on Quantum Addition Circuits
We investigate the theoretical limits of the effect of the quantum
interaction distance on the speed of exact quantum addition circuits. For this
study, we exploit graph embedding for quantum circuit analysis. We study a
logical mapping of qubits and gates of any -depth quantum adder
circuit for two -qubit registers onto a practical architecture, which limits
interaction distance to the nearest neighbors only and supports only one- and
two-qubit logical gates. Unfortunately, on the chosen -dimensional practical
architecture, we prove that the depth lower bound of any exact quantum addition
circuits is no longer , but . This
result, the first application of graph embedding to quantum circuits and
devices, provides a new tool for compiler development, emphasizes the impact of
quantum computer architecture on performance, and acts as a cautionary note
when evaluating the time performance of quantum algorithms.Comment: accepted for ACM Journal on Emerging Technologies in Computing
System
Generalizations of the Kolmogorov-Barzdin embedding estimates
We consider several ways to measure the `geometric complexity' of an
embedding from a simplicial complex into Euclidean space. One of these is a
version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove
inequalities relating the thickness and the number of simplices in the
simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved
for graphs. We also consider the distortion of knots. We give an alternate
proof of a theorem of Pardon that there are isotopy classes of knots requiring
arbitrarily large distortion. This proof is based on the expander-like
properties of arithmetic hyperbolic manifolds.Comment: 45 page
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