41 research outputs found
Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem
We show that the NP-hard quadratic unconstrained binary optimization (QUBO)
problem on a graph can be solved using an adiabatic quantum computer that
implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding
of in the quantum hardware graph . There are two components to this
reduction: embedding and parameter setting. The embedding problem is to find a
minor-embedding of a graph in , which is a subgraph of
such that can be obtained from by contracting edges. The
parameter setting problem is to determine the corresponding parameters, qubit
biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper,
we focus on the parameter setting problem. As an example, we demonstrate the
embedded Ising Hamiltonian for solving the maximum independent set (MIS)
problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system.
We close by discussing several related algorithmic problems that need to be
investigated in order to facilitate the design of adiabatic algorithms and AQC
architectures.Comment: 17 pages, 5 figures, submitte
Small Complete Minors Above the Extremal Edge Density
A fundamental result of Mader from 1972 asserts that a graph of high average
degree contains a highly connected subgraph with roughly the same average
degree. We prove a lemma showing that one can strengthen Mader's result by
replacing the notion of high connectivity by the notion of vertex expansion.
Another well known result in graph theory states that for every integer t
there is a smallest real c(t) so that every n-vertex graph with c(t)n edges
contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an
n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of
order at most C(\epsilon)log n. We use our extension of Mader's theorem to
prove that such a graph G must contain a K_t-minor of order at most
C(\epsilon)log n loglog n. Known constructions of graphs with high girth show
that this result is tight up to the loglog n factor
Merlin: A Language for Provisioning Network Resources
This paper presents Merlin, a new framework for managing resources in
software-defined networks. With Merlin, administrators express high-level
policies using programs in a declarative language. The language includes
logical predicates to identify sets of packets, regular expressions to encode
forwarding paths, and arithmetic formulas to specify bandwidth constraints. The
Merlin compiler uses a combination of advanced techniques to translate these
policies into code that can be executed on network elements including a
constraint solver that allocates bandwidth using parameterizable heuristics. To
facilitate dynamic adaptation, Merlin provides mechanisms for delegating
control of sub-policies and for verifying that modifications made to
sub-policies do not violate global constraints. Experiments demonstrate the
expressiveness and scalability of Merlin on real-world topologies and
applications. Overall, Merlin simplifies network administration by providing
high-level abstractions for specifying network policies and scalable
infrastructure for enforcing them
On combinatorial structures in linear codes
In this work we show that given a connectivity graph of a
quantum code, there exists , such that , and the 's are
-expander. If the codes are classical we show
instead that the 's are -expander.
We also show converses to these bounds. In particular, we show that the BPT
bound for classical codes is tight in all Euclidean dimensions. Finally, we
prove structural theorems for graphs with no "dense" subgraphs which might be
of independent interest