13,031 research outputs found
On the unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial
of a graph, including unimodality, log-concavity and reality of zeros. We
establish recurrence relations and give factorizations of independence
polynomials for certain classes of graphs. As applications we settle some
unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric
Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
At its core a -design is a method for sampling from a set of unitaries in
a way which mimics sampling randomly from the Haar measure on the unitary
group, with applications across quantum information processing and physics. We
construct new families of quantum circuits on -qubits giving rise to
-approximate unitary -designs efficiently in
depth. These quantum circuits are based on a relaxation of technical
requirements in previous constructions. In particular, the construction of
circuits which give efficient approximate -designs by Brandao, Harrow, and
Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys.
(2016).) required choosing gates from ensembles which contained inverses for
all elements, and that the entries of the unitaries are algebraic. We reduce
these requirements, to sets that contain elements without inverses in the set,
and non-algebraic entries, which we dub partially invertible universal sets. We
then adapt this circuit construction to the framework of measurement based
quantum computation(MBQC) and give new explicit examples of -qubit graph
states with fixed assignments of measurements (graph gadgets) giving rise to
unitary -designs based on partially invertible universal sets, in a natural
way. We further show that these graph gadgets demonstrate a quantum speedup, up
to standard complexity theoretic conjectures. We provide numerical and
analytical evidence that almost any assignment of fixed measurement angles on
an -qubit cluster state give efficient -designs and demonstrate a quantum
speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in
newest version. new References added.Proofs unchanged. Results unchange
Homological symbols and the Quillen conjecture
We formulate a "correct" version of the Quillen conjecture on linear group
homology for certain arithmetic rings and provide evidence for the new
conjecture. In this way we predict that the linear group homology has a direct
summand looking like an unstable form of Milnor K-theory and we call this new
theory "homological symbols algebra". As a byproduct we prove the Quillen
conjecture in homological degree two for the rank two and the prime 5
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