220 research outputs found
Entangling gates in even Euclidean lattices such as the Leech lattice
The group of automorphisms of Euclidean (embedded in ) dense
lattices such as the root lattices and , the Barnes-Wall lattice
, the unimodular lattice and the Leech lattice
may be generated by entangled quantum gates of the corresponding
dimension. These (real) gates/lattices are useful for quantum error correction:
for instance, the two and four-qubit real Clifford groups are the automorphism
groups of the lattices and , respectively, and the three-qubit
real Clifford group is maximal in the Weyl group . Technically, the
automorphism group of the lattice is the set of
orthogonal matrices such that, following the conjugation action by the
generating matrix of the lattice, the output matrix is unimodular (of
determinant , with integer entries). When the degree is equal to the
number of basis elements of , then also acts on basis
vectors and is generated with matrices such that the sum of squared entries
in a row is one, i.e. may be seen as a quantum gate. For the dense lattices
listed above, maximal multipartite entanglement arises. In particular, one
finds a balanced tripartite entanglement in (the two- and three- tangles
have equal magnitude 1/4) and a GHZ type entanglement in BW. In this
paper, we also investigate the entangled gates from and
, by seeing them as systems coupling a qutrit to two- and
three-qubits, respectively. Apart from quantum computing, the work may be
related to particle physics in the spirit of \cite{PLS2010}.Comment: 11 pages, second updated versio
Finite reflection groups and graph norms
Given a graph on vertex set and a function , define \begin{align*} \|f\|_{H}:=\left\vert\int
\prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*}
where is the Lebesgue measure on . We say that is norming if
is a semi-norm. A similar notion is defined by
and is said to be weakly norming if
is a norm. Classical results show that weakly norming graphs
are necessarily bipartite. In the other direction, Hatami showed that even
cycles, complete bipartite graphs, and hypercubes are all weakly norming. We
demonstrate that any graph whose edges percolate in an appropriate way under
the action of a certain natural family of automorphisms is weakly norming. This
result includes all previously known examples of weakly norming graphs, but
also allows us to identify a much broader class arising from finite reflection
groups. We include several applications of our results. In particular, we
define and compare a number of generalisations of Gowers' octahedral norms and
we prove some new instances of Sidorenko's conjecture.Comment: 29 page
Efficient Bayesian Learning in Social Networks with Gaussian Estimators
We consider a group of Bayesian agents who try to estimate a state of the
world through interaction on a social network. Each agent
initially receives a private measurement of : a number picked
from a Gaussian distribution with mean and standard deviation one.
Then, in each discrete time iteration, each reveals its estimate of to
its neighbors, and, observing its neighbors' actions, updates its belief using
Bayes' Law.
This process aggregates information efficiently, in the sense that all the
agents converge to the belief that they would have, had they access to all the
private measurements. We show that this process is computationally efficient,
so that each agent's calculation can be easily carried out. We also show that
on any graph the process converges after at most steps, where
is the number of agents and is the diameter of the network. Finally, we
show that on trees and on distance transitive-graphs the process converges
after steps, and that it preserves privacy, so that agents learn very
little about the private signal of most other agents, despite the efficient
aggregation of information. Our results extend those in an unpublished
manuscript of the first and last authors.Comment: Added coauthor. Added proofs for fast convergence on trees and
distance transitive graphs. Also, now analyzing a notion of privac
- …