16 research outputs found
The isoperimetric constant of the random graph process
The isoperimetric constant of a graph on vertices, , is the
minimum of , taken over all nonempty subsets
of size at most , where denotes the set of
edges with precisely one end in . A random graph process on vertices,
, is a sequence of graphs, where
is the edgeless graph on vertices, and
is the result of adding an edge to ,
uniformly distributed over all the missing edges. We show that in almost every
graph process equals the minimal degree of
as long as the minimal degree is . Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically , the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value
Entanglement bounds on the performance of quantum computing architectures
There are many possible architectures of qubit connectivity that designers of
future quantum computers will need to choose between. However, the process of
evaluating a particular connectivity graph's performance as a quantum
architecture can be difficult. In this paper, we show that a quantity known as
the isoperimetric number establishes a lower bound on the time required to
create highly entangled states. This metric we propose counts resources based
on the use of two-qubit unitary operations, while allowing for arbitrarily fast
measurements and classical feedback. We use this metric to evaluate the
hierarchical architecture proposed by A. Bapat et al. [Phys. Rev. A 98, 062328
(2018)], and find it to be a promising alternative to the conventional grid
architecture. We also show that the lower bound that this metric places on the
creation time of highly entangled states can be saturated with a constructive
protocol, up to a factor logarithmic in the number of qubits.Comment: 9 pages, 5 figures, 1 table (journal version
Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs
It is known that many different types of finite random subgraph models
undergo quantitatively similar phase transitions around their percolation
thresholds, and the proofs of these results rely on isoperimetric properties of
the underlying host graph. Recently, the authors showed that such a phase
transition occurs in a large class of regular high-dimensional product graphs,
generalising a classic result for the hypercube.
In this paper we give new isoperimetric inequalities for such regular
high-dimensional product graphs, which generalise the well-known isoperimetric
inequality of Harper for the hypercube, and are asymptotically sharp for a wide
range of set sizes. We then use these isoperimetric properties to investigate
the structure of the giant component in supercritical percolation on
these product graphs, that is, when , where is the
degree of the product graph and is a small enough constant.
We show that typically has edge-expansion . Furthermore, we show that likely contains a linear-sized
subgraph with vertex-expansion . These
results are best possible up to the logarithmic factor in .
Using these likely expansion properties, we determine, up to small
polylogarithmic factors in , the likely diameter of as well as the
typical mixing time of a lazy random walk on . Furthermore, we show the
likely existence of a path of length .
These results not only generalise, but also improve substantially upon the
known bounds in the case of the hypercube, where in particular the likely
diameter and typical mixing time of were previously only known to be
polynomial in