2,763 research outputs found
Partition Expanders
We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any pair of sufficiently large sets is close to the expected number, we consider partitions and require this condition only for most of the pairs of blocks. As a result, the blocks can be substantially smaller.
We show that for some range of parameters, to be a partition expander a random graph needs exponentially smaller degree than any expander would require in order to achieve similar expanding properties.
We apply the concept of partition expanders in communication complexity. First, we give a PRG for the SMP model of the optimal seed length, n+O(log(k)). Second, we compare the model of SMP to that of Simultaneous Two-Way Communication, and give a new separation that is stronger both qualitatively and quantitatively than the previously known ones
High-Dimensional Expanders from Expanders
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders.
In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004]
Efficient Quantum Tensor Product Expanders and k-designs
Quantum expanders are a quantum analogue of expanders, and k-tensor product
expanders are a generalisation to graphs that randomise k correlated walkers.
Here we give an efficient construction of constant-degree, constant-gap quantum
k-tensor product expanders. The key ingredients are an efficient classical
tensor product expander and the quantum Fourier transform. Our construction
works whenever k=O(n/log n), where n is the number of qubits. An immediate
corollary of this result is an efficient construction of an approximate unitary
k-design, which is a quantum analogue of an approximate k-wise independent
function, on n qubits for any k=O(n/log n). Previously, no efficient
constructions were known for k>2, while state designs, of which unitary designs
are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe
Multi-way expanders and imprimitive group actions on graphs
For n at least 2, the concept of n-way expanders was defined by various
researchers. Bigger n gives a weaker notion in general, and 2-way expanders
coincide with expanders in usual sense. Koji Fujiwara asked whether these
concepts are equivalent to that of ordinary expanders for all n for a sequence
of Cayley graphs. In this paper, we answer his question in the affirmative.
Furthermore, we obtain universal inequalities on multi-way isoperimetric
constants on any finite connected vertex-transitive graph, and show that gaps
between these constants imply the imprimitivity of the group action on the
graph.Comment: Accepted in Int. Math. Res. Notices. 18 pages, rearrange all of the
arguments in the proof of Main Theorem (Theorem A) in a much accessible way
(v4); 14 pages, appendix splitted into a forthcoming preprint (v3); 17 pages,
appendix on noncommutative L_p spaces added (v2); 12 pages, no figure
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