Amnon Ta-Shma

Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators

We show that the size of the smallest depth-two

Loss-less condensers, unbalanced expanders, and extractors

N

Dense quantum coding and quantum finite automata

-superconcentrator is

Quantum bit escrow

\Theta(N\log^2 N/\log\log N).

Interaction in quantum communication and the complexity of set disjointness

Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. For obtaining lower bounds, we present two different methods. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs of Kovari, Sos, and Turan, this gives an almost optimal lower bound of

Dense quantum coding and a lower bound for 1-way quantum automata

\Omega( N (\log N/\log \log N)^2)

The Hidden Subgroup Problem and Quantum Computation Using Group Representations

on the size of

Tight bounds for depth-two superconcentrators

N