Improved constructions of quantum automata
  We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use \frac{4}{\epsilon} \log 2p + O(1) states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of \log p than the previously known construction of Ambainis and Freivalds (quant-ph/9802062).   Similarly to Ambainis and Freivalds, our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction. 
Andris Ambainis, Nikolajs Nahimovs