Browse Theses

The description of the state of an n -bit quantum system requires 2 ⁿ-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dissertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measurement. This also sheds light on the power of computing with a finite number of quantum bits—using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton. We then consider the complexity of solving certain problems in the quantum black-box model, an information theoretic model that has been a rich source of insights into the nature of quantum computation. We derive nearly optimal lower bounds for several problems in this model, including that of approximating the median. We also give new, optimal algorithms for approximate medians and other order statistics.