Efficient Simulation of Random Quantum States and Operators

We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.