  Randomized benchmarking is an experimental procedure intended to demonstrate
control of quantum systems. The procedure extracts the average error introduced
by a set of control operations. When the target set of operations is intended
to be the set of Clifford operators, the randomized benchmarking algorithm is
particularly easy to perform and its results have an important interpretation
with respect to quantum computation. The aim of the benchmark is to provide a
simple, useful parameter describing the quality of quantum control with an
experiment that can be performed in a standard way on any prospective quantum
computer. This parameter can be used to fairly compare different experiments or
to mark improvement in a single experiment.
  In this thesis I discuss first the original randomized-benchmarking procedure
and the importance of the Clifford operators for its implementation. I develop
the statistical analysis of the results and the physical assumptions that are
required for the simplest analysis to apply. The original procedure does not
extend in an obvious way to benchmarking of more than one qubit, so I introduce
a standardized procedure for randomized benchmarking that applies to any number
of qubits. This new procedure also enables the benchmarking of an individual
control operation. I describe two randomized-benchmarking experiments I helped
to design: one involved a single qubit and utilized a variation of the original
procedure and the second involved two qubits and demonstrated the new
procedure. I conclude with several potential extensions to the original and new
procedures that give them reduced experimental overhead, the ability to
describe encoded operations, and fairer comparisons between experiments.
