  We develop the concept of a unitary t-design as a means of expressing
operationally useful subsets of the stochastic properties of the uniform (Haar)
measure on the unitary group U(2^n) on n qubits. In particular, sets of
unitaries forming 2-designs have wide applicability to quantum information
protocols. We devise an O(n)-size in-place circuit construction for an
approximate unitary 2-design. We then show that this can be used to construct
an efficient protocol for experimentally characterizing the fidelity of a
quantum process on n qubits with quantum circuits of size O(n) without
requiring any ancilla qubits, thereby improving upon previous approaches.
