  The task of finding a correctable encoding that protects against some
physical quantum process is in general hard. Two main obstacles are that an
exponential number of experiments are needed to gain complete information about
the quantum process, and known algorithmic methods for finding correctable
encodings involve operations on exponentially large matrices. However, we show
that in some cases it is possible to find such encodings with only partial
information about the quantum process. Such useful partial information can be
systematically extracted by averaging the channel under the action of a set of
unitaries in a process known as "twirling". In this paper we prove that
correctable encodings for a twirled channel are also correctable for the
original channel. We investigate the particular case of twirling over the set
of Pauli operators and qubit permutations, and show that the resulting quantum
operation can be characterized experimentally in a scalable manner. We also
provide a postprocessing scheme for finding unitarily correctable codes for
these twirled channels which does not involve exponentially large matrices.

Quantum Physics