  We study optimal synthesis of Clifford circuits, and apply the results to
peep-hole optimization of quantum circuits. We report optimal circuits for all
Clifford operations with up to four inputs. We perform peep-hole optimization
of Clifford circuits with up to 40 inputs found in the literature, and
demonstrate the reduction in the number of gates by about 50%. We extend our
methods to the optimal synthesis of linear reversible circuits, partially
specified Clifford functions, and optimal Clifford circuits with five inputs up
to input/output permutation. The results find their application in randomized
benchmarking protocols, quantum error correction, and quantum circuit
optimization.
