  In a recent paper, we show that an S-matrix functional for graviton
amplitudes can be described by an N=8 supersymmetric gravitational holonomy
operator in twistor space. In this paper, we obtain an alternative expression
for the gravitational holonomy operator such that it can be interpreted as a
square of an N=4 holonomy operator for frame fields, by taking a sum of certain
shuffles over ordered indices. The new expression leads to amplitudes of not
only spin-2 gravitons but also spin-0 massless particles. We discuss that the
squared model is favored as a theory of quantum gravity.
