  We obtain an explicit error expansion for the solution of Backward Stochastic
Differential Equations (BSDEs) using the cubature on Wiener spaces method. The
result is proved under a mild strengthening of the assumptions needed for the
application of the cubature method. The explicit expansion can then be used to
construct implementable higher order approximations via Richardson-Romberg
extrapolation. To allow for an effective efficiency improvement of the
interpolated algorithm, we introduce an additional projection on finite grids
through interpolation operators. We study the resulting complexity reduction in
the case of the linear interpolation.
