  In a recent preprint by Deutsch et al. [1995] the authors suggest the
possibility of polynomial approximability of arbitrary unitary operations on
$n$ qubits by 2-qubit unitary operations. We address that comment by proving
strong lower bounds on the approximation capabilities of g-qubit unitary
operations for fixed g. We consider approximation of unitary operations on
subspaces as well as approximation of states and of density matrices by quantum
circuits in several natural metrics. The ability of quantum circuits to
probabilistically solve decision problem and guess checkable functions is
discussed. We also address exact unitary representation by reducing the upper
bound by a factor of n^2 and by formalizing the argument given by Barenco et
al. [1995] for the lower bound. The overall conclusion is that almost all
problems are hard to solve with quantum circuits.
