  The complexity of the following numerical problem is studied in the quantum
model of computation: Consider a general elliptic partial differential equation
of order 2m in a smooth, bounded domain Q\subset \R^d with smooth coefficients
and homogeneous boundary conditions. We seek to approximate the solution on a
smooth submanifold M\subseteq Q of dimension 0\le d_1 \le d. With the right
hand side belonging to C^r(Q), and the error being measured in the L_\infty(M)
norm, we prove that the n-th minimal quantum error is (up to logarithmic
factors) of order n^{-min((r+2m)/d_1,r/d+1)}. For comparison, in the classical
deterministic setting the n-th minimal error is known to be of order n^{-r/d},
for all d_1, while in the classical randomized setting it is (up to logarithmic
factors) n^{-min((r+2m)/d_1,r/d+1/2)}.
