The Quantum Query Complexity of Elliptic PDE

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)}.