The quantum query complexity of elliptic PDE

The query 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 ⊂ Rd with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M ⊆ Q of dimension 0 ≤ d1 ≤ d. With the right-hand side belonging to Cr (Q), and the error being measured in the L∞(M) norm, we prove that the nth minimal quantum error is (up to logarithmic factors) of order n-min((r+2m)/d1,r/d+1). For comparison, in the classical deterministic setting the nth minimal error is known to be of order n-r/d, for all d1, while in the classical randomized setting it is (up to logarithmic factors) n-min((r+2m)/d1,r/d+1/2).

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