[@Kac05], while in [@Kwa05] certain parabolic problems were studied in this setting. The complexity of elliptic problems in the quantum model of computation has not been analyzed before. This is the topic of the present paper. We consider a general elliptic partial differential equation given on a smooth domain in ${{\mathbb R}}^d$, with smooth coefficients and homogeneous boundary conditions. We seek to find an approximation to the solution on a given, $d_1$-dimensional smooth submanifold, where $0\le d_1\le d$. Thus, we consider the whole range of problems from local solution (find the solution in a single point, $d_1=0$) up to global solution (find the full solution, in the whole domain, $d_1=d$). Our analysis is carried out in the quantum setting of information-based complexity theory, as developed in [@Hei01]. For a study of other basic numerical problems in this framework we refer to [@Nov01; @Hei01b; @Hei04a; @Hei04b; @Kac05; @Kwa05; @TW01; @Wie04], see also the surveys [@Hei03; @Hei04c]. For general background on quantum computation we refer to the surveys [@Aha98], [@EHI00], [@Sho00], and the monographs [@Pit99], [@Gru99], [@NC00]. For the classical settings of information-based complexity theory we refer to [@TWW88; @Nov88; @Hei93].

This paper can be considered as a continuation of [@Hei05a; @Hei05b]. 