Kirchhoff's rule for quantum wires. II: The inverse problem with possible applications to quantum computers

In this article we continue our investigations of one particle quantum scat tering theory for Schrodinger operators on a set of connected (idealized on e-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linea r boundary conditions at the vertices (the local Kirchhoff law). In "Kirchh off's rule for quantum wires" [J. Phys. A: Math. Gen. 32, 595-630 (1999)] w e provided an explicit algebraic expression for the resulting (on-shell) S- matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unit ary n x n matrix may be realized as the S-matrix at a given energy by choos ing appropriate (unique) boundary conditions. This might possibly be used f or the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a g iven time associated to a quantum mechanical Hamiltonian. We also suggest a n approach by which the S-matrix in our context may be obtained from "scatt ering experiments", another aspect of the inverse problem. Finally we exten d our previous discussion, how our approach is related to von Neumann's the ory of selfadjoint extensions.