A Many Body Eigenvalue Problem for Quantum Computation

A one dimensional many body Hamiltonian is presented whose eigenvalues are related to the order of $G_N$. This is the same order of $G_N$ used to decode the RSA algorithm. For some values of $N$ the Hamiltonian is a noninteracting fermion problem. For other values of $N$ the Hamiltonian is a quantum impurity problem with fermions interacting with a spin-like object. However, the generic case has fermions or spins interacting with higher order interactions beyond two body interactions. Because this is a mapping between two different classes of problems, one of interest in quantum computing and the other a more traditional condensed matter physics Hamiltonian, we will show (i) how knowledge of the order of $G_N$ can be used to solve some novel one dimensional strongly correlated problems and (ii) how numerical techniques, particularly for quantum impurity limit, can be used to find the order of $G_N$.