The 1994 discovery by P. Shor of a polynomial quantum factoring algorithm has started initial efforts into the development of quantum computers. This opened new avenues of research for mathematicians, computer scientists, and physicists alike. The quantum paradigm of computing induced renewed interest in complexity classes, (quantum) error correction, searches for other algorithms, and the search for physical implementations of the basic building blocks, such as qubits and quantum gates, of a quantum computer.

We have applied multigrid methods to solve for a two-dimensional Schrödinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for eigenfunctions of a Schrödinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential.

Length and energy scales of the trapping potential and interaction potential are too disparate to allow for a naive discretization of the differential operator. The size of the resulting linear systems would prohibit reasonable calculations. An adaptive grid approach automatically finds a spatial representation that is refined when the atoms are close together and the interaction potential is very strong and is coarse at larger separation where the trapping potential is the largest energy scale. The finite element discretization of the Schrödinger equation on this grid produces an eigenproblem with the matrix A^-1B where A is the usual finite element discretization of a second order linear elliptic operator and B contains the L₂ inner products of the finite element basis functions. Selected eigenvalues and eigenvectors are computed by an iterative method that requires only multiplication of a vector by the matrix. Multiplication by A^-1 is achieved by solving a linear system with the matrix A by a multigrid method.