Subspace Iteration Methods in terms of Matrix Product States

When dealing with quantum many-body systems one is faced with problems growing exponentially in the number of particles to be considered. To overcome this curse of dimensionality one has to consider representation formats which scale only polynomially. Physicists developed concepts like matrix product states (MPS) to represent states of interest and formulated algorithms such as the density matrix renormalization group (DMRG) to find such states. We consider the standard Lanczos algorithm and formulate it for vectors given in the MPS format. It turns out that a restarted version which includes a projection onto the MPS manifold gives the same approximation quality as the well-established DMRG method. Moreover, this variant is more flexible and provides more information about the spectrum.