Computations in Quantum Tensor Networks

Abstract

The computation of the ground state (i.e. the eigenvector related to the smallest eigenvalue) is an important task in the simulation of quantum many-body systems. As the dimension of the underlying vector space grows exponentially in the number of particles, one has to consider appropriate subsets promising both convenient approximation properties and efficient computations. The variational ansatz for this numerical approach leads to the minimization of the Rayleigh quotient. The Alternating Least Squares technique is then applied to break down the eigenvector computation to problems of appropriate size, which can be solved by classical methods. Efficient computations require fast computation of the matrix-vector product and of the inner product of two decomposed vectors. To this end, both appropriate representations of vectors and efficient contraction schemes are needed. Here approaches from many-body quantum physics for one-dimensional and two-dimensional systems (Matrix Product States and Projected Entangled Pair States) are treated mathematically in terms of tensors. We give the definition of these concepts, bring some results concerning uniqueness and numerical stability and show how computations can be executed efficiently within these concepts. Based on this overview we present some modifications and generalizations of these concepts and show that they still allow efficient computations such as applicable contraction schemes. In this context we consider the minimization of the Rayleigh quotient in terms of the {\sc parafac} (CP) formalism, where we also allow different tensor partitions. This approach makes use of efficient contraction schemes for the calculation of inner products in a way that can easily be extended to the mps format but also to higher dimensional problems.

Presented in part at the 26th GAMM Seminar on Tensor Approximations at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig, February 2010

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