Computational Algorithm for Covariant Series Expansions in General Relativity

Abstract

Abstract. We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . . < n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. We show that this computational problem belongs to the complexity class LEXP.

A Geometric View on Quantum Tensor Networks

“…In the latter case the computations become much more complicated and extended, but this approach guarantees that the state transformation will be unitary for all t ∈ [0, 1]. It should be stressed that one has to compute the components in the basis {σ i }, not in the coordinate basis of normal coordinates; a computation algorithm was proposed in [12]. Note that in a system of n qubits, one has 2 n − 1 partitions into two complementary pairs of subsystems, and for each pair the von Neumann entropy can be easily computed in terms of the coefficients a i 1 ...i n .…”