The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces

TL;DR

This paper analyzes the complexity of linear tensor product problems in (anti-) symmetric Hilbert spaces, providing explicit algorithms, optimality results, and complexity characterizations, with applications to quantum chemistry.

Contribution

It introduces an explicit optimal algorithm for symmetric and anti-symmetric tensor problems and characterizes their tractability, extending classical results to symmetric settings.

Findings

Explicit formula for worst case error in terms of singular values

Optimality of the proposed algorithm among a broad class

Characterizations of polynomial tractability based on symmetry

Abstract

We study linear problems defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the singular values of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems we give characterizations for polynomial tractability and strong polynomial tractability in terms of the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schr\"odinger equation.