The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces
Provides complexity bounds and optimal algorithms for linear problems in symmetric Hilbert spaces, relevant for quantum chemistry and high-dimensional approximation.
The paper studies linear problems on (anti-)symmetric tensor product Hilbert spaces, constructing an optimal linear algorithm with explicit error formula and characterizing polynomial tractability in terms of symmetry. For the electronic Schrödinger equation, the results show that symmetry can reduce complexity compared to unrestricted problems.
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 Schrödinger equation.