Complexity of quantum cohomology

TL;DR

This paper investigates the complexity of quantum cohomology in 2D topological quantum field theories, providing bounds and characterizations for specific classes of symplectic manifolds, including Fano complete intersections and homogeneous varieties.

Contribution

It introduces bounds on the dimension of states with finite complexity in quantum cohomology and characterizes this space for certain varieties, including a sharp bound for Gr(2, n).

Findings

Bound on the dimension of states with finite complexity.

Sharp bound and description for Gr(2, n).

Positivity of eigenvalues of quantum multiplication by the quantum Euler class.

Abstract

Circuit complexity for two-dimensional topological quantum field theories (2D TQFT) was defined by Couch, Fan, and Shashi in [12]. In this paper, we study complexity for the 2D TQFT given by quantum cohomology of compact symplectic manifolds. We will estimate the number of states with finite approximate complexity of arbitrarily small tolerance for Fano complete intersections and (co)minuscule homogeneous varieties. We will give an upper bound for the dimension of the space spanned by states with finite complexity. In the case of Gr(2, n), this bound is sharp and we also obtain a precise description for this subspace. For (co)minuscule homogeneous varieties, we prove a positivity result for the eigenvalues of quantum multiplication by the handle element (also called the quantum Euler class) divided by the class of a point.