Sectional curvatures distribution of complexity geometry

A bstract In the geometric approach to defining complexity, operator complexity is defined as distance in space. this paper, based on analogy with circuit size adopted metric of space where path length complexity. The typical sectional curvatures geometry are positive. It further proved that always positive if an arbitrary function size, while usually expected be negatively curved manifolds. By analyzing distribution for N -qubit system, it shown surfaces generated by Hamiltonians smaller than can have negative curvatures. large limit, form uniquely constrained up constant corrections we require order 1 /N 2 . With knowledge states, should modified due redundant action operators, and thus generalized state-dependent. Then use state-dependent Hilbert define state also space, 2-surfaces operators much acting states