Sectional curvatures distribution of complexity geometry

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