  We provide a framework to determine the upper bound to the complexity of a
computing a given observable with respect to a Hamiltonian. By considering the
Heisenberg evolution of the observable, we show that each Hamiltonian defines
an equivalence relation, causing the operator space to be partitioned into
equivalence classes. Any operator within a specific class never leaves its
equivalence class during the evolution. We provide a method to determine the
dimension of the equivalence classes and evaluate it for various models, such
as the $ XY $ chain and Kitaev model on trees. Our findings reveal that the
complexity of operator evolution in the $XY$ model grows from the edge to the
bulk, which is physically manifested as suppressed relaxation of qubits near
the boundary. Our methods are used to reveal several new cases of simulable
quantum dynamics, including a $XY$-$ZZ$ model which cannot be reduced to free
fermions.
