Polynomially restricted operator growth in dynamically integrable models

We provide a framework to determine the upper bound for the complexity of computing the exact Heisenberg representation of a given operator with respect to a Hamiltonian. Working in the Heisenberg picture, 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 it 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. For classes with a dimensionality that is not excessively large, one can use our framework to compute the exact Heisenberg representation of any operator within that class. Our findings reveal that the complexity of operator evolution in these models grows from the edge to the bulk, which is physically manifested as suppressed relaxation of qubits near the boundary. Our methods serve to reveal several new cases of simulable quantum dynamics, such as $XY$ evolution augmented with certain non-Clifford single-qubit gates. We also introduce an $XY\text{\ensuremath{-}}ZZ$ model where the dimension of an equivalence class, while exponential in the system size, grows considerably slower than might be naively anticipated, which entails a practical computational advantage when simulating finite systems. Furthermore, we demonstrate how to apply our method to time-dependent Hamiltonians and dissipative systems.