Polynomially restricted operator growth in dynamically integrable models

TL;DR

This paper introduces a framework to analyze the complexity of observable evolution in Hamiltonian systems, revealing operator class structures and their implications for quantum dynamics and simulability.

Contribution

It develops a method to determine equivalence class dimensions in operator space, applied to models like XY and Kitaev, identifying new simulable quantum dynamics cases.

Findings

Operator complexity in XY model varies from edge to bulk.

Boundary qubits exhibit suppressed relaxation.

Identifies new simulable quantum models, including XY-ZZ.

Abstract

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.