  Operator growth, or operator spreading, describes the process where a
"simple" operator acquires increasing complexity under the Heisenberg time
evolution of a chaotic dynamics, therefore has been a key concept in the study
of quantum chaos in both single-particle and many-body systems. An explicit way
to quantify the complexity of an operator is the Shannon entropy of its
operator coefficients over a chosen set of operator basis, dubbed "coefficient
entropy". However, it remains unclear if the basis-dependency of the
coefficient entropy may result in a false diagnosis of operator growth, or the
lack thereof. In this paper, we examine the validity of coefficient entropy in
the presence of hidden symmetries. Using the quantum cat map as an example, we
show that under a generic choice of operator basis, the coefficient entropy
fails to capture the suppression of operator growth caused by the symmetries.
We further propose "symmetry-resolved coefficient entropy" as the proper
diagnosis of operator complexity, which takes into account robust unknown
symmetries, and demonstrate its effectiveness in the case of quantum cat map.

Statistical Mechanics;Quantum Physics