  The ETH ansatz for matrix elements of a given operator in the energy
eigenstate basis results in a notion of thermalization for a chaotic system. In
this context for a certain quantity - to be found for a given model - one may
impose a particular condition on its matrix elements in the energy eigenstate
basis so that the corresponding quantity exhibit linear growth at late times.
The condition is to do with a possible pole structure the corresponding matrix
elements may have. Based on the general expectation of complexity one may want
to think of this quantity as a possible candidate for the quantum complexity.
We note, however, that for the explicit examples we have considered in this
paper, there are infinitely many quantities exhibiting similar behavior.
