  A Lie system is a non-autonomous system of ordinary differential equations
describing the integral curves of a $t$-dependent vector field taking values in
a finite-dimensional Lie algebra of vector fields. Lie systems have been
generalised in the literature to deal with $t$-dependent Schr\"odinger
equations determined by a particular class of $t$-dependent Hamiltonian
operators, the quantum Lie systems, and other differential equations through
the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and
quantum Lie systems to cope with $t$-dependent Schr\"odinger equations
associated with the here called quantum quasi-Lie systems. To illustrate our
methods, we propose and study a quantum analogue of the classical nonlinear
oscillator searched by Perelomov and we analyse a quantum one-dimensional fluid
in a trapping potential along with quantum $t$-dependent
Smorodinsky--Winternitz oscillators.
