  Recent experiments on the dipolar-octupolar pyrochlore compound
Ce$_2$Zr$_2$O$_7$ indicate that it may realize a three-dimensional quantum spin
liquid (QSL). In particular, the analyses of available data suggest that the
system is in a region of parameter space, where the so-called $\pi$-flux $U(1)$
octupolar quantum spin ice (QSI) with an emergent photon could be realized. On
the other hand, because the system is far from the perturbative classical spin
ice regime, it is unclear whether the quantum ground state is primarily a
coherent superposition of 2-in-2-out configurations as in the canonical QSI
phases. In this work, we explore other possible competing quantum spin liquid
states beyond QSI using the Schwinger boson parton construction of the
dipolar-octupolar pseudospin-1/2 model. After classifying all symmetric $U(1)$
and $\mathbb{Z}_2$ QSLs using the projective symmetry group (PSG), we construct
a mean-field phase diagram that possesses a number of important features
observed in an earlier exact diagonalization study. In the experimentally
relevant frustrated region of the phase diagram, we find two closely competing
gapped $\mathbb{Z}_2$ QSLs with a narrow spinon dispersion. The equal-time and
dynamical spin structure factors of these states show key features similar to
what was reported in neutron scattering experiments. Hence, these QSLs are
closely competing ground states in addition to the $\pi$-flux $U(1)$ QSI, and
they should be taken into account on equal footing for the interpretation of
experiments on Ce$_2$Zr$_2$O$_7$.
