  Investigating Shubnikov-de Haas (SdH) oscillations in high magnetic fields,
we experimentally infer the electronic band structure of the
quasi-two-dimensional electron gas (2DEG) at the ionic-liquid gated amorphous
(a)-LaAlO$_3$/KTaO$_3$ interface. The angular dependence of SdH oscillations
indicates a 2D confinement of a majority of electrons at the interface.
However, additional SdH oscillations with an angle-independent frequency
observed at high tilt angles indicate the coexistence of 3D charge carriers
extending deep into the KTaO$_3$. The SdH oscillations measured in magnetic
fields perpendicular to the interface show four frequencies corresponding to
four 2D subbands with different effective masses (0.20 $m_e$ - 0.55 $m_e$). The
single-frequency oscillations originating from 3D electrons yields a larger
effective mass of $\sim$ 0.70 $m_e$. Overall, the inferred subbands are in good
agreement with the theoretical-calculations and angle-resolved photoemission
spectroscopy studies on 2DEG at KTaO$_3$ surface.
