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.
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