  SrTiO$_3$-based heterointerfaces support quasi-two-dimensional (2D) electron
systems that are analogous to III-V semiconductor heterostructures, but also
possess superconducting, magnetic, spintronic, ferroelectric, and ferroelastic
degrees of freedom. Despite these rich properties, the relatively low
mobilities of 2D complex-oxide interfaces appear to preclude ballistic
transport in 1D. Here we show that the 2D LaAlO$_3$/SrTiO$_3$ interface can
support quantized ballistic transport of electrons and (non-superconducting)
electron pairs within quasi-1D structures that are created using a
well-established conductive atomic-force microscope (c-AFM) lithography
technique. The nature of transport ranges from truly single-mode (1D) to
three-dimensional (3D), depending on the applied magnetic field and gate
voltage. Quantization of the lowest $e^2/h$ plateau indicate a ballistic
mean-free path $l_{MF}\sim$ 20 $\mu$m, more than two orders of magnitude larger
than for 2D LaAlO$_3$/SrTiO$_3$ heterostructures. Non-superconducting electron
pairs are found to be stable in magnetic fields as high as $B=11$ T, and
propagate ballistically with conductance quantized at 2$e^2/h$. Theories of
one-dimensional (1D) transport of interacting electron systems depend crucially
on the sign of the electron-electron interaction, which may help explain the
highly ballistic transport behavior. The 1D geometry yields new insights into
the electronic structure of the LaAlO$_3$/SrTiO$_3$ system and offers a new
platform for the study of strongly interacting 1D electronic systems.
