  Motivated by the recent realization of an artificial quantum spin ice in an
array of superconducting qubits with tunable parameters [King {\it et al.},
Science 373, 576 (2021)], we scrutinize a quantum six vertex model on the
square lattice that distinguishes type-I and type-II vertices. We map the
zero-temperature phase diagram using numerical (exact diagonalization) and
analytical (perturbation expansion, Gerschgorin theorem) methods. Following a
symmetry classification, we identify three crystalline phases alongside a
subextensive manifold of isolated configurations. Monte Carlo simulations at
the multicritical Rokhsar-Kivelson point provide evidence for a quantum phase
exhibiting a cascade of transitions with increasing flux. By comparing
structure factors, we find evidence for the emergence of the fully flippable
and plaquette phases in the artificial quantum spin ice.
