  Controlling the current-phase relation (CPR) of Josephson elements is
essential for tailoring the eigenstates of superconducting qubits, tuning the
properties of parametric amplifiers, and designing nonreciprocal
superconducting devices. Here, we introduce the hybrid Josephson rhombus, a
highly tunable superconducting circuit containing four
semiconductor-superconductor hybrid Josephson junctions embedded in a loop.
Combining magnetic frustration with gate-voltage-controlled tuning of
individual Josephson couplings provides deterministic control of the harmonic
content of the rhombus CPR. We show that for balanced Josephson couplings at
full frustration, the hybrid rhombus displays a $\pi$-periodic $\cos(2\varphi)$
potential, indicating coherent charge-$4e$ transport. Tuning away from the
balanced configuration, we observe a superconducting diode effect with
efficiency exceeding 25%. These results showcase the potential of hybrid
Josephson rhombi as fundamental building blocks for noise-resilient qubits and
quantum devices with custom transport properties.
