  We present a way to achieve fully directional, quantum-limited
phase-preserving amplification in a four-port, four-mode superconducting
Josephson circuit by utilizing interference between six parametric processes
that couple all four modes. Full directionality, defined as the reverse
isolation surpassing forward gain between the matched input and output ports of
the amplifier, ensures its robustness against impedance mismatch that might be
present at its output port during applications. Unlike existing directional
phase-preserving amplifiers, both the minimal back-action and the
quantum-limited added noise of this amplifier remains unaffected by noise
incident on its output port. In addition, the matched input and output ports
allow direct on-chip integration of these amplifiers with other circuit QED
components, facilitating scaling up of superconducting quantum processors.
