  We report on the design and implementation of a Field Programmable Josephson
Amplifier (FPJA) - a compact and lossless superconducting circuit that can be
programmed \textit{in situ} by a set of microwave drives to perform reciprocal
and nonreciprocal frequency conversion and amplification. In this work we
demonstrate four modes of operation: frequency conversion ($-0.5~\mathrm{dB}$
transmission, $-30~\mathrm{dB}$ reflection), circulation ($-0.5~\mathrm{dB}$
transmission, $-30~\mathrm{dB}$ reflection, $30~\mathrm{dB}$ isolation),
phase-preserving amplification (gain $>20~\mathrm{dB}$, $1~\mathrm{photon}$ of
added noise) and directional phase-preserving amplification ($-10~\mathrm{dB}$
reflection, $18~\mathrm{dB}$ forward gain, $8~\mathrm{dB}$ reverse isolation,
$1~\mathrm{photon}$ of added noise). The system exhibits quantitative agreement
with theoretical prediction. Based on a gradiometric Superconducting Quantum
Interference Device (SQUID) with Nb/Al-AlO$_x$/Nb Josephson junctions, the FPJA
is first-order insensitive to flux noise and can be operated without magnetic
shielding at low temperature. Due to its flexible design and compatibility with
existing superconducting fabrication techniques, the FPJA offers a
straightforward route toward on-chip integration with superconducting quantum
circuits such as qubits or microwave optomechanical systems.
