Enumeration of all superconducting circuits up to 5 nodes

Nonlinear superconducting circuits can be used as amplifiers, transducers, and qubits. Only a handful have been analyzed or built, so useful configurations likely remain undiscovered. In this paper, we enumerate all unique nonlinear superconducting circuits, up to five nodes in size, built of capacitors, inductors, and Josephson junctions. We catalog the resulting design space by sorting circuits into "Hamiltonian classes" using a set of reference coordinates. Finally, we search for novel superconducting qubits by explicitly considering all three-node circuits, showing how the results of our enumeration can be used as a starting point for circuit design tasks.