  The quantum mechanics of superconducting circuits is derived by starting from
a classical Hamiltonian dynamical system describing a dissipationless circuit,
usually made of capacitive and inductive elements. However, standard approaches
to circuit quantization treat fluxes and charges, which end up as the
canonically conjugate degrees of freedom on phase space, asymmetrically. By
combining intuition from topological graph theory with a recent symplectic
geometry approach to circuit quantization, we present a theory of circuit
quantization that treats charges and fluxes on a manifestly symmetric footing.
For planar circuits, known circuit dualities are a natural canonical
transformation on the classical phase space. We discuss the extent to which
such circuit dualities generalize to non-planar circuits.
