Flux-Charge Symmetric Quantization of Nonreciprocal Superconducting Circuits

Superconducting circuits are a leading platform for quantum simulation and computation, renewing interest in circuit theory and necessitating Hamiltonian formalisms that describe the dynamics that can be naively realized by lumped element circuits. Using homological algebra, we incorporate nonreciprocal elements into a theory of quantizing reciprocal circuits that explicitly encodes the circuit's topology into its Lagrangian. We prove that arbitrarily nonlinear and nonreciprocal circuits admit a Hamiltonian description assuming only the solubility of Kirchoff's laws, the conservation of energy, and the form of the constitutive relations. Additionally, we have derived nonreciprocal circuit simplification rules that act as canonical (unitary) transformations of classical (quantum) circuit Hamiltonians, preserving their spectra while combining and occasionally removing gyrators. These rules also extend Tellegen's discovery that gyrators convert circuit elements to their electrical dual to nonlinear elements. We find that some nonreciprocal circuits are unitarily related to reciprocal circuits while stating the conditions under which a circuit's dynamics may only be represented by nonreciprocal elements.