Universal Formalism and Software for Obtaining and Analyzing Hamiltonians of an Arbitrary Superconducting Circuits and Qubits.

We propose a mathematical formalism and software that allow to express Hamiltonian of any superconducting circuit in a standardized way. The process allows for efficient/quick diagonalization of Hamiltonians of complex circuits. With vastly increasing number of various complicated qubit designs (0-pi qubits, gmons,...) it becomes necessary to be able to describe various LCJ circuits analytically and model them numerically. Normally, this requires thorough study of different degrees of freedom in the circuit in order to decide which variables are essential to understanding the properties of a circuit and which are secondary. Proposed formalism automatically classifies different degrees of freedom of a circuit based on their essential properties (charge quantization, mode frequencies) and role in the circuit. Hamiltonian of any qubit circuit can be automatically obtained and the generalized coordinates can be quantized in a most natural way: through charge number operators for charge islands and photon number operators for oscillatory circuit modes. This, in turn allows us to automatically and efficiently find eigenstates of various quantum qubit circuits. We show how the formalism is used for advancing practical qubit applications.