Superconducting qubits beyond the dispersive regime

Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.