Probing localization and long-range couplings in circuit QED with matrix product states

The accurate characterization of the low-energy spectrum of circuit-QED systems is central to quantum control, calibration, and circuit design. However, exact-diagonalization techniques are restricted to around a dozen modes. In this work we develop a variant of the density matrix renormalization group (DMRG) algorithm for targeting specific excited states in a 2d transmon array system, without solving for all eigenstates with energy lower than the target(s). Crucially, this method is robust in regimes of strong hybridization. We demonstrate this method by solving for all single-photon excitations of a 2d transmon array with up to 65 transmons; we analyze eigenstate localization and the effects of both intentional and spurious couplings. Our technique is a useful tool in the determination of context-dependent gate infidelities and optimal operating frequencies.