  Chains of superconducting circuit devices provide a natural platform for
studies of synthetic bosonic quantum matter. Motivated by the recent
experimental progress in realizing disordered and interacting chains of
superconducting transmon devices, we study the bosonic many-body localization
phase transition using the methods of exact diagonalization as well as matrix
product state dynamics. We estimate the location of transition separating the
ergodic and the many-body localized phases as a function of the disorder
strength and the many-body on-site interaction strength. The main difference
between the bosonic model realized by superconducting circuits and similar
fermionic model is that the effect of the on-site interaction is stronger due
to the possibility of multiple excitations occupying the same site. The phase
transition is found to be robust upon including longer-range hopping and
interaction terms present in the experiments. Furthermore, we calculate
experimentally relevant local observables and show that their temporal
fluctuations can be used to distinguish between the dynamics of Anderson
insulator, many-body localization, and delocalized phases. While we consider
unitary dynamics, neglecting the effects of dissipation, decoherence and
measurement back action, the timescales on which the dynamics is unitary are
sufficient for observation of characteristic dynamics in the many-body
localized phase. Moreover, the experimentally available disorder strength and
interactions allow for tuning the many-body localization phase transition, thus
making the arrays of superconducting circuit devices a promising platform for
exploring localization physics and phase transition.
