  Transmon arrays are one of the most promising platforms for quantum
information science. Despite being often considered simply as qubits, transmons
are inherently quantum mechanical multilevel systems. Being experimentally
controllable with high fidelity, the higher excited states beyond the qubit
subspace provide an important resource for hardware-efficient many-body quantum
simulations, quantum error correction, and quantum information protocols. Alas,
dissipation and dephasing phenomena generated by couplings to various
uncontrollable environments yield a practical limiting factor to their
utilization. To quantify this in detail, we present here the primary
consequences of single-transmon dissipation and dephasing to the many-body
dynamics of transmon arrays. We use analytical methods from perturbation theory
and quantum trajectory approach together with numerical simulations, and
deliberately consider the full Hilbert space including the higher excited
states. The three main non-unitary processes are many-body decoherence,
many-body dissipation, and heating/cooling transitions between different
anharmonicity manifolds. Of these, the many-body decoherence -- being
proportional to the squared distance between the many-body Fock states -- gives
the strictest limit for observing effective unitary dynamics. Considering
experimentally relevant parameters, including also the inevitable site-to-site
disorder, our results show that the state-of-the-art transmon arrays should be
ready for the task of demonstrating coherent many-body dynamics using the
higher excited states. However, the wider utilization of transmons for
ternary-and-beyond quantum computing calls for improving their coherence
properties.
