  Recent quantum simulation by Google [Nature 612, 240 (2022)] has demonstrated
the formation of bound states of interacting photons in a quantum-circuit
version of the XXZ spin chain. While such bound states are protected by
integrability in a one-dimensional chain, the experiment found the bound states
to be unexpectedly robust when integrability was broken by decorating the
circuit with additional qubits, at least for small numbers of qubits ($\leq
24$) within the experimental capability. Here we scrutinize this result by
state-of-the-art classical simulations, which greatly exceed the experimental
system sizes and provide a benchmark for future studies in larger circuits. We
find that the bound states consisting of a small and finite number of photons
are indeed robust in the non-integrable regime, even after scaling to the
infinite time and infinite system size limit. Moreover, we show that such
systems possess unusual spectral properties, with level statistics that
deviates from the random matrix theory expectation. On the other hand, for low
but finite density of photons, we find a much faster onset of thermalization
and significantly weaker signatures of bound states, suggesting that anomalous
dynamics may only be a property of dilute systems with zero density of photons
in the thermodynamic limit. The robustness of the bound states is also
influenced by the number of decoration qubits and, to a lesser degree, by the
regularity of their spatial arrangement.
