  Systems of correlated particles appear in many fields of science and
represent some of the most intractable puzzles in nature. The computational
challenge in these systems arises when interactions become comparable to other
energy scales, which makes the state of each particle depend on all other
particles. The lack of general solutions for the 3-body problem and acceptable
theory for strongly correlated electrons shows that our understanding of
correlated systems fades when the particle number or the interaction strength
increases. One of the hallmarks of interacting systems is the formation of
multi-particle bound states. In a ring of 24 superconducting qubits, we develop
a high fidelity parameterizable fSim gate that we use to implement the periodic
quantum circuit of the spin-1/2 XXZ model, an archetypal model of interaction.
By placing microwave photons in adjacent qubit sites, we study the propagation
of these excitations and observe their bound nature for up to 5 photons. We
devise a phase sensitive method for constructing the few-body spectrum of the
bound states and extract their pseudo-charge by introducing a synthetic flux.
By introducing interactions between the ring and additional qubits, we observe
an unexpected resilience of the bound states to integrability breaking. This
finding goes against the common wisdom that bound states in non-integrable
systems are unstable when their energies overlap with the continuum spectrum.
Our work provides experimental evidence for bound states of interacting photons
and discovers their stability beyond the integrability limit.
