  We study topological properties of bound pairs of photons in
spatially-modulated qubit arrays (arrays of two-level atoms) coupled to a
waveguide. While bound pairs behave like Bloch waves, they are topologically
nontrivial in the parameter space formed by the center-of-mass momentum and the
modulation phase, where the latter plays the role of a synthetic dimension. In
a superlattice where each unit cell contains three two-level atoms (qubits), we
calculate the Chern numbers for the bound-state photon bands, which are found
to be $(1,-2,1)$. For open boundary condition, we find exotic topological
bound-pair edge states with radiative losses. Unlike the conventional case of
the bulk-edge correspondence, these novel edge modes not only exist in gaps
separating the bound-pair bands, but they also may merge with and penetrate
into the bands. By joining two structures with different spatial modulations,
we find long-lived interface states which may have applications in storage and
quantum information processing.
