  We propose an analog superconducting quantum simulator for a one-dimensional
model featuring momentum-dependent (nonlocal) electron-phonon couplings of
Su-Schrieffer-Heeger and "breathing-mode" types. Because its corresponding
vertex function depends on both the electron- and phonon quasimomenta, this
model does not belong to the realm of validity of the Gerlach-L\"{o}wen theorem
that rules out any nonanalyticities in single-particle properties. The
superconducting circuit behind the proposed simulator entails an array of
transmon qubits and microwave resonators. By applying microwave driving fields
to the qubits, a small-polaron Bloch state with an arbitrary quasimomentum can
be prepared in this system within times several orders of magnitude shorter
than the typical qubit decoherence times. We demonstrate that in this system --
by varying the circuit parameters -- one can readily reach the critical
coupling strength required for observing the sharp transition from a
nondegenerate (single-particle) ground state corresponding to zero
quasimomentum ($K_{\textrm{gs}}=0$) to a twofold-degenerate small-polaron
ground state at nonzero quasimomenta $K_{\textrm{gs}}$ and $-K_{\textrm{gs}}$.
Through exact numerical diagonalization of our effective Hamiltonian, we show
how this nonanalyticity is reflected in the relevant single-particle properties
(ground-state energy, quasiparticle residue, average number of phonons). The
proposed setup provides an ideal testbed for studying quantum dynamics of
polaron formation in systems with strongly momentum-dependent electron-phonon
interactions.
