  Motivated by the revitalized interest in the digital quantum simulation of
coupled fermion-boson models from medium- and high-energy physics, we
investigate the nonequilibrium dynamics following a Yukawa-interaction quench
on IBM Q. Adopting -- due to current quantum-hardware limitations -- a
single-site (zero-dimensional) version of the scalar Yukawa-coupling model as
our point of departure, we design low-depth quantum circuits that emulate its
dynamics with up to three bosons. In particular, using advanced
circuit-optimization techniques, in the one-boson case we demonstrate circuit
compression, i.e.~design a constant-depth circuit containing only two CNOT
gates, regardless of the total simulation time. In the three-boson case --
where such a compression is not possible -- we design a circuit in which one
Trotter step entails $8$ CNOTs, this number being far below the maximal
CNOT-cost of a generic three-qubit gate. Using an analogy with the travelling
salesman problem, we also provide a CNOT-cost estimate for quantum circuits
emulating the system dynamics for higher boson-number truncations. Based on the
designed circuits, we quantify the system dynamics for the initial vacuum state
by evaluating the expected boson number at an arbitrary time after a
Yukawa-interaction quench, as well as the survival probability of the initial
state (the Loschmidt echo). Finally, we make use of the designed circuits to
drive adiabatic transitions and compute the energies of the ground- and first
excited state of the model under consideration. We validate our results by
performing error mitigation in the form of zero-noise extrapolation, finding an
excellent agreement of our obtained results with the exact ones obtained
through classical benchmarking.

Quantum Physics