  We implement the discrete-time quantum walk model using the continuous-time
evolution of the Hamiltonian that includes both the shift and the coin
generators. Based on the Trotter-Suzuki first-order approximation, we consider
an optimization problem in which the Hellinger distance between the walker
probability distributions resulted from the evolution of such Hamiltonian and
the quantum walk dynamics is minimized. The phase space implementation of the
quantum walk is considered where the walker state is encoded on the coherent
state of a resonator and the coin on the two-level state of a qubit. In this
approach, no mechanism for switching between the coin and the shift operators
is included. We show the Hellinger distance is bounded for large number of time
steps. The distance is small when we deviate from the standard quantum walk
model, namely, when the walker is allowed to move in between the sites. In
simulating the standard quantum walk model, the distance is large but bounded
by $26\%$ for a relevant number of time steps. Even so, the system evolution
shows the essential characteristics of the standard quantum walk dynamics,
namely, the ballistic evolution of the probability distribution and the linear
growth of the corresponding standard deviation. Moreover, the entanglement
generated in this approach is approximately the same as the entanglement
generated in the standard quantum walk dynamics. Finally, the system dynamics
under the influence of the decoherence also shows the similar
quantum-to-classical transition as in the standard quantum walk dynamics.
