  We derive macroscopically deterministic flow equations with regard to the
order parameters of the ferromagnetic $p$-spin model with infinite-range
interactions. The $p$-spin model has a first-order phase transition for $p>2$.
In the case of $p\geq5$ ,the $p$-spin model with anti-ferromagnetic XX
interaction has a second-order phase transition in a certain region. In this
case, however, the model becomes a non-stoqustic Hamiltonian, resulting in a
negative sign problem. To simulate the $p$-spin model with anti-ferromagnetic
XX interaction, we utilize the adaptive quantum Monte Carlo method. By using
this method, we can regard the effect of the anti-ferromagnetic XX interaction
as fluctuations of the transverse magnetic field. A previous study derived
deterministic flow equations of the order parameters in the quantum Monte Carlo
method. In this study, we derive macroscopically deterministic flow equations
for the magnetization and transverse magnetization from the master equation in
the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter
decomposition, we consider the Glauber-type stochastic process. We solve these
differential equations by using the Runge-Kutta method and verify that these
results are consistent with the saddle-point solution of mean-field theory.
Finally, we analyze the stability of the equilibrium solutions obtained by the
differential equations.
