  Introducing a nonstoquastic catalyst is a promising avenue to improve quantum
annealing with the transverse field. In the present paper, we propose a
nonstoquastic catalyst for bifurcation-based quantum annealing described by the
spin-1 operators to improve the efficiency of a ground-state search. To
investigate the effect of the nonstoquastic catalyst, we study the
ferromagnetic $p$-spin model, which has difficulty with finding the ground
state due to the first-order phase transition for quantum annealing. A
semiclassical analysis shows that the problematic first-order phase transition
can be eliminated by introducing the proposed nonstoquastic catalyst with the
appropriate amplitude. We also numerically calculate the minimum energy gap for
a finite-size system by diagonalizing the Hamiltonian. We find that while the
energy gap decreases exponentially with increasing system size for the original
Hamiltonian, it decreases polynomially against the system size for the
Hamiltonian with the nonstoquastic catalyst. This result implies that the
proposed nonstoquastic catalyst has the potential to improve the performance of
bifurcation-based quantum annealing.
