  According to the adiabatic theorem of quantum mechanics, a system initially
in the ground state of a Hamiltonian remains in the ground state if one slowly
changes the Hamiltonian. This can be used in principle to solve hard problems
on quantum computers. Generically, however, implementation of this Hamiltonian
dynamics on digital quantum computers requires scaling Trotter step size with
system size and simulation time, which incurs a large gate count. In this work,
we argue that for classical optimization problems, the adiabatic evolution can
be performed with a fixed, finite Trotter step. This "Floquet adiabatic
evolution" reduces by several orders of magnitude the gate count compared to
the usual, continuous-time adiabatic evolution. We give numerical evidence
using matrix-product-state simulations that it can optimally solve the Max-Cut
problem on $3$-regular graphs in a large number of instances, with surprisingly
low runtime, even with bond dimensions as low as $D=2$. Extrapolating our
numerical results, we estimate the resources needed for a quantum computer to
compete with classical exact or approximate solvers for this specific problem.
