  We introduce a novel hybrid algorithm to simulate the real-time evolution of
quantum systems using parameterized quantum circuits. The method, named
"projected - Variational Quantum Dynamics" (p-VQD) realizes an iterative,
global projection of the exact time evolution onto the parameterized manifold.
In the small time-step limit, this is equivalent to the McLachlan's variational
principle. Our approach is efficient in the sense that it exhibits an optimal
linear scaling with the total number of variational parameters. Furthermore, it
is global in the sense that it uses the variational principle to optimize all
parameters at once. The global nature of our approach then significantly
extends the scope of existing efficient variational methods, that instead
typically rely on the iterative optimization of a restricted subset of
variational parameters. Through numerical experiments, we also show that our
approach is particularly advantageous over existing global optimization
algorithms based on the time-dependent variational principle that, due to a
demanding quadratic scaling with parameter numbers, are unsuitable for large
parameterized quantum circuits.
