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We consider the manifold of all quantum many-body states that can begenerated by arbitrary time-dependent local Hamiltonians in a time that scalespolynomially in the system size, and show that it occupies an exponentiallysmall volume in Hilbert space. This implies that the overwhelming majority ofstates in Hilbert space are not physical as they can only be produced after anexponentially long time. We establish this fact by making use of atime-dependent generalization of the Suzuki-Trotter expansion, followed by acounting argument. This also demonstrates that a computational model based onarbitrarily rapidly changing Hamiltonians is no more powerful than the standardquantum circuit model.

Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space