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

Understand

We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space.

• This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time.
• We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument.
• This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.