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It is well known that any quantum time-evolution of local, time-dependent Hamiltonians can be described using a poly-depth (in number of qubits) quantum circuit (DOI:10.1103/PhysRevLett.106.170501; DOI: 10.1126/science.273.5278.1073). Since poly-depth quantum circuits can be described with a polynomial number of variables, doesn't this mean that the space of quantum states that are reachable in polynomial time is only a polynomially large submanifold of the full Hilbert space? Thus, the actual submanifold of realistic states is only polynomially large, and that all realistic wavefunctions that lives within this space only contain a polynomial amount of information, despite being embedded in an exponentially large Hilbert space?

If the above is true, that the actual informational content of realistic wavefunctions is polynomial, then surely we must conclude that describing quantum wavefunctions as vectors in Hilbert space is a major “overkill” and, therefore, an unphysical description. In my opinion, a much more reasonable description of quantum states would be in the form of quantum circuits, or (in some specific cases) tensor networks. What do you think?

Note: whenever I say "polynomial" or "exponential" I mean in the number of quantum particles (e.g. qubits).

Niko
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1 Answers1

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The quantum information of the physically accessible states is only polynomial, but bear in mind that the physically accessible states are still a Hilbert space. The authors of the PRL paper IMO go too far in their conclusion with the following statement:

This raises the question of whether it makes sense to describe many-body quantum systems as vectors in a linear Hilbert space.

The physical states still form a Hilbert space, nothing is changed by the fact that it's a tiny fraction of the "full Hilbert space". Their result only shows that the commonly used bases are a computationally very inefficient way to represent the physical Hilbert space.

The recent advances in real-space renormalization group methods [3,23,24] indeed seem to suggest that a viable approach consists of parametrizing quantum many-body states using tensor networks and quantum circuits.

The authors for some reason seem to be under the impression that quantum circuits are not described by a Hilbert space, which is nonsensical. What they've argued is in fact that a much more useful representation of the Hilbert space is using tensor networks and quantum circuits.

AwkwardWhale
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