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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).

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  • $\begingroup$ I don't quite understand what you mean when you say that the submanifold of "realistic states" is polynomially large. If you have $N$ qubits, then the relevant Hilbert space is $2^N$-dimensional. That's a finite number, and every linear superposition of those $2^N$ basis vectors corresponds to a physically allowed pure state. Which of them are you advocating be thrown away as unphysical? $\endgroup$
    – J. Murray
    Commented Apr 12, 2022 at 15:24
  • $\begingroup$ What does 'polynomial time' here mean? Typically you can use tensor networks to write ground states of gapped systems but after time evolution the entanglement entropy rises too fast to probe intermediate time dynamics this way. $\endgroup$
    – jacob1729
    Commented Apr 12, 2022 at 15:37
  • $\begingroup$ @jacob1729 Polynomial time here means a time-evolution that is of the order of poly(N) length, with N being the number of qubits. $\endgroup$
    – Niko
    Commented Apr 12, 2022 at 16:00
  • $\begingroup$ @J.Murray I'm advocating to throw out all states which are not reachable in polynomial time. See the first reference. They there say that only an “exponentially small” submanifold of the full Hilbert space is reachable during a poly(N) length time-evolution. But I'm not entirely sure what this “exponentially small” means – do they mean a submanifold that is poly(N) volume, or one that's just an exponentially small fraction of the Hilbert space? (though this could still be exponentially large itself: e.g. 2^N divided by 1/2^{N/2} is 2^{N/2}). $\endgroup$
    – Niko
    Commented Apr 12, 2022 at 16:00

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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.

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  • $\begingroup$ Thanks for the reply, I gave an upvote. I suppose the authors' point is that the “Hilbert space” description is misleading in the sense that it implies quantum many-body systems contain exponential amounts of information, when in fact they don't (unless you decide to time-evolve the system for an exponential amount of time). ––– But one thing I don't understand, is whether the PRL authors are saying the volume of “physical states” is polynomially large or not? I mean, they claim it's an exponentially small fraction, but what does that mean exactly? $\endgroup$
    – Niko
    Commented Apr 13, 2022 at 9:24
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    $\begingroup$ The conceptual problem with counting arguments is that the Hilbert space is uncountable, while a finite set of unparameterized gates gives you a countable number of possible states as output. So the most you can do is approximate a state to within some accuracy $\epsilon$. Therefore each output state in your circuit corresponds to a patch in Hilbert space of radius $\epsilon$, and the number of distinct patches scales super-exponentially with system size. The actual number of physically reachable patches they showed to scale exponentially, its given by $N_{circuits}=(MK^2)^{K^\alpha}$. $\endgroup$ Commented Apr 13, 2022 at 14:20
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    $\begingroup$ Also maybe its worth considering a concrete example of how physical Hilbert spaces can be far smaller than the full: take for instance permutationally symmetric Hamiltonians on $n$ qubits. Any state evolving from the ground state can be described using symmetric Dicke states, which form an $n$ dimensional Hilbert space, while the tensor product basis gives you a $2^n$ dimensional Hilbert space. $\endgroup$ Commented Apr 13, 2022 at 14:25
  • $\begingroup$ Why is the Hilbert space uncountable? If I have $n$ qubits, then surely the cardinality of my Hilbert space is $2^n$? ––– I appreciate that if you have a certain number of gates available, you'll be able to combine them in a certain number of different ways, allowing you to project your initial state into a certain number of different states. That makes sense to me, but I can't say I understand what a “patch” is or why there are super-exponentially many of them? $\endgroup$
    – Niko
    Commented Apr 13, 2022 at 14:44
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    $\begingroup$ The basis for the Hilbert space is countable, but the set of all states that can be generated by the basis set is uncountable, since it's parameterised by complex numbers. So in the "true" Hilbert space, we can't talk about the number of states, much like we can't talk about "how many real numbers" there are in an interval. On the other hand, if we only care about approximating states to within some $\epsilon$, we can ask how many of these patches fit in our Hilbert space. This is what the authors are counting. $\endgroup$ Commented Apr 13, 2022 at 15:04

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