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If we can use PEPS (Projected Entangled Pair State) to represent a many body quantum state, can we generate it by a quantum computer?

As far as I can understand, PEPS is dual to a quantum computer with postselection: any PEPS can be created by a postselected quantum circuit, and any output of such a circuit can be written as a Peps (arXiv:quant-ph/0611050). But quantum computer with post selection is not a physical device.

Does this mean PEPS can be used to represent certain high complexity state that can not be efficiently prepared by a quantum computer from a simple initial state?

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You are absolutely right: The result about PEPS = postselection, together with the fact that postselection is considerably more powerful than polynomial-time quantum computation, implies that it is impossible to prepare a general PEPS efficiently on a quantum computer. So in that sense, PEPS can describe high-complexity quantum states (just as, for instance, certain Hamiltonians can).

Note, however, that it might be that the complexity is in the translation from the PEPS description to the state, rather than in the preparation procedure itself. This is the case, for instance, in variants of the construction in the cited paper (which basically yields a product state). Note that the same is true, e.g., for preparing the ground state of a classical spin glass Hamiltonian (which is a product state), when starting from the Hamiltonian.

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  • $\begingroup$ Thanks. Sorry I have to add another comment. If the so-called 'efficient representation' of a quantum state like PEPS does not mean the state is 'physical', then PEPS seems to only work as a numerical tool but its structure itself does not encode the reality of the physical world, which is in a state that can be physically realized. Does this means PEPS (maybe also neural network) can not be used to explain spacetime geometry as in MERA/AdS duality? $\endgroup$ – XXDD Apr 11 '18 at 4:13
  • $\begingroup$ @X.Dong What you are saying is not logical. In analogy, the set of all quantum states contains plenty of states which are not physical, yet it can be used to encode the reality of the physical world. $\endgroup$ – Norbert Schuch Apr 11 '18 at 6:27
  • $\begingroup$ Your statement sounds strange. If a certain quantum state is too complex to be physical, then it should not be part of the real physical world, or the physical world is not in such a state . Why we need to bother the complex state to describe the reality? Even there is a possibility that both a complex and a simple quantum state describe (generate) the same physical (observed) world, we can use (or prefer to use) the simple state to understand the world. Is our world in an unphysical quantum state? $\endgroup$ – XXDD Apr 12 '18 at 11:18

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