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Shorter unentangled proofs for ground state connectivity

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Abstract

Can one considerably shorten a proof for a quantum problem by using a protocol with a constant number of unentangled provers? We consider a frustration-free variant of the \(\textsf {QCMA}\)-complete ground state connectivity (GSCON) problem for a system of size n with a proof of superlinear size. We show that we can shorten this proof in \(\textsf {QMA}(2)\): There exists a two-copy, unentangled proof with length of order n, up to logarithmic factors, while the completeness–soundness gap of the new protocol becomes a small inverse polynomial in n.

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Notes

  1. In general, this could be also l-local unitaries; we choose \(l=2\). This variant of the problem is still QCMA complete [11].

  2. This inverse polynomial is quite small, as shown in Sect. 4.4: \(c'-s' = \Omega \left( \Delta ^{13} m^{-32} G^{-10} \right) \), with \(\Delta \) from the definition of GSCON and G the gate set size.

  3. Note that there are squares in the expression, while [3], Lemma 3.3, has a typo, missing the squares.

  4. Our proof would also go through for a very small \(\eta _1\), or could be avoided with more copies of the proof. However, we have not found a good enough way of performing a single measurement of energy for non-frustration-free Hamiltonians, that would with high enough probability tell if an energy of a single copy of a state (a superposition of eigenstates with various energies) is below or above thresholds that could be very close together.

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Acknowledgements

We thank Bill Fefferman, Cedric Lin, and Jens Eisert for interesting discussions. LC has done the work on this paper during his PhD studies at the Faculty of Mathematics, Physics and Informatics of the Comenius University in Bratislava. DN’s research has received funding from the People Programme (Marie Curie Actions) EU’s 7th Framework Programme under REA Grant Agreement No. 609427. This research has been further co-funded by the Slovak Academy of Sciences. LC and DN were also supported by the Slovak Research and Development Agency Grant QETWORK APVV-14-0878. MS thanks the Alexander von Humboldt Foundation for support.

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Authors and Affiliations

  1. Research Center for Quantum Information, Institute of Physics, Slovak Academy of Sciences, Dbravsk cesta 9, 84511, Bratislava, Slovakia

    Libor Caha & Daniel Nagaj

  2. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195, Berlin, Germany

    Martin Schwarz

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  1. Libor Caha
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  2. Daniel Nagaj
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Correspondence to Libor Caha.

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Caha, L., Nagaj, D. & Schwarz, M. Shorter unentangled proofs for ground state connectivity. Quantum Inf Process 17, 174 (2018). https://doi.org/10.1007/s11128-018-1944-4

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