Pirsa: 13110065 - A universal Hamiltonian simulator: the full characterization


Gemma De Las Cuevas

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We show that if the ground state energy problem of a classical spin model is NP-hard, then there exists a choice parameters of the model such that its low energy spectrum coincides with the spectrum of \emph{any} other model, and, furthermore, the corresponding eigenstates match on a subset of its spins. This implies that all spin physics, for example all possible universality classes, arise in a single model. The latter property was recently introduced and called ``Hamiltonian completeness'', and it was shown that several different models had this property. We thus show that Hamiltonian completeness is essentially equivalent to the more familiar complexity-theoretic notion of NP-completeness. Additionally, we also show that Hamiltonian completeness implies that the partition functions are the same. These results allow us to prove that the 2D Ising model with fields is Hamiltonian complete, which is substantially simpler than the previous examples of complete Hamiltonians. Joint work with Toby Cubitt.
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