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- A universal Hamiltonian simulator: the full characterization

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13110065

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.

©2012 Perimeter Institute for Theoretical Physics