S. Iblisdir, M. Cirio, O. Boada, G. K. Brennen
A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided and shown to be compatible with previous results. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allow to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that wavefunction overlaps for ground states of quantum Hamiltonians, which can serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetisations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.
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http://arxiv.org/abs/1208.3918
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