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A scheme for measuring complex temperature partition functions of Isingmodels is introduced. In the context of ordered qubit registers this schemefinds a natural translation in terms of global operations, and single particlemeasurements on the edge of the array. Two applications of this scheme arepresented. First, through appropriate Wick rotations, those amplitudes can beanalytically continued to yield estimates for partition functions of Isingmodels. Bounds on the estimation error, valid with high confidence, areprovided through a central-limit theorem, which validity extends beyond thepresent context. It holds for example for estimations of the Jones polynomial.Interestingly, the kind of state preparations and measurements involved in thisapplication can in principle be made "instantaneous", i.e. independent of thesystem size or the parameters being simulated. Second, the scheme allows toaccurately estimate some non-trivial invariants of links. A third resultconcerns the computational power of estimations of partition functions for realtemperature classical ferromagnetic Ising models on a square lattice. Weprovide conditions under which estimating such partition functions allows oneto reconstruct scattering amplitudes of quantum circuits making the problemBQP-hard. Using this mapping, we show that fidelity overlaps for ground statesof quantum Hamiltonians, which serve as a witness to quantum phase transitions,can be estimated from classical Ising model partition functions. Finally, weshow that the ability to accurately measure corner magnetizations on thermalstates of two-dimensional Ising models with magnetic field leads to fullypolynomial random approximation schemes (FPRAS) for the partition function.Each of these results corresponds to a section of the text that can beessentially read independently.

Low Depth Quantum Circuits for Ising Models