  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 through a central-limit theorem, which validity extends beyond the
present context. It holds for example for estimations of the Jones polynomial.
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 allows one
to reconstruct scattering amplitudes of quantum circuits making the problem
BQP-hard. Using this mapping, we show that fidelity overlaps for ground states
of quantum Hamiltonians, which 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 magnetizations 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.
