  We study the problem of approximating the Ising model partition function with
complex parameters on bounded degree graphs. We establish a deterministic
polynomial-time approximation scheme for the partition function when the
interactions and external fields are absolutely bounded close to zero.
Furthermore, we prove that for this class of Ising models the partition
function does not vanish. Our algorithm is based on an approach due to Barvinok
for approximating evaluations of a polynomial based on the location of the
complex zeros and a technique due to Patel and Regts for efficiently computing
the leading coefficients of graph polynomials on bounded degree graphs.
Finally, we show how our algorithm can be extended to approximate certain
output probability amplitudes of quantum circuits.
