  We present fully polynomial approximation schemes for a broad class of Holant
problems with complex edge weights, which we call Holant polynomials. We
transform these problems into partition functions of abstract combinatorial
structures known as polymers in statistical physics. Our method involves
establishing zero-free regions for the partition functions of polymer models
and using the most significant terms of the cluster expansion to approximate
them.
  Results of our technique include new approximation and sampling algorithms
for a diverse class of Holant polynomials in the low-temperature regime and
approximation algorithms for general Holant problems with small signature
weights. Additionally, we give randomised approximation and sampling algorithms
with faster running times for more restrictive classes. Finally, we improve the
known zero-free regions for a perfect matching polynomial.
