Computing the partition function of the Ising model on a graph has been investigated from both sides of computer science and statistical physics, with producing fertile results of P cases, FPTAS/FPRAS cases, inapproximability and intractability. Recently, measurement-based quantum computing as well as quantum annealing open up another bridge between two fields by relating a tree tensor network representing a quantum graph state to a rank decomposition of the graph. This paper makes this bridge wider in both directions. An $O^*(2^{ \frac{\omega}{2} bw(G)})$-time algorithm is developed for the partition function on n-vertex graph G with branch decomposition of width bw(G), where O^* ignores a polynomial factor in n and ω is the matrix multiplication parameter less than 2.37287. Related algorithms of $O^*(4^{rw(\tilde{G})})$ time for the tree tensor network are given which are of interest in quantum computation, given rank decomposition of a subdivided graph $\tilde{G}$ with width $rw(\tilde{G})$. These algorithms are parameter-exponential, i.e., O^*(c^p) for constant c and parameter p, and such an algorithm is not known for a more general case of computing the Tutte polynomial in terms of bw(G) (the current best time is O^*(min{2ⁿ, bw(G)^O(bw(G))})) with a negative result in terms of the clique-width, related to the rank-width, under ETH.