Calculation of partition function of Ising model on quantum computer

А. А. Зайцев

Raymond Saturnino Diesposti

Keisuke Fujii, Tomoyuki Morimae

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.

Chikao Kawabata, Masaki Takeuchi, Yukihiko Karaki

Partition function of finite Ising model (spin=1/2) was exactly obtained by a computer for three dimensional even lattice 4×4×2=32 with periodic boundary condition in the presence of a magnetic field. Specific heat was calculated as a function of temperature. The zeros of the partition functions of the complex temperature plane and magnetic field plane were investigated.

I. R. Yukhnovskii