Diagonalizing Hamiltonian for Multi-Qubit Ising Model

[mehdi86]

Hello,I am doing a research related to Ising Model
in m research, evolution of a multi-qubit Ising system with the initial and final
Hamiltonian is given by:
(1) H_i=(-1/2)$\sum\sigma^{(i)}_{x}$
(2) H_f=(-1/2)[itex]\sumh_i\sigma^{(i)}_{z}[/itex]+(1/2)$\sumJ_{ij}\sigma^{(i)}_{z}\sigma^{(j)}_{z}$
(3) H_s(t)=(1-s(t))H_i+s(t)H_f
and in H_f must i>j in the second summation, and h_i and J_ij from{-1,0,1}. please help me how can I diagonalize the Hamiltonian (3) with (1) and (2) with random instances for 20 qubits by randomly choosing h_i and J_ij. mostly I have problem with multiplying two sigmas in H_f.
Thanks a lot.

 

[DrDu]

Maybe you should first repair the display of your formulas so that they can be read by everybody in the forum.
I don't think it is a good idea to diagonalize the hamiltonian (which is a matrix of size 2^20 times 2^20 or 10^6 times 10^6) if you are only interested in the time evolution of selected states.
Your initial hamiltonian is diagonal (or practically so, if you interchange the roles of sigma_x and sigma_z) and it should be easy to follow the time evolution of a single state.

 

[mehdi86]

excuse me,our Hamiltonian is:
H_i=(-1/2)$\sum\sigma^{(i)}_{x}$
H_f=(-1/2)$\sum\sigma^{(i)}_{z}$h_i
+(1/2)$\sum\sigma^{(i)}_{z}*sigma^{(j)}_{z}$J_ij
Moreover if we want diagonalize h_s(t) for two first states(i.e. 0 &1 states), what shall we do?
thank you very much.

 

[DrDu]

Finding the lowest states for fixed t is quite easy: It is easy to see that applying $\exp(-\tau H)$ on any starting function which is not orthogonal to the ground state. will converge to the eigenstate corresponding to the lowest eigenvalue in the limit tau to infinity. For small tau, you can write
$\exp(-\tau H)\approx (1-\tau H)$. So you just have to apply the last expression with a small value of tau (smaller than the norm of H) repeatedly to some starting function. Eventually you have to rescale the function every now an then to avoid it to become to small. Once you have found the ground state, you can start again from a state orthogonal to it and repeat the procedure to find the first excited state. If you want to find more states, there exist more elaborate methods. Consider a book on numerical mathematics, e.g. "Numerical recipes" is a good starting point.