# Heisenberg XXX time evolution operator for three qubits

I've a problem to reproduce the result in equation (4) on page three of this paper: http://arxiv.org/abs/0802.2588.

So far I've understood that they apply a Heisenberg XXX interaction between 2&4, 4&6, 1&3 and 3&5. After this they measure with

$\frac{1}{2}(|0000><0000|+|1111><1111|)$

and calculate the fidelity. I think I calculate the last two steps correctly but I'm not sure about my implementation of the Heisenberg-Interaction.

At the moment I apply four matrices $U(\alpha)\otimes 1\otimes 1\otimes 1\otimes 1$, $1\otimes U(\alpha)\otimes 1\otimes 1\otimes 1$, $1\otimes 1\otimes 1\otimes 1\otimes U(\alpha)$ and $1\otimes 1\otimes 1\otimes U(\alpha)\otimes 1$ one after another (1 for the neutral element) after I sorted the qubits in the right order with swap gates. That means the initial state is $\rho_F\otimes\rho_F\otimes\rho_F$ with order "1-2-3-4-5-6" and I sort them to "1-3-5-2-4-6". But that doesn't seem right. Or is my measurement projector wrong?

Edit: Maybe I should add what I mean by $U(\alpha)$: When you calculate the time evolution operator from the Heisenberg Hamiltonian you get a term $\int dt J(t)$ in the e-function where J is the strength of the interaction. It depends on $t$ because you turn the interaction `on and off'. Now set $\alpha=\int dt J(t)$. That is $U(\alpha)$ just means time evolution is created by turning the interaction on and off.

Greets

First, my measurement was wrong. In the paper they use $|0011><0011$ and $|1100><1100$. But that was not the main problem of the question.
My first ansatz of applying an unitary operator on the qubits 1 and 2 at first and the same on the qubits 2 and 3 was wrong. One has to calculate the matrix exponential $exp(-i \alpha ((\frac{1}{4} 1_4 - \psi^-) \otimes 1_2 + 1_2 \otimes (\frac{1}{4} 1_4 - \psi^-))$ (the Hamilton operators come from http://arxiv.org/abs/1401.5670) because the interaction is applied to all three qubits at the same time. So the problem was to understand that I have to calculate the unitary operator from the sum of the hamilton operators which is not the same as the product of the unitary operators of both hamilton operators as I've learned. However I wasn't able to calculate this analyticaly but used an CAS to do this for me.