New answers tagged nielsen-and-chuang
1
In the first summation for $U$:
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}(it\Omega \frac{H_{2}}{\Omega})^{2n}=\cos(\Omega t)\begin{pmatrix} 0&0&0\\0&1&0\\0&0&1\end{pmatrix}(\text{This is wrong})$$
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}(it\Omega \frac{H_{2}}{\Omega})^{2n}=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\...
4
You seem to be overcomplicating this somewhat!
You are right to split it up into the two terms $H_1$ and $H_2$. So, we have
$$
e^{-i(H_1+H_2)t}=e^{-iH_1t}e^{-iH_2t}.
$$
Now, straightforwardly,
$$
e^{-iH_1t}=I+(e^{-i\delta t}-1)|00\rangle\langle 00|.
$$
Next, we need to think about the $e^{-iH_2t}$ term. Of course, it maps $|00\rangle$ to $|00\rangle$. So, ...
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