Timeline for Controlled U gate with arbitrary U
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 11 at 7:44 | comment | added | DaftWullie | $1+e^{i\theta}=e^{i\theta/2}(e^{-i\theta/2}+e^{i\theta/2})$. The global factor disappears in the norm. | |
| Nov 9 at 16:55 | comment | added | Daniel Hoekwater | We can simplify $\cos\theta^2+\sin\theta^2+2\cos\theta+1$ to either $4\cos(\frac{\theta}{2})^2$ or $2(\cos\theta + 1)$, so our answer becomes either $a^2\cos^2\frac{\theta_1}{2}+b^2\cos^2\frac{\theta_2}{2} = a^2(\cos^2\frac{\theta_1}{2}-\cos^2\frac{\theta_2}{2})+\cos^2\frac{\theta_2}{2}$ OR $\frac{1}{2}(a^2(\cos\theta_1+1)+b^2(\cos\theta_2 + 1) = \dfrac{a^2(\cos\theta_1 - \cos\theta_2) + \cos\theta_2 + 1}{2}$ | |
| Nov 9 at 16:51 | comment | added | Daniel Hoekwater | I used a more roundabout method, but I arrived at the same solution: $\frac{1}{4}(|a(1+e^{i\theta_1}|^2+|b(1+e^{i\theta_2})|^2) = \frac{1}{4}(|a(1+e^{i\theta_1})|^2||u_1\rangle|^2+|b(1+e^{i\theta_2})|^2||u_2\rangle|^2)=\frac{1}{4}(|a(1+e^{i\theta_1})|^2|+|b(1+e^{i\theta_2})|^2|)$. | |
| Nov 9 at 16:40 | comment | added | Daniel Hoekwater | Thank you! I was trying to work it out in matrix form, which was where I ultimately went wrong. I think I understand all your steps here except for one: how did you go from $1+e^{i\theta_1}$ to $e^{-i\theta_1/2}+e^{i\theta_1/2}$? | |
| Nov 9 at 16:35 | vote | accept | Daniel Hoekwater | ||
| Nov 4 at 8:25 | history | answered | DaftWullie | CC BY-SA 4.0 |