# Constructing a pure state in Bloch sphere using 3 gates

We have the 3 following gates :

$$H = \dfrac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix}$$ $$R(\varphi) = \begin{bmatrix}1 & 0 \\ 0 & e^{-i\varphi} \end{bmatrix}$$ $$R(\psi) = \begin{bmatrix}1 & 0 \\ 0 & e^{i\psi} \end{bmatrix}$$

and we want to construct a one-bit circuit that produces the final state $$|\Xi\rangle = \cos {\varphi\over{2}} |0\rangle + e^{i\psi}\sin {\varphi\over{2}} |1\rangle$$

I do not understand how a factor $$\cos {\varphi\over{2}}$$ can appear in front of $$|0\rangle$$, can someone help me ?

• Note that, the two states $|\psi_1\rangle=a|0\rangle+b|1\rangle$, and $|\psi_2\rangle=e^{i\phi}(a|0\rangle+b|1\rangle)$ are the same up to a global phase. quantumcomputing.stackexchange.com/a/13765/9474 Apr 18, 2022 at 11:15

## 1 Answer

Starting from $$|0\rangle$$ if you 'mix' the amplitudes with $$H$$ then rotate by $$R(\phi)$$ and then 'unmix' using $$H$$ again you'll have transfered the phase $$\phi$$ to the amplitude of $$|0\rangle$$. i.e. starting with $$H|0\rangle$$:

$$|0\rangle\overset{H}{\to}\frac1{\sqrt{2}}\Bigl(|0\rangle + |1\rangle\Bigr)$$ then you rotate in $$Z$$ by $$\phi$$: $$\frac1{\sqrt{2}}\Bigl(|0\rangle + |1\rangle\Bigr)\overset{R(\phi)}{\to}\frac1{\sqrt{2}}\Bigl(|0\rangle + e^{-i\phi}|1\rangle\Bigr)$$ then H again: $$\frac1{\sqrt{2}}\Bigl(|0\rangle + e^{-i\phi}|1\rangle\Bigr)\overset{H}{\to}\frac12\Bigl((1+e^{-i\phi})|0\rangle + (1-e^{-i\phi})|1\rangle\Bigr)$$

Now the amplitude of $$|0\rangle$$ is \begin{align} \frac14\|1 + e^{-i\phi}\|^2 &= \frac14\Bigl(1+\cos(\phi)-i\sin(\phi)\Bigr) \Bigl(1+\cos(\phi)+i\sin(\phi)\Bigr)\\ &= \frac12\Bigl(\cos(\phi) + 1\Bigr)\\ &= \cos^2(\frac{\phi}2) \end{align} and similarly the amplitude of $$|1\rangle$$ is $$\sin^2(\frac{\phi}2)$$ so you can write this state as $$\cos\bigl(\frac{\phi}2\bigr)|0\rangle + e^{i\theta}\sin\bigl(\frac{\phi}2\bigr)|1\rangle$$ where $$e^{i\theta}$$ is their relative phase which ends up being $$i = e^{i\frac{\pi}2}$$.

• Thank you very much ! Great explanation !
– Bozu
Apr 26, 2022 at 12:56