3

I am new to quantum gates but do not understand the connection between the $RX$ and $X$ gates. I know that

$$R X(\theta)=\exp \left(-i \frac{\theta}{2} X\right)=\left(\begin{array}{cc} \cos \frac{\theta}{2} & -i \sin \frac{\theta}{2} \\ -i \sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{array}\right)$$

Meanwhile the $X$ gate is given by

$$X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)$$

Is there a value of $\theta$ such that the two are the same? I see that choosing $\theta = \pi/2$ gives the result upto an overall factor of $-i$. Is that it or is there a deeper connection between the two gates? Is there a similar connection between the $Y$ and $RY$ gates and the $Z$ and $RZ$ gates such that the rotated gates are more general than the $X, Y$ and $Z$ gates?

Jason Fring
  • 33
  • 3

1 Answers1

3

You're almost correct - choosing $ \theta = \pi$ does yield $$ \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix} $$

Because this differs from the $X$ gate by a constant factor global phase ($ -i$), the gates are equivalent. (See here to learn more about the global phase).

This connection holds similarly for $ RY$ and $Y$, and $RZ$ and $Z$. A way to visualize this is the Bloch sphere: in essence, these gates are rotations about the $X, Y, Z$ axes (respectively):

Bloch sphere

So essentially our Pauli primitives are $\pi$ rotations over the respective axis.

C. Kang
  • 1,716
  • 8
  • 23
  • 1
    I'd clarify this a bit: equivalent, unless you want to use a controlled version of them. The habit of ignoring the global phase of the gate can come back and bite painfully: https://quantumcomputing.stackexchange.com/questions/5973/counting-in-q-number-of-solutions/6446#6446 – Mariia Mykhailova Sep 11 '20 at 16:24
  • Hi, may you explain why I can't reproduce such a property within wolframalpha.com, inputing e^{i{{0,1},{1,0}}(-pi/2)}. – Daniele Cuomo Sep 19 '22 at 10:12
  • Hi @DanieleCuomo, it might be valuable to create a new question with the full details of what you're attempting! – C. Kang Oct 02 '22 at 14:35
  • I actually did it! https://quantumcomputing.stackexchange.com/questions/28224/equivalence-check-between-rotational-gates-and-pauli-gates – Daniele Cuomo Oct 02 '22 at 15:07