Linked Questions

1 vote
1 answer
125 views

Quantum gates with respect to phase angles [duplicate]

We can say that $X (\cos \frac{\theta}{2} |0\rangle + e^{i \phi}\sin \frac{\theta}{2} |1\rangle) = \cos \frac{\pi-\theta}{2} |0\rangle + e^{-i \phi}\sin \frac{\pi-\theta}{2} |1\rangle$, a fact that ...
Fabian's user avatar
  • 35
3 votes
3 answers
1k views

Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

I remember reading this somewhere... Is there an elegant proof for this?
Quantum Guy 123's user avatar
4 votes
3 answers
4k views

How to represent the Hadamard gate as a rotations on the Bloch sphere?

I am new to Quantum Computing, and I have decided to try and learn the quantum gates. I am trying to understand how to represent some basic gates as rotations on the Bloch Sphere. I was able to ...
William Ryman's user avatar
6 votes
2 answers
625 views

Can we rotate Bloch vectors for qudits like we do with qubits in the Bloch sphere?

I have been looking into the Bloch vectors for qudits and have been wondering if we can do rotations that are similar to the rotations in the qubit Bloch sphere. Like, once we create a Bloch vector ...
Parmeet Singh EP 066's user avatar
1 vote
1 answer
1k views

What is the general formula for unitary rotations in terms of Pauli spin operators?

Recently I have read a paper in which they have used a unitary transformation as follows: $$U_{\frac{7\pi}{16}}=\cos\left(\frac{7\pi}{8}\right)\sigma_{z}+\sin\left(\frac{7\pi}{8}\right)\sigma_{x}$$ ...
Jasmine's user avatar
  • 161
3 votes
1 answer
473 views

How to derive the rotations caused by the H gate?

In Nielsen and Chuang, there's the following paragraph: The Hadamard operation is just a rotation of the sphere about the ˆy axis by 90◦, followed by a rotation about the ˆx axis by 180◦. I am ...
Claire's user avatar
  • 669
1 vote
1 answer
454 views

Rotation angles of unitary operator

Given a complex unitary $2*2$ matrix $A$ that represents some quantum gate on a single qubit. What is the formula to extract to $\theta_X, \theta_Y, \theta_Z $ rotations around each one of the axes in ...
Ron Cohen's user avatar
  • 1,482
1 vote
1 answer
409 views

Finding the rotation axis and angle in a Bloch sphere for a arbitrary quantum gate

I have an arbitrary single qubit quantum gate $U_3(t,f,l)$ that transforms (rotates) a given qubit $q$ into the target qubit $p$. $$\begin{align}U_3(t,f,l) q = p && t,l,f \in \mathbb{R}; q,p \...
Rufus Buschart's user avatar
2 votes
0 answers
104 views

Geometric representation of rotation operator on Bloch sphere

I'm studying Nielsen&Chuang Book and need some clarification on mapping arbitrary rotation operator onto geometric tranformation of vector on Bloch sphere. I thought that $R_{\vec{n}}(\theta)$ ...
Михаил Горчаков's user avatar
0 votes
0 answers
47 views

Axis and Angle of rotation of $\frac{1}{\sqrt{2}}\begin{bmatrix}-i&-1\\1&i\end{bmatrix}$

I have made use of the following formulas, \begin{align} \theta&=2\cos^{-1}\bigg(\frac{e^{-i\alpha}Tr(X)}{2}\bigg)\\ n_i&=\frac{e^{-i\alpha}Tr(X\sigma_x)}{2\sin\theta/2}\\ e^{i\alpha}&=\...
Sooraj S's user avatar
  • 831
2 votes
0 answers
39 views

What do the values of θ, ɸ, and λ represent visually in the Bloch Sphere when defining a unitary gate? [duplicate]

In IBM Quantum Docs, it is stated that a unitary matrix can be defined as $U = \begin{bmatrix} \cos(\theta/2) & -e^{j\lambda}\sin(\theta/2) \\ e^{j\phi}\sin(\theta/2) & e^{j\lambda+j\phi}\cos(\...
Satvik Duddukuru's user avatar