1 vote
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 ...
• 35
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?
• 1,371
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 ...
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 ...
1 vote
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: $$U_{\frac{7\pi}{16}}=\cos\left(\frac{7\pi}{8}\right)\sigma_{z}+\sin\left(\frac{7\pi}{8}\right)\sigma_{x}$$ ...
• 161
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 ...
• 669
1 vote
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 ...
• 1,482
1 vote
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 \...
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 $R_{\vec{n}}(\theta)$ ...