All Questions
Tagged with textbook-and-exercises bloch-sphere
29
questions
5
votes
2
answers
403
views
Represent Hadamard gate in terms of rotations and reflections in Bloch sphere
I read in a book that any single qubit operation can be decomposed as
$$
\bf{U} =e^{i\gamma}\begin{pmatrix}e^{-i\phi/2}&0\\ 0&e^{i\phi/2}\end{pmatrix}\begin{pmatrix}\cos{\theta/2}&-\sin{\...
- 51
2
votes
1
answer
158
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What is the expression for $|\psi\rangle\!\langle\psi|$ if $|\psi\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)e^{i\phi}|1\rangle$?
Let $|\psi\rangle = \alpha|0\rangle + \beta |1\rangle$. In Bloch sphere representation, this is
$\cos\frac{\theta}{2}|0\rangle + \sin\frac{\theta}{2}e^{i\phi}|1\rangle$.
In matrix representation:
$|\...
- 518
2
votes
5
answers
470
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How to eliminate the global phase of a state vector?
Say that I have a qubit that began in the $|0\rangle$ state and then the
Hadamard gate is applied, resulting in the following state:
$ \begin{bmatrix}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}}
\end{...
- 143
3
votes
2
answers
183
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What is $HTHTH\left| 0 \right>$?
I'm currently going through Introduction to Classical and Quantum Computing, by Thomas Wong, and I'm struggling with exercise 2.33 (page 108):
Exercise 2.33. Answer the following:
(a) Calculate $...
- 164
2
votes
2
answers
332
views
What is the Bloch sphere representation of $\rho\to\mathcal{E}(\rho) = |+\rangle\langle+|ρ|+\rangle\langle+| + |−\rangle\langle−|ρ|−\rangle\langle−|$?
Suppose a projective measurement is performed on a single qubit in the basis $|+\rangle, |−\rangle$, where $|±\rangle \equiv (|0\rangle\pm |1\rangle)/\sqrt{2}$. In the event that we are ignorant of ...
- 831
5
votes
1
answer
829
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Why are orthogonal quantum states represented as collinear in the Bloch sphere?
We know that the angle between two orthogonal qubit states is 90 degrees. Why then, when we use the Bloch sphere, the angle becomes 180 degrees?
- 143
2
votes
1
answer
211
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Writing a Density matrix in terms of the magnitude of the Bloch Vector
Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
- 472
1
vote
0
answers
29
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How can one visual a transformation which affects a component of density matrix?
How to visualize a transformation that looks like $$\rho = \frac{\textbf{I} + r_1 \sigma_1 + r_2 \sigma_2 + r_3 \sigma_3}{2} \rightarrow \frac{\textbf{I} + r_1 \sigma_1 + \lambda ~r_2 \sigma_2 + r_3 \...
- 159
-1
votes
1
answer
140
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Which angle convention is used in general equation of quantum state?
In this post, I am unsure which angles are denoted in general equation of quantum state.
I realize that $\theta$ is azimuthal angle, while $\phi$ is the ...
- 289
3
votes
2
answers
107
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Find a set of vectors on the Bloch sphere such that $\langle \psi_i | \psi_j \rangle = \frac{1}{\sqrt{n}}$
How can I find a set of multiple vectors on the block sphere which satisfies
$$\langle \psi_i | \psi_j \rangle = \frac{1}{\sqrt{n}}$$
where $n$ is any natural number greater than $2$?
I think I have ...
- 141
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 ...
- 165
0
votes
2
answers
143
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Calculate qubit state in terms of two states that are opposite points on Bloch sphere
I am new to quantum computing and reading the book "Introduction to Classical and Quantum Computing", by Wong (link).
I do not understand how to calculate the qubit state for the below ...
- 11
0
votes
0
answers
171
views
Evaluation of Wigner function representation of a Bloch Sphere
Consider Wigner function representation of a qubit in the basis labeled by $\sigma_z$ and $\sigma_x$ eigenvalues. A general single qubit mixed state has the Bloch representation,$\rho = 1/2 (I + r.\...
0
votes
1
answer
453
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Show that the trace of squared density matrix gives ${\rm tr}(\rho^2)=\frac12(1+\|\mathbf n\|^2)$ [duplicate]
Equation 7.7 is given below:
$$\hat\rho = \frac12(I +n_x(\hat X)+n_y(\hat Y)+n_z(\hat Z)) $$
Where $I$ is the identity matrix and $\hat X,\hat Y,\hat Z$ are Pauli matrices.
Now my attempt of this was ...
- 121
0
votes
1
answer
373
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How to calculate the coefficients of a qubit from the angles of its Bloch representation?
A quantum bit $|\psi\rangle=a|0\rangle+b|1\rangle$ is represented on the Bloch sphere as a point on the spherical surface with $\theta = 40^°$ and $\phi = 245^°$.
Calculate the (complex) coefficients $...
- 79