Questions tagged [bloch-sphere]
For questions related to the Bloch sphere. In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. (Wikipedia)
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closeness between two unitaries on the bloch sphere
The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this ...
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Sufficient conditions for a single-qubit unitary to be the identity
Say I have a unitary $U = e^{-iHt}$ where $H = \alpha X + Z$.
First, suppose $U = I$. Then it rotates a set of initial states to themselves. Say I'm working on a computational basis, then on the Bloch ...
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Can you twist a qubit?
Is it possible to operate on a single qubit by a map which has a nonzero degree?
Let $|c\rangle=c_0|0\rangle + c_1|1\rangle$ represent a qubit state where $c_0,c_1 \in \mathbb{C}$ and $|c_0|^2+|c_1|^2=...
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What does the product of two density matrices represent physically?
A quantum state, pure or mixed, can be described by a density matrix that encodes the Bloch vector $\hat{m}$ analog of a quantum state like
$\rho = \frac{1}{2}[\mathbb{I} + \hat{m}.\vec{\sigma}]$.
Let ...
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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{...
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Representing networks with qubits as edges
I am looking to take a classical non-negative real valued network and generalize it to the quantum case for processing. A network is given by an adjacency matrix, essentially edge weights $e_{ij}$ for ...
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How to represent an entangle state on Bloch Sphere [duplicate]
If states are not entangle, both bloch sphere represent the state of qubit but if they are entangle, why bloch sphere show nothing??
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Plotting the Bell state on a Bloch sphere
How can you plot the bell state on a Bloch sphere?
bell = QuantumCircuit(2)
bell.h(0)
bell.x(1)
bell.cx(0,1)
Is there any good reference for understanding how ...
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Is the Bloch sphere the same as the electron spin?
A single qubit is represented with a bloch sphere and implemented on an electron with only two energy values from lots of them.
So, I am confused about why an energy range is represented as a ...
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Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2
$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $
I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ...
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Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
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get statevector from bloch coordinates for one qubit
I need to get a state vector for one qubit from bloch coordinates no matter if there could be many states that describes the same bloch coordinates, and it does not have to be normalized, because the ...
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Finding a unitary transformation for a qubit mixed state that projects onto a pure state
Suppose we have a single qubit mixed state described by a density matrix $\rho$, and we want to find a unitary transformation that brings $\rho$ to the pure state $|0\rangle\langle 0|$. Is there a ...
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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 $...
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Why are probabilities represented with alpha^2 and beta^2?
To preserve the Complementary Rule of probability, the sum of the probabilities of the outcomes (measured |0> or measured |1>) must equal 1 or 100%. That's why alpha^2+beta^2=1. However, why the ...
user24663
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Why is the conjugate being used when rewriting a qubit state in another basis?
I'm currently going through Introduction to Classical and Quantum Computing, by Thomas Wong, and I'm struggling with an example given on page 88 (point 4).
The author gives the two following states, ...
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Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?
Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like ...
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Is the function PU(2) and SO(3) induced by the Bloch sphere bijective?
I have difficulty understanding the fact that, as written in this reference,
every single-qubit
unitary corresponds to a unique rotation of R3 and vice versa.
If I understand well, this means there ...
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Finding the polar and azimuthal angles of the Bloch vector corresponding to $\frac1{\sqrt2}(1-i)|0\rangle-\frac i{\sqrt2}|1\rangle$ [duplicate]
I need to find the polar angles and azimuthal angles of the following bloch vector:
$$
\frac{1-i}{2}|0\rangle - \frac{i}{\sqrt{2}}|1\rangle
$$
I just couldn't figure it out, and I could also not find ...
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Clarification defining/finding the relative phase of a qubit
Let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$.
According to p. 22 of ...
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How many parameters do we need to characterize a pure state?
Suppose I have a pure qubit. I can think of starting with the state $\vert 0\rangle$ and apply some unitary to it. Such a unitary has three parameters according to this link. In $d$ dimensions, the ...
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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 $R_{\vec{n}}(\theta)$ ...
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Where does 1/sqrt(2) come from in the state of i
I’m trying to learn about calculating coordinates for $\theta$ and $\varphi$ in a Bloch-sphere.
I came accross this book about it, including example questions.
At question 2.12b, they ask to give the ...
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Significance of angle $\phi$ on bloch sphere
So far I learned that a qubit can be written as $| \psi \rangle = \alpha | 0 \rangle + \beta | 1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$ and reparametrized as $| \psi \rangle = cos( \theta / 2) + e^{...
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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 ...
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Affine map of single qubit quantum operations
In my reference, Page 375, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
Lemma: The Pauli matrices, along with the identity matrix $I$, form an ...
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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(\...
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Solving a Bloch sphere where alpha is imaginary
$$
\begin{array}{l}
|\psi\rangle=\frac{\sqrt{3 i}}{2}|0\rangle-\frac{1}{2}|1\rangle \\
|\psi\rangle=0.924|0\rangle-0.382 i |1\rangle
\end{array}
$$
Basically I’m trying to convert these to a standard ...
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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?
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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 ...
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How to interpret QSphere visualisation
AFAIK, $n$ qubits together can be described by using a column vector (also called State vector) of $2^n$ complex amplitudes such that sum of the squares of mod of all these complex amplitudes will be $...
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Decomposition of unitary operator into rotations around Bloch sphere
I apologize in advance for any mistakes as I am new to this field and come from a programming, rather than mathematical/physical background.
I am looking for a way to decompose a given operator $U$ ...
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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 $$\rho = \frac{\textbf{I} + r_1 \sigma_1 + r_2 \sigma_2 + r_3 \sigma_3}{2} \rightarrow \frac{\textbf{I} + r_1 \sigma_1 + \lambda ~r_2 \sigma_2 + r_3 \...
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Finding the rotation axis and angle in a Bloch sphere for a arbitrary quantum gate
I have an arbitrary single qubit quantum gate $U_3(t,f,l)$ that transforms (rotates) a given qubit $q$ into the target qubit $p$.
$$\begin{align}U_3(t,f,l) q = p && t,l,f \in \mathbb{R}; q,p \...
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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 ...
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Set relative phase of qubit to zero without measurement?
Is there a quantum way to set the unknown relative phase of a qubit (assumed in a pure state) to zero, without measurement? The relative phase is not known, otherwise I would subtract it using a phase ...
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Is there a general parametric transformation matrix form in bloch-space corresponding to the unitary operations on qutrits?
I've been looking into the structure of the Bloch sphere for qudits, and I am wondering if there is a transformation matrix (or rotation matrix) formula corresponding to high-dimensional quantum ...
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How is the surface of a Bloch sphere a Hilbert space?
In the linear algebra section of the Qiskit textbook appears the following claim regarding the Bloch sphere:
The surface of this sphere, along with the inner product between qubit
state vectors, is a ...
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Stepwise $SU(2)$ Rotations on the Bloch Sphere from $\pi$ to $2\pi$
Based on the useful straight-forward answers to both of my former questions, on multiple rotations of a qubit and bloch sphere subplots, I was able to implement the following $SU(2)$ rotations:
At ...
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How to implement subplots (several blochsphere plots) using qiskit?
Qiskit seems to use matplotlib for rendering bloch spheres under the hood. Therefore, it would be nice if we could also make use of matplotlib's subplot technique.
I would like to implement subplots, ...
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A fundamental question on multiple rotation of a qubit using $SU(2)$
I am trying to implement qubit rotations using $SU(2)$ from scratch in order to understand and debug what happens under the (physical) hood. The reason why gates and high-level APIs are omitted here ...
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Why do H and T gates generate rotations of an (irrational * pi)
My quantum information lecture notes state that neither $H$ nor $T$ is a rotation of (irrational * $\pi$).
However it states that both $TH$ and $HT$ are.
Could somebody explain why?
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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 ...
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How to implement SU(2) rotation with qiskit?
Before I start, I would like to say sorry for this possibly stupid question of mine.
I just recently got into the topic of quantum computing and try me there.
I'm currently trying to display a SU(2) ...
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Can a many-qubit quantum state be represented on Bloch spheres? [duplicate]
Can a general multi-qubit quantum state be represented on Bloch spheres? If not, what is the maximal number? Are there any corresponding constraints?
Thanks.
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Rxx gate as a set of rotations
I'm trying to represent Rxx gate as a set of physical rotations of two qubits in 3D space (or as rotations of Bloch Spheres that is the same). In some simple cases it works well:
If q0 is in the ...
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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 ...
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When using Qiskit's plot_bloch_vector function, how can I set colors for vectors?
I am able to visualize vectors on a bloch sphere using Qiskit as follows:
...
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Are qubits just analog, continuous classical bits? [duplicate]
Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply ...
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Is there a way to rotate an unknown state towards another known state?
I would like to rotate my $|\Psi\rangle$ state towards $|1\rangle$:
$$ |\Psi\rangle= a|0\rangle + b|1\rangle \ \rightarrow \ |\Psi'\rangle= a'|0\rangle + b'|1\rangle$$
with $|a'| < |a|$, $|b'| > ...
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