Questions tagged [pauli-gates]
For questions about Pauli matrices in general or Pauli gates in particular, as relevant to quantum computing and/or quantum information theory. The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. The three Pauli gates are: Pauli-X gate, Pauli-Y gate & Pauli-Z gate. X = {{0,1},{1,0}}; Y = {{0,-i},{i,0}}; Z = {{1,0},{0,-1}}.
188
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Gate Y returns wrong phase in IBM's circuit composer
One can check that, with IBM's circuit composer, $Y$ gate acted on $|0\rangle$ or on $|1\rangle$ returns the same phase of $\pi/2$. Is this a bug?
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What does the notation $\sigma_j^z$ mean for Pauli matrices?
In multiples papers or online article on the QAOA algorithm (such as this one), I found notation for the Hamiltonian similar to this one :
$$ \sum_{ij} \frac{1}{2} (I-\sigma_i^z \sigma_j^z)$$
I don'...
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Qubit in a mix sin/cosine state
The question is pretty simple. How can I get an input qubit $|0⟩$ in the state, say $$\cos{\frac{\pi}{10}}|0⟩ + \sin{\frac{\pi}{10}}|1⟩$$ Or any other sine/cosine mix state? Which gates do I need to ...
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Similarity Transformations on Pauli Operators in 2-qubit states (eq. 11 - Farhi's QNN Paper)
Again, I am new to quantum computing and have a CS background, so apologies if this seems like an obvious question or if I seem unclear. $\newcommand{\braket}[1]{\langle #1 \rangle}\newcommand{\bra}[1]...
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1
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Generate the state $\frac{-|0\rangle + |1\rangle}{\sqrt{2}}$ with qiskit: problem with Pauli-Z behavior
I want to construct the following state of a qubit using a quantum circuit: $\frac{-|0\rangle + |1\rangle}{\sqrt{2}}$
When I use the following qiskit code in Python:
...
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What is the difference between the action of $Z$ and $\exp(-i Z t)$ on a state?
What is the difference between performing $Z$ operation and performing $e^{-i Zt}$ operation on a state, given that $e^{-i Zt}= \mathbb{1} + (-i Zt) + ...$ is not equal to $Z$ for any value of $t$?
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Intuitive link between clifford group and gottesman-knill theorem
Elements of the Pauli group are the n-Pauli matrices with $\pm 1$ or $\pm i$ on front of them. They all commute or anti-commute between them.
The Clifford group are element that preserve the n-Pauli ...
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How to get specific state applying $e^{-i\phi \sigma_2/2}$ to $|0\rangle$ or $|1\rangle$?
I try to solve problems from Problems in Quantum Computing.
I stuck with problem #3:
I do the following:
Because:
$$ \sigma_2 = \begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}$$
Then:
$$ -i \...
4
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1
answer
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Expectation Value of Stabilizer
Given that operator $S_M$, which consists entirely of $Y$ and $Z$ Pauli operators, is a stabilizer of some graph state $G$ i.e. the eigenvalue equation is given as $S_MG = G$.
In the paper 'Graph ...
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Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]
Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?
For example, I have a matrix 16x16 and want it to be decomposed into something ...
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How does a single-qubit gate affect other qubits?
An instructional quantum computing article I'm reading (How the quantum search algorithm works) states that the following circuit takes $\vert x\rangle\vert 0\rangle$ to $−\vert x\rangle\vert 0\rangle$...
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How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?
How can I demonstrate on the exponential part equality of the Hadamard matrix:
$$H=\frac{X+Z}{\sqrt2}\equiv\exp\left(i\frac{\pi}{2}\frac{X+Z}{\sqrt2}\right).$$
In general, how can I demonstrate on:
$\...
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Matrix Index and multiplication rules for Hermitian Pauli group products
Given the Hermitian Pauli group products
$$
\Omega_{a,b}=\{\pm 1,\pm i\}_{a,b}\cdot \{I,X,Y,Z\}_{a,b}^{\otimes n}
$$
composed of $n$ 2x2 pauli matrices $(I,X,Y,Z)$ in tensor product, such that they ...
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Stabilizer state QFI lower limit query
On page 1 of this paper it states that the QFI (Quantum Fisher Information) for pure states $\psi$ is $$\mathcal{Q}(\psi) = \sum_{i,j=1}^n\text{Tr}(X_iX_j\psi)-\text{Tr}(X_i \psi)\text{Tr}(X_j \psi)~~~...
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Physical Interpretation of Pauli Matrices as Polarization Check
We know that the Pauli matrices are:
$$\sigma_x = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \sigma_y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}, \sigma_z = \begin{bmatrix}1 & ...
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The solution when we transmit a qubit through a Pauli channel?
A Pauli channel is defined as a convex combination of Pauli operators, i.e. $\epsilon_{\text{Pauli}} (\rho)=\sum_{j} q_j\sigma_j\rho \sigma_j$, where $0 \leq q_j \leq 1$ and $\sum_j q_j=1$. Now, I ...
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Should a Pauli $X$ matrix equal the identity matrix to be unitary?
My understanding is that any unitary matrix must have its inverse be equal to its conjugate transpose.
Looking at the pauli x gate as shown here: $$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$
It ...
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Is $M = a \mathbb{I} - ib \sigma_Z$ a valid representation in terms of logic gates?
I have a matrix $M= \begin{pmatrix} a - ib & 0 \\ 0 & a + ib \end{pmatrix}$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$. I need to represent this matrix in terms of the quantum ...
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Expected value of a product of the Pauli matrices in different bases
I'm trying to reproduce the results of this article https://arxiv.org/abs/1801.03897, using Qiskit and Xanadu PennyLane.
Particularly, this part with expected values of the Pauli operators:
For ...
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Can there be multiple energy eigenstates corresponding to the same eigenvalue of a Hamiltonian (Pauli-X)?
all. I am a high-school student who has recently familiarized himself with linear algebra and is looking to understand quantum computing. So, I bought the classic textbook "Quantum Computation and ...
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How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?
I read in this article (Apendix III p.8) that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis.
$$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+...
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Definition of the Pauli group and the Clifford group
There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as
\begin{align*}
\mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z,...
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CRZ and CRY Gates
I am trying to understand the function of the $CR_z$ and $CR_y$ gates.
Why are these gates used in practice? What advantage is gained by rotating a qubit around the $y$- or $z$-axis?
For example, this ...
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Expressing CNOT in the eigenbasis of $X$ (Preskill lecture notes eq. 7.6)
In chapter 7, equation 7.6 says CNOT works as follows:
CNOT: $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle )\otimes |x\rangle \rightarrow
\frac{1}{\sqrt{2}} (|0\rangle + (-1)^x |1\rangle ) \otimes |x\...
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How do physical implementations of Z gate selectively affect $\lvert1\rangle $ basis vector?
The Pauli Z gate inverts the phase of $\lvert1\rangle $ while leaving $\lvert0\rangle$ unaffected.
When I think about how $\lvert1\rangle $ and $\lvert0\rangle$ are physically realized, however, as ...
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How does the actual measurement collapsing an error to an orthogonal basis look like?
An error can be written as a linear combination of $\Bbb I$, $X$, $Z$, $XZ$ Pauli matrices. So when measuring an errand state we aim at collapsing the error into one of these four possibilities. How ...
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Rotation operator on Pauli parity gates $XX$, $YY$ and $ZZ$
If we suppose that $XX$ is the tensor product of $X$ with $X$ such as $XX = X \otimes X$
How would we calculate the rotation operator of this $XX$ gate.
Does this work? If so why?
$$
R(XX)_\theta = ...
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Where is the factor of $-i$ in rotation gates coming from?
As I understand it the Pauli-X, Y and Z gates are the same as their rotational gates with a rotation of $\pi$. But given the expression for those gates, I find that there is a factor of $-i$ in each ...
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Simulate hamiltonian evolution
I'm trying to figure out how to simulate the evolution of qubits under the interaction of Hamiltonians with terms written as a tensor product of Pauli matrices in a quantum computer. I have found the ...
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Fast way to check if two state vectors are equivalent up to Pauli operations
I'm looking for fast code, or a fast algorithm, for checking if a given state vector $A$ can be transformed into another state vector $B$ using only the Pauli operations $X$, $Y$, $Z$.
The naive ...
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How to construct matrix of regular and "flipped" 2-qubit CNOT?
When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning:
"If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix ...
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How does the stated Pauli decomposition for $\operatorname{CP\cdot A\cdot CP}$ arise?
I'm having a bit of trouble understand @DaftWullie's answer here.
I understood that the $4\times 4$ matrix $A$
$$ \frac{1}{4} \left[\begin{matrix}
15 & 9 & 5 & -3 \\
9 & 15 & 3 &...
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Is there a simple rule for the inverse of a Clifford circuit's stabilizer table?
In Improved Simulation of Stabilizer Circuits by Aaronson and Gottesman, it is explained how to compute a table describing which Pauli tensor products the X and Z observable of each qubit get mapped ...
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Shorthand notation for the sign flip gate
I need to use the following matrix gate in a quantum circuit:
$$\text{Sign Flip}=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$$
$\text{Sign Flip}$ can be decomposed as (in terms ...
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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Why is the decomposition of a qubit-qutrit Hamiltonian in terms of Pauli and Gell-Mann matrices not unique?
If I have the $X$ gate acting on a qubit and the $\lambda_6$ gate acting on a qutrit, where $\lambda_6$ is a Gell-Mann matrix, the system is subjected to the Hamiltonian:
$\lambda_6X=
\begin{pmatrix}0 ...
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Obtaining gate $e^{-i\Delta t Z}$ from elementary gates
I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't ...
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How to compute the average value $\langle X_1 Z_2\rangle$ for a two-qubit system?
Show that the average value of the observable $X_1Z_2$ in a two-qubit system measured in the state $(|00\rangle + |11\rangle)/\sqrt{2}$ is zero.
How would we approach this question? I understand that ...
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