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}}.

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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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1answer
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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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1answer
52 views

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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2answers
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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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1answer
60 views

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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2answers
788 views

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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1answer
50 views

Decomposition of a matrix 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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2answers
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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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What is the need of Operator module from qiskit.aqua?

My question is two-fold: a) Why is it that Pauli takes two arguments (z,x). What is the role of x? In the max_cut module, z is initialized as an np array of size=no of nodes in graph with False dtype ...
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1answer
148 views

CRZ and CRY Gates

I am trying to understand the CRZ and CRY gates functions : what do they actually used for ? why we require a certain angle rotation around z or y Axis on a Qubit ? and what does it yield ? ...
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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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1answer
52 views

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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1answer
158 views

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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1answer
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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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3answers
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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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2answers
122 views

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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3answers
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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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258 views

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 ...
8
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1answer
264 views

Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The Pauli group for $n$-qubits is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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246 views

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}...
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1answer
411 views

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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428 views

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 ...