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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Is there a way to present conjugate transpose of a Y Pauli rotation as a Cirq Operator?
Given:
Ry(theta) acting on one qubit
I'm trying to use existing Cirq Operators to build the conjugate transpose of the above gate. I need the operator to produce the exact unitary of the given gate ...
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1
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Why can any density operator be written this way? (quantum tomography)
From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product:
\begin{aligned}
\rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
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Expanding two rotation operators
I have the following operators acting on two qubits, denoted as $(1)$ and $(2)$:
$$T_1=\displaystyle\exp\left(-i\frac{\pi}{4}Z\otimes Z\right)\cdot R_z^{(1)}\left(\frac{\pi}{2}\right)\cdot R_z^{(2)}\...
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What is the procedure of finding z-y decomposition of unitary matrices?
The title explains it all. Suppose one needs to find z-y decomposition of unitary matrix H or T. What is the step by step process to find it?
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How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices? [duplicate]
I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices:
\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\...
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Is there a different way to represent Pauli gates in X basis?
It's easy to see that in computational basis, Pauli matrices could be represented in the outer product form:
$$
X=|0\rangle\langle1|+|1\rangle\langle0|\\
Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\
Z=...
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If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"?
The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate
\begin{equation}
Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
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How to generalize the relationship HXH = Z for higher dimensions
Concerning the Hadamard gate and the Pauli $X$ and $Z$ gates for qubits, it is straightforward to show the following relationship via direct substitution:
$$ HXH = Z.\tag{1}$$
And I would like to ...
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How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?
I am trying to write $|0\rangle+i|1\rangle$ in terms of elementary gates like H, CNOT, Pauli Y, using the IBM QE circuit composer.
I was thinking some kind of combination of H and Y since $Y|0\rangle=...
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1
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Single-qubit rotations on a subspace within two-qubit unitary
I would like to implement the operation
$$
U(a,b) = \exp\left(i \frac{a}{2} (XX + YY) + i \frac{b}{2} (XY - YX) \right)
$$
($a,b \in \mathbb{R}$) without using Baker-Campbell-Hausdorf expansion, ...
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How to find the normalization factor of the eigenvectors of the $\sigma_x$ Pauli gate?
I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes ...
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Qiskit PauliWeightedOperator in the matrix representation?
Suppose we have a PauliWeightedOperator object from Qiskit. Is there any built-in method to convert it to the matrix representation in the computational basis? My search in the docs was not successful....
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Understanding the EPR argument with a simple description using Pauli matrices
Can someone explain the EPR argument with a simple description using Pauli matrices?
Two non-commuting physical quantity are being discussed philosophically whether there is an element of reality ...
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How do I create an inverse identity gate?
Is it possible for me to construct a gate that inverse everything ($|0\rangle \rightarrow -|0\rangle, |1\rangle \rightarrow -|1\rangle$, etc. basically like a $-I$ gate) from the basic $X, Y, Z, CX,......
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How can I verify that the Pauli group is a group? And is it abelian? [duplicate]
So how can I verify that the Pauli Group is a Group? Then furthermore, Abelian? And then to sum it up, the order of the group. Trying to do some research into the group but I can't find much about it.
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Quick question about Two-qubit SWAP gate from the Exchange interaction
I am reading the following paper: Optimal two-qubit quantum circuits using exchange interactions.
I have a problem with the calculation of the unitary evolution operator $U$ (Maybe it is stupid):
I ...
user13341
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Definitions of $D_y$ gate in Hamiltonian simulation: are they the same?
I'm reading a Hamiltonian simulation example proposed in this paper. From their notation, the operator $D_y$ (sometimes it's called $H_y$) serves the function to diagonalize the Pauli matrix $\sigma_y(...
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Simulate Hamiltonians with Pauli operations (controlled time evolution)
I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ...
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Transforming $|01 \rangle + |10 \rangle - |11 \rangle \to |01 \rangle - |10 \rangle + |11 \rangle$
How to convert from current state: $$|\psi \rangle =\dfrac{ |01 \rangle + |10 \rangle - |11 \rangle}{\sqrt{3}}$$
into a target state
$$|\phi \rangle = \dfrac{|01 \rangle - |10 \rangle + |11 \rangle}{\...
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How can I simulate Hamiltonians composed of Pauli matrices?
Suppose I want to perform the time-evolution simulation on the following Hamiltonians:
$$
H_{1} = X_1+ Y_2 + Z_1\otimes Z_2
\\
H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2
$$
Where $X,Y,Z$ are Pauli ...
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How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere?
For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
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Benefit of phase shift in quantum computing
I am new to quantum computing.
I compare Pauli-X gate and Pauli-Y gate as equivalent to NOT gate in classical computers. Though I am not very sure when to use Pauli-X and Pauli-Y gates as the result ...
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Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer?
Quantum annealers are single purpose machines allowing to solve quadratic unconstrained binary optimization (QUBO) problems. QUBO problems have following objective function:
$$
F=-\sum_{i<j}J_{ij}...
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How do I visualize the action of Pauli $X$ and $Y$ gates on $|0\rangle$?
I visualize Pauli Gates making rotations by π radians about the x, y, and z axes on the Bloch sphere.
If this is the case, then how Pauli-X and Pauli-Y gates applied to $|0\rangle$ differ? As both ...
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In Variational Quantum Eigensolvers, what does "grouping Pauli operators into tensor products requiring the same post-rotations" mean?
In this paper (nature version), the authors state
We group the Pauli operators into tensor product basis sets that require the same post-rotations.
As a result, they have the table S2 in the suppl. ...
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CNOT expressed with CZ and H gates by taking into account HZH =X
From this link:
Where equation 1 is:
I can probably brute-force this by explicitly calculating this quantum circuit's effective 4x4 matrix and seeing that its equivalent to this teleportation ...
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Principal square root of Pauli Y gate in Qiskit?
I've seen a similar question asked (How do I compute the square root of the $Y$ gate?) but I'm trying to understand how I can use the gates $Y^{\frac{1}{2}}$ or $Y^{\frac{1}{4}}$ in Qiskit in terms of ...
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controlled-Z rotation gates in symmetrical fashion
I was going through the qiskit textbook and in this chapter I came across a statement under the topic "Kickback with the T-gate" related to the Controlled-Z gate that
the controlled-Z ...
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0
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Changing the Basis
I am attempting to use a VQE algorithm to find the ground state of a deuterium nucleus by applying a constructed hamiltonian to an ansatz state with one parameter created by a circuit. While I am ...
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Is there a function in Qiskit to measure tensor products of Pauli operators?
Is there a function in Qiskit similar to this function in Q# which measures strings of Pauli operators applied to different qubits?
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Can you take infinitely many square roots of Pauli-X?
I am trying to find the cost for a n-bit Toffoli gate based on the recurrent circuit presented on Barenco's Work in Lemma 7.5 (Elementary gates for quantum computation)
The construction requires that ...
2
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1
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Why do we transform a Boolean variable into a a Pauli Z matrix
Under Qiskit QAOA's tutorial (https://qiskit.org/textbook/ch-applications/qaoa.html), the authors specify that a cost function $C(x)$ representing the optimization objective of a Binary Combinatorial ...
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2
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What are boost and shift operators and why are they called so?
In some texts I see $X$ and $Z$ Pauli operators as being said as boost and shift operators respectively.
But I came across some text that defines its own operators, namely:
$$
X \vert j\rangle = \...
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What do coupling coefficients mean in terms of Pauli gates, and why are they time dependent?
I am reading this error mitigation paper by the IBM team and I am slightly confused about the meaning of "coupling coefficients" when describing multi-qubit Hamiltonian.
I have only seen ...
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0
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Generalized set of Pauli elements for a basis for the linear transformations on the vector space [duplicate]
I have been doing some practice problems from "Gentle introduction to Quantum Computing". I am a little bit lost with this one:
The generalized Pauli group $\mathcal G_n$ is defined by all elements ...
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What would the CHSH inequality be equal to if the two qubits were separable?
I am writing an Excel spreadsheet to work through the matrix algebra for a simple Bell's game with the following parameters.
$$\newcommand{\bra}[1]{\langle #1|}\newcommand{\ket}[1]{|#1\rangle}A_\pm = ...
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Pauli Identity Using Tensor Network Notation
I am trying to understand the meaning of the equation shown in the above image taken from this paper, but I am unfamiliar with the tensor network notation. My current strategy is trying to write down ...
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VQE: Can I build a non-hermitian Hamiltonian with just Pauli matrices?
From the VQE paper they claim that a Hamiltonian can be expressed as a polynomial series of pauli operators (equation 1).
While coding up VQE from scratch I made a function which would allow me to ...
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What is the state of a qubit after measuring some a Pauli operator?
I'm confused about the state of a system after a measurement. Say we have a particle $v$ in the state: $ |\psi\rangle= \sqrt{1/4} \ |0\rangle + \sqrt{3/4} \ |1\rangle $.
From my understanding, if one ...
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How to build a circuit for simulation of a simple Hamiltonian?
Consider very simple Hamiltonian $\mathcal{H} = Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$. It has eigenvalues 1 and -1 with coresponding eigenstates $|0\rangle$ and $|1\rangle$, ...
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Cannot interpret transformations on the bloch sphere as matrix multiplications
I understand that X,Y and Z gates are rotations around the axes with the respective letters, but I cannot understand how can Y gate multiply the amplitude of 0 with unreal number and have it landing ...
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What kind of transformation does the Y-gate do on the bloch sphere?
I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ...
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Effect of Pauli X gate on minus state using bloch sphere
As I understood, the X gate flips the state around : $X(|0\rangle) = |1\rangle$. It can also be visualized with a $\pi$ rotation around the $x$ axis in the Bloch sphere. I have no problem with that.
...
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How can I decompose a matrix in terms of Pauli matrices?
I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices.
I would prefer an option to do this in larger than 2 dimensions, ...
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Why can I apply $HS^\dagger$ and then measure in the computational basis to measure $Y$?
I come from a CS background
I was reading Neven and Farhi's paper ("Classification with Quantum Neural Networks on near Term Processors"), and I am trying to implement the subset parity problem using ...
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Qiskit flipped representation of qubits in CNOT gate?
The conventional CNOT gate is shown on the right, and the Qiskit version is on the left. Since Qiskit defines it has a flipped representation kindly explain what is happening to the 11 position?
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Question Regarding Simulating Hamiltonian With Quantum Circuit
There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/...
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How is the ground state of a Hamiltonian defined?
I'm studying VQE, but there is something I don't get.
We know (I think) that for a given Hamiltonian the minimum eigenvalue is associated with the ground state. But if we take the Hamiltonian to be ...
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2
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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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