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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55 views

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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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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300 views

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 ...
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1answer
36 views

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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1answer
90 views

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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1answer
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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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71 views

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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60 views

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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1answer
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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-...
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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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125 views

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 ...
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1answer
120 views

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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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?

Hi 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 coupling ...
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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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1answer
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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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Iterative qubit coupled cluster (iQCC) ansatz (Efficient screening procedure)

The paper Iterative Qubit Coupled Cluster approach with efficient screening of generators describes a new screening procedure for generators of the QCC ansatz. The paper states: 'In the absence of ...
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2answers
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Confusion about the state of a system after a measurement

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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1answer
108 views

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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1answer
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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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1answer
56 views

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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1answer
106 views

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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1answer
419 views

Example of Hamiltonian decomposition into Pauli matrices [closed]

I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices. Please show me how this is written in Python. What I have tried ...
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2answers
290 views

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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1answer
120 views

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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1answer
123 views

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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1answer
515 views

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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2answers
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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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1answer
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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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1answer
66 views

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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629 views

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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1answer
139 views

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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1answer
90 views

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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1answer
99 views

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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2answers
158 views

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 \...
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3answers
368 views

How is partial trace related to operator sum representation?

In Quantum Computation and Quantum Information by Nielsen and Chuang, the authors introduce operator sum representation in Section 8.2.3. They denote the evolution of a density matrix, when given an ...
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1answer
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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 ...