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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15
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1answer
633 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 ...
12
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3answers
908 views

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 ...
10
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3answers
3k views

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 ...
10
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1answer
891 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 ...
9
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3answers
323 views

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 ...
8
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2answers
1k views

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 ...
8
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2answers
179 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 ...
7
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4answers
269 views

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: $\...
7
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2answers
346 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}...
6
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2answers
153 views

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 = \...
5
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2answers
697 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 that ...
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2answers
1k 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 ...
5
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2answers
772 views

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,...
5
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1answer
88 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 $t$?
5
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1answer
276 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 ...
5
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1answer
118 views

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 ...
5
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1answer
82 views

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 ...
4
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2answers
131 views

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 $−\vert x\rangle\vert 0\rangle$...
4
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1answer
307 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 ...
4
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3answers
299 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 ...
4
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1answer
73 views

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 ...
4
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2answers
300 views

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 & ...
4
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1answer
62 views

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 ...
3
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3answers
622 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]...
3
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2answers
49 views

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 ...
3
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2answers
103 views

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\...
3
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1answer
449 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 = ...
3
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1answer
56 views

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 ...
3
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1answer
134 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: ...
3
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1answer
45 views

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)~~~...
3
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1answer
65 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 ...
3
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1answer
40 views

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 ...
3
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1answer
86 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 ...
3
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1answer
85 views

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 ...
2
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2answers
157 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 \...
2
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2answers
210 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 ...
2
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1answer
68 views

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'...
2
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1answer
226 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 ...
2
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2answers
40 views

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 ...
2
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1answer
102 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?
2
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1answer
92 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/...
2
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1answer
674 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+...
2
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1answer
48 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}...
2
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1answer
71 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 ...
2
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1answer
187 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 ...
2
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2answers
111 views

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?
2
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1answer
71 views

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 ...
2
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1answer
77 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?
2
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0answers
41 views

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 ...
1
vote
1answer
54 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 ...