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 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 ...
Josu Etxezarreta Martinez's user avatar
17 votes
1 answer
1k 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 ...
brzepkowski's user avatar
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16 votes
2 answers
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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 ...
C-Roux's user avatar
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3 answers
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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 ...
Thomas Hubregtsen's user avatar
15 votes
3 answers
2k 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 ...
Apo's user avatar
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11 votes
1 answer
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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, ...
yishairasowsky's user avatar
10 votes
2 answers
3k 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,...
snsunx's user avatar
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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 ...
Craig Gidney's user avatar
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9 votes
4 answers
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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: $\...
walid's user avatar
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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 ...
Craig Gidney's user avatar
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8 votes
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What is (formally) a transversal operator?

This question concerns about a formal definition of transversal operator. I understood that transversal operator are a group of operators which are efficient in terms of circuit depth and can be used ...
Daniele Cuomo's user avatar
8 votes
1 answer
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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+...
lufydad's user avatar
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Why isn't $Ry(\pi/2)$ gate equivalent to Hadamard gate?

I've been experimenting with quantum circuits and can't quite fathom how the difference between states comes together. Speaking in terms of simulations using qiskit,...
Ricardo's user avatar
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2 answers
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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 ...
Sorin Bolos's user avatar
7 votes
2 answers
554 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}0 ...
user1271772's user avatar
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
Quantum Guy 123's user avatar
7 votes
1 answer
1k views

Does the controlled Pauli Z gate cause entanglement?

I'm trying to understand the relationship between the factorability of a 2 qubit gate and that gate's ability to cause entanglement. I've begun by considering the controlled Pauli Z gate. After ...
Flipper's user avatar
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3 answers
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Is there a convention for denoting $Y$ eigenstates?

Two common shorthands for eigenstates of the $Z$ operator are $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|-1\rangle\}$, where in the first case we have $Z|z\rangle=(-1)^z|z\rangle$ and in the second ...
Jahan Claes's user avatar
6 votes
2 answers
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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 ...
Alk's user avatar
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2 answers
841 views

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\...
Sire's user avatar
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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 = \...
Divy's user avatar
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6 votes
3 answers
117 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=...
Natasha's user avatar
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2 answers
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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 ...
Coconut's user avatar
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1 answer
201 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 $t$?
Rob's user avatar
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6 votes
1 answer
279 views

What are the relations between the permutation group and the Clifford group?

I'm trying to understand the relation between the permutation group on all the $2^n$ bitstrings and the Clifford group. My question arises from the fact that the Toffoli gate (which can be thought of ...
mavzolej's user avatar
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6 votes
1 answer
491 views

Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
Quantum Guy 123's user avatar
6 votes
1 answer
444 views

Is there a non-Clifford gate preserving both $X$ and $Z$ errors?

I would like to know if there exists an $n$-qubit (for $n \geq 2$) quantum gate $G_n$ that preserves both $X$ and $Z$ errors and that is additionnally non-Clifford. In other words, I would like that $...
Marco Fellous-Asiani's user avatar
6 votes
1 answer
219 views

Getting intuition on the state-injection relations for the generalized $\exp(-iP \pi/8)$ $T$-gates (ideally using ZX calculus)

In Litinsky's paper, there are many circuits relations, like the one below. The left handside represents the "rotation" $\exp(-i P \phi)$ with $\phi=\pi/8$ with similar definitions for the ...
Marco Fellous-Asiani's user avatar
5 votes
2 answers
1k views

How to construct the two qubit gate generated by the Hamiltonian $H= X\otimes X + Y \otimes Y + Z \otimes Z $?

I know that the two qubit gate generated by $H=X\otimes X$ is $\exp\{-\text{i}\theta X\otimes X\}=\cos{\theta} \mathbb1 \otimes \mathbb1 - \text{i} \sin{\theta} X \otimes X$, where $X$ is the $\...
Nehad's user avatar
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5 votes
2 answers
835 views

Commutation rules between Pauli $X$ and controlled-Hadamard

Are there any known commutation rules between the $X$ gate and the $CH$ gate?
Daniele Cuomo's user avatar
5 votes
2 answers
2k 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 ...
QFTUNIverse's user avatar
5 votes
2 answers
713 views

How to perform a controlled Pauli string rotation gate?

I would like to know some circuit decomposition for an arbitrary controlled Pauli string rotation: \begin{equation} |0\rangle\langle 0| \otimes e^{i \theta (P_1\otimes...\otimes P_n)}+ |1\rangle\...
Pablo's user avatar
  • 501
5 votes
2 answers
524 views

Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
Quantum Guy 123's user avatar
5 votes
1 answer
336 views

How many $ \sqrt{X} $ are there?

I was reading Square root of Pauli operators: is there a common convention to define them uniquely? and it got me thinking about square roots. Recall the Pauli $ X $ gate $$ X=\begin{bmatrix} 0 & ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
403 views

In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?

My question: Is the center of $ \overline{\text{Cl}_n} $ trivial? Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group must ...
Ian Gershon Teixeira's user avatar
5 votes
1 answer
2k views

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 ...
ZR-'s user avatar
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5 votes
2 answers
435 views

Why are rotations represented by exponentials of Pauli matrices?

I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
slimmerikko's user avatar
5 votes
1 answer
185 views

What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?

Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as $$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
Nichola's user avatar
  • 392
5 votes
1 answer
513 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}...
Martin Vesely's user avatar
5 votes
1 answer
224 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 ...
Saxodrum's user avatar
5 votes
2 answers
2k 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 & ...
QuestionEverything's user avatar
5 votes
2 answers
165 views

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

Basically the title. If I have a $2^N\times 2^N$ Hamiltonian $H$ of random numbers (we can take the Hamiltonian as normalized if we want) and $N$ is an integer, is there an efficient way of writing $$ ...
Physics Penguin's user avatar
5 votes
2 answers
2k views

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 ...
neel.spartacus's user avatar
5 votes
2 answers
163 views

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 ...
Calvin Liu's user avatar
5 votes
1 answer
238 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 ...
Dragonsheep's user avatar
5 votes
1 answer
295 views

Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
user3493403's user avatar
4 votes
3 answers
856 views

Can we write Pauli-Y gate without even complex part?

I was just curious, why is the quantum gate Y-gate (Pauli-Y gate) written in terms of complex numbers? We can actually write Pauli-Y gate as $$ Y = i * \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{...
Gahan's user avatar
  • 159
4 votes
2 answers
536 views

What is the largest number of stabilizers a pure state can have?

What is the largest number of stabilizers a pure state can have? Elaborately put: Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
Quantum Guy 123's user avatar
4 votes
1 answer
118 views

How to prove the fundamental equation in the theory of angular momentum $\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$?

How to prove the inequality$$\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$$ where $J_l = \mathop{\Sigma}_{i=1}^N \frac{1}{2}\sigma_l^{i}$, and $\sigma_l^i$ is pauli matrix acting on the $i$th ...
narip's user avatar
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4 votes
2 answers
562 views

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{...
Oilobobolus's user avatar

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