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}}.
204
questions
18
votes
1
answer
4k
views
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 ...
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 ...
- 1,029
15
votes
2
answers
8k
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 ...
- 838
15
votes
3
answers
10k
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 ...
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 ...
- 543
10
votes
1
answer
3k
views
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, ...
- 211
10
votes
2
answers
2k
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,...
- 303
10
votes
3
answers
837
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 ...
- 32.4k
9
votes
4
answers
866
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:
$\...
- 335
9
votes
2
answers
310
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 ...
- 32.4k
8
votes
2
answers
445
views
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 ...
- 1,414
7
votes
3
answers
2k
views
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,...
- 179
7
votes
2
answers
4k
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 ...
- 601
7
votes
1
answer
2k
views
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+...
- 461
7
votes
2
answers
538
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 ...
- 13.4k
7
votes
1
answer
226
views
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 ...
- 1,351
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 ...
- 155
6
votes
3
answers
186
views
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 ...
- 647
6
votes
2
answers
2k
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 ...
- 163
6
votes
2
answers
679
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\...
- 63
6
votes
2
answers
917
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 = \...
- 61
6
votes
3
answers
116
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=...
- 121
6
votes
2
answers
2k
views
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 ...
- 95
6
votes
1
answer
186
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 $t$?
- 371
6
votes
1
answer
202
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 ...
- 1,770
6
votes
1
answer
440
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 $\sigma$ ...
- 1,351
6
votes
1
answer
433
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 $...
- 1,272
6
votes
1
answer
187
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 ...
- 1,272
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 $\...
- 71
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 ...
- 151
5
votes
2
answers
448
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 ...
- 1,351
5
votes
1
answer
205
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
$$
X=\begin{bmatrix}
0 & ...
- 2,820
5
votes
1
answer
336
views
In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?
My question:
Is the center of $ \overline{\text{Cl}_n} $ trivial?
Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group must ...
- 2,820
5
votes
1
answer
1k
views
How can I simulate Hamiltonians composed of Pauli matrices?
Suppose I want to perform the time-evolution simulation on the following Hamiltonians:
$$
H_{1} = X_1+ Y_2 + Z_1\otimes Z_2
\\
H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2
$$
Where $X,Y,Z$ are Pauli ...
- 2,368
5
votes
2
answers
122
views
Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
- 103
5
votes
1
answer
170
views
What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?
Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as
$$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
- 360
5
votes
1
answer
470
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:
$$
F=-\sum_{i<j}J_{ij}...
- 13.4k
5
votes
1
answer
204
views
Where is the factor of $-i$ in rotation gates coming from?
As I understand it the Pauli-X, Y and Z gates are the same as their rotational gates with a rotation of $\pi$. But given the expression for those gates, I find that there is a factor of $-i$ in each ...
- 53
5
votes
2
answers
1k
views
Physical Interpretation of Pauli Matrices as Polarization Check
We know that the Pauli matrices are:
$$\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 & ...
- 1,776
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 ...
5
votes
2
answers
148
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 ...
- 183
5
votes
1
answer
200
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 ...
- 233
5
votes
0
answers
205
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 ...
- 51
4
votes
3
answers
774
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
$$
Y = i * \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{...
- 159
4
votes
2
answers
521
views
What is the largest number of stabilizers a pure state can have?
What is the largest number of stabilizers a pure state can have? Elaborately put:
Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
- 1,351
4
votes
1
answer
116
views
How to prove the fundamental equation in the theory of angular momentum $\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$?
How to prove the inequality$$\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$$
where $J_l = \mathop{\Sigma}_{i=1}^N \frac{1}{2}\sigma_l^{i}$, and $\sigma_l^i$ is pauli matrix acting on the $i$th ...
- 2,842
4
votes
2
answers
446
views
If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"?
The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate
\begin{equation}
Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
- 133
4
votes
2
answers
196
views
How does a single-qubit gate affect other qubits?
An instructional quantum computing article I'm reading (How the quantum search algorithm works) states that the following circuit takes $\vert x\rangle\vert 0\rangle$ to $−\vert x\rangle\vert 0\rangle$...
- 85
4
votes
1
answer
463
views
Shorthand notation for the sign flip gate
I need to use the following matrix gate in a quantum circuit:
$$\text{Sign Flip}=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$$
$\text{Sign Flip}$ can be decomposed as (in terms ...
- 17.1k
4
votes
2
answers
185
views
Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as
$$
P=VDV^{-1}
$$
for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily ...
- 2,820