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}}.
188
questions
0
votes
0
answers
18
views
How is the Pauli X, Y, Z implemented in order to "flip" the complex phase of the state probability?
Let's suppose that a 2 qubit system has the following wavefunction:
$$ \Psi = C_{00}\Psi_0\Psi_0 + C_{01}\Psi_0\Psi_1 + C_{10}\Psi_1\Psi_0 + C_{11}\Psi_1\Psi_1 $$
I suppose that what is meant by the &...
- 231
2
votes
1
answer
106
views
Are $n$-qubit Pauli channels $\mathcal E(\rho)=\sum_j p_j P_j \rho P_j$ invertible?
An $n$-qubit Pauli channel $\mathcal E$ acting on a quantum state $\rho$ is of the form
\begin{equation}\label{PauliChannel}
\mathcal E(\rho)=\sum_jp_jP_j\rho P_j
\end{equation}
where $p_j\in[0,1]$ ...
- 457
0
votes
0
answers
34
views
Phase estimation of the Pauli-Y matrix
I'm trying to use the phase estimation algorithm to extract the eigen value for both eigen vectors of the Pauli-Y matrix using the ibm quantum experiance.
So far I have this for the possitive state |+&...
0
votes
0
answers
22
views
Uncorrectable error due to error on ancilla qubit
Consider a controlled-NOT (CX) gate between the two qubits, implemented with an interaction of the form
$
\widehat{H}_{\mathrm{CX}}=V\left[\left(\frac{\hat{I}_1+\widehat{Z}_1}{2}\right) \otimes \hat{I}...
- 197
3
votes
2
answers
184
views
Proof that $\frac{1}{d} | \text{Tr}(\mathcal{P}_i^\dagger \mathcal{E}(\mathcal{P}_j))| \leq 1$ for superoperators
I have two Pauli operators $\frac{1}{\sqrt{d}} \mathcal{P}_i$, $\frac{1}{\sqrt{d}} \mathcal{P}_j$, and an arbitrary quantum channel $\mathcal{E}$ (in the superoperator/Liouville representation) all ...
- 33
1
vote
0
answers
21
views
Is it possible to receive the optimal commuting partitioning of pauli strings (MIN-COMMUTING-PARTITION)
In this paper the authors describe that the MIN-COMMUTING-PARTITION problem is NP-hard. This problem is about finding sets of Pauli strings (of length $n$) in which the strings mutually commute - the ...
- 85
0
votes
1
answer
44
views
What state do you get applying the pauli Y gate to $|\pm\rangle$? [duplicate]
I know it's a basic question but what state gives when you apply pauli $Y$ gate over states $+$ and $-$?
If I apply $Y|+i⟩ = |+i⟩$ or $Y|0⟩ = i|1⟩$, but I don't understand what do you get when you do $...
2
votes
1
answer
53
views
Rotation of qubit - Pauli Gates XYZ
I don't understand how to apply a Pauli Gate on a qubit.
Lets say 8 got a qubit with in state:
$$|\psi\rangle = 0.891 |0\rangle+ 0.454i |1\rangle$$
How would I compute e.g. rotating it 90 degrees ...
1
vote
0
answers
25
views
Conjugating pairs of Paulis to each other with a non-entangling Clifford
This a follow-up question to Conjugating pairs of Paulis to each other with a Clifford
We call a Clifford gate local if it is a tensor product of single qubit Clifford gates.
We call a Clifford gate ...
- 2,556
3
votes
1
answer
50
views
Do the operators $\Phi(\sigma_k)$ form a basis if $\sigma_k$ do?
Assume we have a quantum channel $\Phi$. The single qubit Pauli basis is $\sigma_0, \sigma
_1, \sigma_2, \sigma_3$. Now we apply $\Phi$ to Pauli basis and get $\gamma_0=\Phi(\sigma_0), \gamma_1 = \Phi(...
- 357
2
votes
2
answers
116
views
Measurement in X basis
There is something I don't understand about measurement in other basis than the Z-Pauli Basis.
If measurement fixes the state of a quantum system thus destroying superposition, how can we get a ...
- 266
2
votes
1
answer
65
views
Conjugating pairs of Paulis to each other with a Clifford
Let $ A,B $ be two Paulis with the same order, and neither of which is a multiple of the identity. Then there always exists some Clifford $ C $ such that
$$
CAC^\dagger=B
$$
Let $ A_1,A_2 $ be two ...
- 2,556
0
votes
2
answers
107
views
Transform Pauli basis to other basis
The Pauli basis is
\begin{align}
I=\left[\begin{matrix}
1&0 \\
0&1
\end{matrix}
\right],
\end{align}
\begin{align}
X=\left[\begin{matrix}
0&1 \\
1&0
\end{matrix}
\right],
\...
- 357
0
votes
0
answers
100
views
Pauli decomposed Hamiltonian as Diagonal U gate
While trying to implement a quantum circuit, I had to apply Hadamard gates to all qubits to achieve equal superposition. Done.
The next operation is decomposing the Hamiltonian into a sum of tensor ...
- 81
1
vote
1
answer
100
views
Express $e^{i\frac{\gamma}{2}Z\otimes Z}$ in terms of CNOT and rotations gate
I have the Hamiltonian evolution operator for a two qubit system :
\begin{equation}
\hat{U}= e^{-i\gamma H}\;,\quad H = \frac{1}{2}(I - Z\otimes Z)
\end{equation}
where the $Z$ gate is applied on both ...
- 31
5
votes
1
answer
157
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
$$
X=\begin{bmatrix}
0 & ...
- 2,556
0
votes
1
answer
67
views
Tensor product of Pauli strings?
We define
\begin{equation}
\sum_{i=1}^{4^l} P_i \otimes P_i,
\sum_{i=1}^{4^m} Q_i \otimes Q_i,
\end{equation}
where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string.
Does ...
- 357
4
votes
2
answers
114
views
Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as
$$
P=VDV^{-1}
$$
for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily ...
- 2,556
1
vote
1
answer
67
views
Given a unitary matrix, output the gate name using Qiskit
Given a unitary matrix of a unknown gate, can we write a program in Qiskit to output the name of the corresponding gate if that is a standard gate?
So suppose I have $U = \begin{pmatrix}
0 & 1\\
...
- 115
0
votes
1
answer
70
views
Propagation rules for the cross-resonance gate of IBM ($R_{zx}$)
The $R_{zx}$ is the fundamental two-qubit gate supported by IBM processors. I'd like to see how Pauli operators propagate over such a gate.
Other well-known cases are shown in the picture below.
- 1,300
1
vote
2
answers
102
views
Generating and executing large Pauli rotations in Python
I am interested in generating collective Pauli X, Y and Z spin operators for the purpose of rotating $2^N$ dimensional state vectors $|\psi\rangle$ (in the computational basis) for a quantum protocol. ...
- 747
2
votes
1
answer
124
views
What does "commuting operators can be measured simultaneously" mean?
I want to understand better what it means by any commuting set of operators can be measured simultaneously.
Suppose I have an $n$-qubit arbitrary pure state $\rho = \lvert \psi \rangle \langle \psi \...
- 435
1
vote
2
answers
56
views
Vanishing expectation value $|\langle Z_1Z_2...Z_N \rangle|$
I'm doing a research involving expectation values of different observables.
I've observed that, given a random Quantum Circuit $U$ with $N$ qubits acting on an inital state $|0\rangle$ in such a way ...
- 333
1
vote
2
answers
73
views
How to find angle for Pauli-Z in U3
What are the angles (lambda, phi and theta) for Pauli-Z gate in U3 ?
2
votes
1
answer
129
views
Tracking the signs of the inverse tableau
Given a tableau and its inverse, how can the signs of the inverse tableau be updated when an operation is appended?
For example, if a $S$ gate is applied on qubit $i$, then the sign of a given ...
user22310
2
votes
1
answer
85
views
How VQE is scalable if the dimension of the Pauli basis of the given Hamiltonian grows exponentially with the number of qubits?
For a given Hamiltonian operator $H$, It's possible to approximate its smallest eigenvalue using VQE. Any Hamiltonian is a Hermitian operator. Therefore, for a system with $n$ qubits, the set $S$ of ...
- 1,624
6
votes
1
answer
171
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 ...
- 1,740
6
votes
1
answer
425
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 $...
- 1,142
1
vote
1
answer
71
views
Why use conjugate not transpose complex conjugate in superoperator？
For the n-qubit depolarizing noise, I want to know why it uses $\sigma_{0}^{i*}$ instead of $\sigma_{0}^{i}$ or $\sigma_{0}^{i\dagger}$.
- 457
1
vote
2
answers
136
views
Does applying a random Pauli matrix to a density matrix result in the identity?
Nielsen and Chuang's textbook, Equation 8.101 (section 8.3.4 'Depolarizing Channel') shows that applying a random Pauli to a density matrix representing one qubit equals the identity (times one half):
...
- 1,331
3
votes
0
answers
64
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
$$
...
2
votes
1
answer
111
views
Algorithm for finding Pauli stabilizers of a code
Given the zero logical $ |0_L\rangle $ and one logical $ |1_L\rangle $ for an $ [[n,1,d]] $ code is there a well known/ efficient algorithm for determining which Pauli operators stabilize the code?
- 2,556
0
votes
1
answer
60
views
Expectation values of non-local operators in Qiskit
Is there a convenient way in Qiskit to calculate the expectation value for a non-local operator, i.e. I would like to calculate:
$$ \langle \Psi|O|\Psi \rangle $$
More precisely, I would like to ...
- 143
1
vote
2
answers
107
views
Check if a Pauli string belongs to a stabilizer tableau
Given a Pauli string and a stabilizer tableau, how do I know that the Pauli string belongs to the tableau, i.e. can be written as a product of strings already in the tableau. Thanks.
- 13
0
votes
0
answers
65
views
Equivalence check between rotational gates and Pauli gates
My question is highly related to this one.
I am trying to understand the relationship between rotational gates $R_P(\theta)$, where $P \in \{X,Y,Z\}$.
As stated here, $\exp(iPx)=\cos(x)I+i\sin(x)P$. ...
- 1,300
1
vote
1
answer
92
views
Decomposing a projector in the computational basis in terms of Pauli matrices
I have a $x \in \mathbb{N}$, and I would like to decompose it in terms of the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ and the identity. My first steps are as follows:
$$
\begin{align}
|x \...
- 327
0
votes
0
answers
32
views
Does anybody know what a low-degree Markov field is?
In the paper Fast Estimation of Sparse Quantum Noise I saw the following description:
quantum devices approaching the fault-tolerant regime will have very
few significant errors (and therefore are ...
- 457
5
votes
1
answer
176
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
2
votes
1
answer
646
views
construction of Y gate from X,Z and H gates
As a part of textbook exercise, Y gate is to be constructed using H,Z and X-gates, just like we have $X = HZH$. is there some way/process/intuition to find such combinations or it is just like we need ...
- 439
3
votes
2
answers
213
views
How to show that a sum of Pauli operators is non-zero?
Suppose we have an $n$-qubit system. Let $Y_i$ and $Z_j$ denote the Pauli-Y and Pauli-Z operators acting on the $i$th and $j$th qubits, respectively.
Suppose we have a finite set of tuples $E = \{(i,j)...
- 2,182
1
vote
2
answers
321
views
What are the Pauli-Y eigenvectors?
The question should be pretty simple, but it turns out that there's more to it with respect to what I initially expected.
Starting from the definition of the gate $Y = \begin{bmatrix} 0 & -i \\ i &...
- 478
3
votes
2
answers
249
views
Why doesn't Z-gate change phase of |0⟩
Since the Pauli Z gate equate to a rotation around z axes of the Bloch sphere by $\pi$ radians, the phase of anything that lies on z axes is expected to change by $\pi$ by applying z-gate. As $|0⟩$ ...
- 439
1
vote
0
answers
481
views
what is Pauli twirling approximation?
In this video, Artur Ekert shows that for a single qubit, 4 Kraus operators can be chosen such that the action on state $\rho$ is given as $\rho \rightarrow \sum_m p_m A_m \rho A_m^\dagger$. We can ...
- 153
2
votes
0
answers
34
views
Efficient quantum algorithms to decompose Hessian matrices into sums of unitaries
Are there efficient quantum algorithms that given a d-sparse hessian $H \in \mathbb{C}^{N \times N}$ decompose it into a sum of unitaries (e.g. Pauli matrices)?
$$H = \sum_i^q a_i U_i$$
If an ...
- 341
1
vote
3
answers
287
views
How to decompose a multi qubit Clifford unitary into a sequence of clifford gates
What are the algorithms that allow to decompose any given multi qubit Clifford unitary into elementary Clifford operations (e.g. Pauli+CNOT, with no T gate)?
- 61
4
votes
1
answer
99
views
In what contexts are different notations used for indicating measurement outcomes?
I have seen a few different notations for denoting measurement outcomes. Does anyone know of which notation is more widely used in various contexts?
For instance, I like referring to this Wikipedia ...
- 1,331
2
votes
1
answer
132
views
Entanglement test in Pauli Representation
A 2-qubit state's density matrix can be written in Pauli representation as:
$$
\rho = \frac{1}{4} \sum_{i, j = 0}^{3} c_{i, j}\; \sigma_i \otimes \sigma_j
$$
For a given $\rho$, to compute the ...
- 873
2
votes
1
answer
335
views
Why do we have only 3 Pauli gates X, Y and Z
This question is out of curiosity thus might not be of much importance.
We have Pauli X, Y, Z gate which rotate the phase by π along X, Y and Z basis. Just wondering why not do we have these 3 gates ...
- 439
1
vote
2
answers
303
views
Expectation value of Pauli strings for VQE
I am studying VQE and have boiled it down to a matter of determining the expectation value of Pauli strings:
$$\langle H \rangle = \sum_i \alpha_i \langle\psi|\hat{P_i}|\psi\rangle.$$
I have been ...
- 11
2
votes
1
answer
164
views
How does the Pauli Y gate act in the $|+\rangle, |-\rangle$ basis?
The X gate in the $|+\rangle$, $|-\rangle$ basis becomes the Z gate and vice versa.
What is the Pauli Y gate as a matrix transformation in the $|+\rangle$, $|-\rangle$ basis?
