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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tricks to finding possible stabilisers for $|GHZ_{3} \rangle$
The famous 3 - qubit Greenberger, Horne and Zeilinger state: $|GHZ_{3} \rangle = \frac{1}{\sqrt{2}}[|000\rangle + |111\rangle]$.
A stabiliser for $|GHZ_{3} \rangle$ is the 3 - tensor product X Pauli ...
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Visualizing Y-gate operation to achieve quantum state
In the below snippet how qc.y(1) helps to achieve the quantum state $i|10\rangle$ ?
...
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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 ...
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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 ...
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Notation: Hamiltonian Simulation of Pauli Gates
Let $\sigma^j_x$ describe the following unitary over $n$ qubits: on the $j$-th qubit, it acts as the Pauli $x$ operator; instead, on any other qubit, it acts as the identity. A paper states now that
\...
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Half Adder using CNOT Gates
As per this schematic of qubits, how this explanation is correct --"If you look again at the four possible sums, you’ll notice that there is only one case for which this is 1 instead of 0: 1+1=10....
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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(...
glS♦
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How to prove the matrix identities $HXH = Z$ and $HZH = X$?
As we know Hadamard gates are used to bring quantum bits into superposition states.
I’m trying to understand how identities $HXH = Z$ & $HZH = X$ w.r.t rotation.
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Can any Qiskit circuit be converted to a gate?
I am trying to convert the following qiskit QuantumCircuit to a gate using to_gate() method.
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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)
On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
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A question on the structure of the Clifford group
Let $P_n$ be the Pauli group on $n$ qubits defined by $P_n=\{i^k \sigma_1...\sigma_n: k=0, 1, 2 \; \mathrm{or}\;3, \sigma_j\in \{I, X, Y, Z\}\}$, where $I, X, Y, Z$ are Pauli matrices. The Clifford ...
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Algorithm for Mutually Unbiased Basis Sets Available?
I'm looking for an implementation or a slightly more efficient algorithm for finding optimal Mutually Unbiased Bases (MUB). What I mean here are MUBs in terms of Pauli Strings as described here. There ...
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Why is the error propagation by the CNOT gate considered without taking into account the state?
In the syndrome measurement circuit of a stabilizer code, I think you would consider that Pauli errors propagate through the CNOT gates. I don't understand why one usually considers the propagation of ...
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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 \...
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How can I implement a Hamiltonian which is sum of tensored pauli operators on qiskit?
I am working with a Tight Binding Hamiltonian with N sites and one orbital at each site in a closed chain. I have converted the fermionic expression to a spin expression using Jordan Wigner ...
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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 ...
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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,...
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Commutation relationship and measurement results
There are things I do not understand about the following circuit, and I would appreciate it if you could explain.
...
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Finding the rotation angle $\theta$ of a 2x2 unitary matrix
We can represent a 2x2 unitary matrix as follows:
$$U = \cos(\theta)I - i \sin(\theta) \vec{n} \cdot \vec{\sigma},$$
where $\vec{n} \in \mathbb{R}^3$ and $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
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G-twisted Pauli twirl circuit
Pauli twirls are obtained by taking a unitary $U$, and finding some Pauli gates $P_1, P_2$ such that $P_1 U P_2$. So, for example, one possible twirl of the $S$ gate would be $YSX$.
In the paper ...
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How is the definition of $n$-qubit Pauli group derived?
The authors give the following definition for the Pauli group in the paper Averaged circuit eigenvalue sampling.
The n-qubit Pauli group $P_n$ consists of n-fold tensor products of single-qubit Pauli ...
glS♦
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Recovering phases in $2n$-bit binary representation of n-qubit Paulis
I am currently going through a paper discussing Pauli sampling strategies for VQE: https://arxiv.org/abs/1908.06942
I want to code and test their strategy.
They explain how to create a circuit ...
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Correctability of X, Y, and Z Errors in Quantum Surface Codes and Color Codes
In surface codes and color codes, when the code distance is $d$, you can correct up to $[(d-1)/2]$ Pauli errors. I would like to know what this $[(d-1)/2]$ Pauli errors means for $X$, $Y$, and $Z$. ...
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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 $p_j\in[0,1]$ ...
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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 |+&...
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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}...
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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 ...
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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 $...
glS♦
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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 ...
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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 ...
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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
$$
X=\begin{bmatrix}
0 & ...
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Measurement in X basis
There is something I don't understand about measurement in other basis than the Z-Pauli Basis.
If measurement fixes the state of a quantum system thus destroying superposition, how can we get a ...
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Conjugating pairs of Paulis to each other with a Clifford
Let $ A,B $ be two Paulis with the same order, and neither of which is a multiple of the identity. Then there always exists some Clifford $ C $ such that
$$
CAC^\dagger=B
$$
Let $ A_1,A_2 $ be two ...
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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],
\...
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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 ...
glS♦
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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):
...
glS♦
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 $U = \begin{pmatrix}
0 & 1\\
...
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Propagation rules for the cross-resonance gate of IBM ($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.
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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. ...
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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 \...
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Commutation rules between Pauli $X$ and controlled-Hadamard
Are there any known commutation rules between the $X$ gate and the $CH$ gate?
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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 $...
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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 ...
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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 $\sigma_{0}^{i\dagger}$.
CommunityBot
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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 ...
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