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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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 $...
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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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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(...
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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 ...
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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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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
$$
X=\begin{bmatrix}
0 & ...
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Tensor product of Pauli strings?
We define
\begin{equation}
\sum_{i=1}^{4^l} P_i \otimes P_i,
\sum_{i=1}^{4^m} Q_i \otimes Q_i,
\end{equation}
where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string.
Does ...
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Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as
$$
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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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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How to find angle for Pauli-Z in U3
What are the angles (lambda, phi and theta) for Pauli-Z gate in U3 ?
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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
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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 ...
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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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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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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}$.
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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):
...
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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
$$
...
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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?
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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