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}}.
254 questions
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Help with a lemma on the argument of a qubit after transformation
From:
King, R. (2023). An improved approximation algorithm for quantum max-cut on triangle-free graphs. Quantum, 7, 1180.
I have trouble understanding item 3 of the above lemma. Here $n_k \cdot \...
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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 $\sigma$ ...
glS♦
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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 ...
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Exponentiating a tensor product of operators acting on disjoint qubit registers
Consider a problem of implementing $\operatorname{e}^{i\bigotimes_j O_j}$, where all the $O_j$ terms act on disjoint sets of qubits.
Assume that efficient circuits implementing individual $\...
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What is the action of $CCZ$ on $X \times I \times I$?
Confused about the action of the $CCZ$ gate on Pauli operators:
I understand the action of the $CZ$ gate: $$CZ: XI \rightarrow XX$$ $$CZ: IX \rightarrow IX$$ $$CZ: ZI \rightarrow ZI$$ $$CZ: IZ \...
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Interested in software helping with projecting multi-qubit states onto irreducible components
My interest in QC comes from a problem in geometry called the Atiyah problem on configurations of points. In short, there is a nice one-to-one correspondence between quantum states of a single qubit ...
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Why is the error propagation by the CNOT gate considered without taking into account the state?
In the syndrome measurement circuit of a stabilizer code, I think you would consider that Pauli errors propagate through the CNOT gates. I don't understand why one usually considers the propagation of ...
CommunityBot
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Is there a convention for denoting $Y$ eigenstates?
Two common shorthands for eigenstates of the $Z$ operator are $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|-1\rangle\}$, where in the first case we have $Z|z\rangle=(-1)^z|z\rangle$ and in the second ...
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Why is the linear combination of Pauli matrices $G =I-XX-YY-ZZ$ PSD?
Define
$$G = I \otimes I - X \otimes X - Y \otimes Y - Z \otimes Z,$$
where $X,Y$ and $Z$ denote the Pauli matrices, and $I$ the identity. I can plug this matrix in my computer and note that
$$G = \...
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Action of below circuit using heisenberg representation
Can someone please explain how the above gate affects logical operators? My understanding is that the circle indicates that we are measuring the second qubit?
My initial guess is that it is equivalent ...
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Can I postpone a Pauli gate $X$ over a conditioned measurement $Y$ or $X$?
The above circuit shows a first measurement, which is $\langle X \rangle$ or $\langle Y \rangle$, depending on the outcome of a second measurement.
Assuming now that a third measurement decides ...
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Why can a quantum code correct $t$ errors only when $d \geq 2t + 1$?
I am working from chapter 7 from notes for ph229 by J. Preskill.
The notes define the distance of a quantum code as:
The distance $d$ is the the minimum weight of a Pauli operator $E$
such that: $$\...
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help understanding gate to hamiltonian and representation
So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
glS♦
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In the phase flip action on standard basis, why do we consider the $-1$ phase only for the $|1\rangle$?
Prof. Watrous in the first lecture of Qiskit summer school 2023, mentions:
"....the significance of putting a minus sign in front of the $|1\rangle$
basis vector and not $|0\rangle$ will be more ...
glS♦
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Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-Transfer Matrices?
I would like to verify something, need a sanity check. Are the quantum channels for different qubits in the Pauli-Liouville basis (Pauli Transfer Matrices) also given by a tensor product?
The Kraus ...
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scaling of error of sum of Pauli strings with number of shots
I have a question which I suppose is quite basic.
Let's say I want to measure the average of an obersvable which is the sum of non-commuting Pauli strings on $N_q$ qubits:
$$
\langle O\rangle =\sum_i^{...
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Process matrix of CNOT gate
The fig below is the process matrix of the CNOT gate from this paper:
where the legend explains that red corresponds to $\frac14$, green to $-\frac14$ and white to zero.
I know the $U_{CNOT} = \frac{...
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Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$
Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory".
The exercise states:
We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We
choose the ordering $(...
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How is Quantum Computing expressed in the language of abstract algebra?
I've lately been taking further coursework in abstract algebra, and it has struck me as fairly reminiscent of quantum computing. Of course, Pauli matrices, etc. have relevant roots within abstract ...
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Expectation value of a given observable computed manually using qiskit.Sampler is different as with qiskit.Estimator
I am trying to calculate the expectation value of a given observable for a certain state $\psi$ using qiskit primitives Sampler (using Sampler requires some further calculations) and Estimator. I ...
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Digitization of errors in QEC
In Nielsen and Chuang, it is stated that any error is given by a quantum channel with Kraus operators $E_i$. A pure state $\vert\psi\rangle\langle\psi\vert$ becomes $\sum_i E_i\vert\psi\rangle\langle\...
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Measuring a single-qubit PauliZ using Qiskit's EstimatorQNN
I am currently working with the EstimatorQNN from Qiskit to construct a Quantum Neural Network using a custom Parametrized Quantum Circuit. But I want to change the ...
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How to find density matrix of 3 qubit W state?
Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?
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How is Pauli twirling so powerful?
So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to:
$\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$
where $\sigma^0 = \mathbb{I}, \...
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Physical description of trace of ancilla state yields a depolarising channel
Let's start with
$Tr_{\Omega}[|0,\Omega_{0}\rangle\langle0,\Omega_{0}|U^{\dagger}] = \sum_{\alpha}E_{\alpha}|0\rangle\langle0|E_{\alpha}^{\dagger}$
where $U$ be a unitary operator. The trace operator ...
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How to interpret the encoding circuit for the 5-qubit QECC
I have a question on circuit which constitutes the sydnrome measurement for the 5-qubit error correcting code. If I focus on just a portion of the circuit:
Reference for image. The full circuit can ...
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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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When is a block diagonal matrix a tensor product of Pauli matrices?
$U = |0\rangle\langle 0|\otimes U_1 + |1\rangle\langle 1|\otimes U_2$ is a block-diagonal unitary matrix. For this question we will assume $U$ acts on qubits. Then for some integer $N\ge 1$, $U$ is a $...
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How to prove that these equations are correct for $CZ$ and $CX$?
How do I prove that the equation on the right is $CX$ and $CZ$ gate? I don't think that reaching the matrix of the CX or CZ is possible with the given equation.
For (b) I keep getting $I \otimes I$ ...
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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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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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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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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 ...
CommunityBot
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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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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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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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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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