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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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 ...
Daniele Cuomo's user avatar
3 votes
2 answers
85 views

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: $$\...
am567's user avatar
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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 ...
Nash's user avatar
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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{...
George's user avatar
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Dimension reduction to subsystem in Pauli-Liouville basis: How to implement partial trace of Pauli-Transfer Matrices?

I have a complementary question to my previous (thankfully answered, but I can't verify for larger systems) question: Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-...
Onur Danaci's user avatar
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33 views

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^{...
Lior's user avatar
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1 vote
1 answer
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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{...
karry's user avatar
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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 ...
Onur Danaci's user avatar
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1 answer
40 views

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 ...
Nurdick's user avatar
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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 ...
ScientiaNatura's user avatar
1 vote
2 answers
55 views

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 ...
yeray142's user avatar
1 vote
1 answer
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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?
Jatin Ghildiyal's user avatar
1 vote
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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}, \...
JoJo P's user avatar
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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 ...
Physkid's user avatar
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1 answer
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Coupling map in QISKIT transpile

I have a 3-qubit unitary represented by a circuit with the following dictionary: {'cx': 30, 'h': 22, 'rz': 15, 's': 4, 'sdg': 4}. I want to use this circuit on IBM ...
R.G.J's user avatar
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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 ...
am567's user avatar
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1 answer
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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 $...
Jonas Anderson's user avatar
2 votes
2 answers
80 views

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 $I \otimes I$ ...
seopr's user avatar
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4 votes
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Efficient way to calculate trace of product of Pauli string and matrix?

Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
Physics Penguin's user avatar
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2 answers
64 views

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 ...
Physkid's user avatar
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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 \...
user20374's user avatar
6 votes
1 answer
219 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 ...
Marco Fellous-Asiani's user avatar
2 votes
1 answer
135 views

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....
Piyush Kumar Sinha's user avatar
1 vote
2 answers
295 views

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. ...
squareroottwo's user avatar
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2 answers
50 views

Visualizing Y-gate operation to achieve quantum state

In the below snippet how qc.y(1) helps to achieve the quantum state $i|10\rangle$ ? ...
Khilesh Chauhan's user avatar
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2 answers
310 views

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.
Khilesh Chauhan's user avatar
1 vote
1 answer
121 views

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 ...
Star21's user avatar
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5 votes
2 answers
452 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 \...
slimmerikko's user avatar
2 votes
2 answers
220 views

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 ...
Cheshta Joshi's user avatar
0 votes
1 answer
63 views

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 \...
kong's user avatar
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1 vote
2 answers
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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 ...
Juri V's user avatar
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2 votes
3 answers
201 views

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 $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
MonteNero's user avatar
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1 vote
3 answers
208 views

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 ...
lassel's user avatar
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1 vote
1 answer
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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 ...
epelaez's user avatar
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1 vote
0 answers
120 views

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 ...
epelaez's user avatar
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0 votes
1 answer
31 views

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 ...
Saturnin's user avatar
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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. ...
david's user avatar
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130 views

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 $Z$. ...
david's user avatar
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2 votes
1 answer
197 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 $p_j\in[0,1]$ ...
karry's user avatar
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0 votes
1 answer
61 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 |+&...
Charlie Plath's user avatar
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0 answers
34 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}...
Aubrey Sharansky's user avatar
3 votes
2 answers
208 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 ...
Jed Burkat's user avatar
0 votes
1 answer
72 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 $...
studen101's user avatar
2 votes
1 answer
229 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 ...
Christian Bernhard's user avatar
1 vote
0 answers
38 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 ...
Ian Gershon Teixeira's user avatar
3 votes
2 answers
72 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(...
Michael.Andy's user avatar
2 votes
2 answers
1k 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 ...
Duen's user avatar
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2 votes
1 answer
104 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 ...
Ian Gershon Teixeira's user avatar

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