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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In context of stabilizer codes, are logical gates and Pauli operators the same?

I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
Hamed's user avatar
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Almost perfect quantum encryption of a mixed state using only $n + O(\log(n) +\log(\frac{1}{\epsilon}))$ shared bits

Alice holds a state $\psi$ of $n$-qubits, and wants to send it to Bob using a single quantum message. Bob and Alice share only $n + O(log(n) +log(\frac{1}{\epsilon}))$ random bits, for some value $\...
Gabi G's user avatar
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Connection between a Pauli measurement and the corresponding Pauli gate?

Suppose I have a qubit and the ability to act a Pauli $Z$ gate on it. This is a black box that does the phase flip and I don't know how it works on the inside. Can I use this black box to implement a ...
Brendan's user avatar
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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 \...
am567's user avatar
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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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Trying to prove Theorem 4.1 from Neilsen and Chuang algebraically

Background Theorem 4.1 of Neilsen and Chuang (10th Anniversary Edition) states how a universal single-qubit unitary can be constructed from Y and Z rotations. Suppose $U$ is a unitary operation on a ...
kaddy's user avatar
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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 ...
Malkoun's user avatar
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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 = \...
Matteo's user avatar
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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 ...
Daniele Cuomo's user avatar
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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: $$\...
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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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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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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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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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
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60 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
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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
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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 ...
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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 ...
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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 vote
1 answer
108 views

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
87 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
1 answer
257 views

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
1 vote
2 answers
70 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
229 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
192 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
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2 answers
400 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
411 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
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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 ...
Star21's user avatar
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5 votes
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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 \...
slimmerikko's user avatar
2 votes
2 answers
273 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
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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 \...
kong's user avatar
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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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3 answers
241 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 $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
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2 votes
3 answers
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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 ...
lassel's user avatar
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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 ...
epelaez's user avatar
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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 ...
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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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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