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# 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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### Confirming locality of a Hamiltonian through decomposition

I was trying to understand Trotterization. The given Hamiltonian is decomposed into a sum of $k$-local Hamiltonians which can be exponentiated in $O(1)$ gate complexity. After which the Trotter ...
340 views

### Direction of rotation for single-qubit unitary operators

The rotation operators for a single qubit are defined as $R_{v}(\theta) = e^{-i \theta X/2}$, with $v \in \{ X,Y,Z\}$. If we look at the direction of rotation of $R_v$ w.r.t. the positive eigenvalue, ...
1 vote
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### How to obtain the state $|0\rangle+|1\rangle$ from $|0\rangle$ via Pauli gates?

Could somebody explain in which way are we able to achieve superposition with Pauli $X$, $Y$, $Z$ matrices? In case of Hadamard gate $H$ we change coefficients to $1/\sqrt{2}$ directly, in case of $X$ ...
343 views

### What is (formally) a transversal operator?

This question concerns about a formal definition of transversal operator. I understood that transversal operator are a group of operators which are efficient in terms of circuit depth and can be used ...
56 views

### Decompose into completely stabilizer preserving channel in surface codes

In the article "Sampling-based quasiprobability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise" they are talking about decomposing (possibly ...
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### Convert Coherent Noise to Clifford Errors with Probability on Surface Codes

Following my question about the equivalence of coherent and no coherent error, in surface codes. Now I understand, it is not equivalent. I tried to read some articles about it, and I couldn't find a ...
507 views

### Commutation rules between Pauli $X$ and controlled-Hadamard

Are there any known commutation rules between the $X$ gate and the $CH$ gate?
425 views

### How to perform a controlled Pauli string rotation gate?

I would like to know some circuit decomposition for an arbitrary controlled Pauli string rotation: \begin{equation} |0\rangle\langle 0| \otimes e^{i \theta (P_1\otimes...\otimes P_n)}+ |1\rangle\...
305 views

### In the Clifford group, is the center of $\overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?

My question: Is the center of $\overline{\text{Cl}_n}$ trivial? Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group must ...
1 vote
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### Can we design a circuit that outputs desired estimates?

If we have state $\lvert\psi\rangle \in (\mathbb{C}^{2})^{\otimes n}$ in an $\textit{n}$-qubit system with Pauli operators $P$ such that $P \in \{I, X, Y, Z\}^{n}$, how can we design a circuit/...
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### General reason behind why $X_L$ and $Z_L$ can be processed on software for error correction

I am reading surface code theory with this paper. It is explained there that the $X_L$ and $Z_L$ (logical $X$ and $Z$ operator) can be pushed at the end of the circuit and they actually do not have to ...
1 vote
84 views

### Is it true that $Ry(\pi/2)\sigma_zRy(-\pi/2)=\sigma_x$?

I saw in a qiskit document that said $Ry(\pi/2)\sigma_zRy(-\pi/2)=\sigma_x$ To confirm this I decided to create the matrix representations of these operations and multiply them together to see if I ...
278 views

### What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?

I'm looking for an algorithm to write an arbitrarily sized tridiagonal matrix as a linear combination of Pauli matrices. The tridiagonal matrix has the form, for example, \begin{pmatrix} 2 & -1 &...
1 vote
149 views

### Heisenberg Uncertainty Principle for BB84 Protocol using Paulis Spin Matrices

I am doing a term project on the BB84 Protocol and it makes use of the Heisenberg Uncertainty Principle. I think I understand the principle in theory. If we have two non-commuting observables, then we ...
1 vote
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### How does the ZZ Feature Map influence the measurement?

I've been look at this Notebook from qiskit and trying to understand whats happening, but can't quite figure it out. From my understanding, rotations around the Z ...
200 views

### How are gates implemented in a transmon qubit?

A transmon qubit is fundamentally in LC circuit. How are gates implemented in a transmon qubit? How do we know what voltage corresponds to the $\sigma_x$ gate for example?
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### What is the largest number of stabilizers a pure state can have?

What is the largest number of stabilizers a pure state can have? Elaborately put: Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
209 views

### Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
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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 $\sigma$ ...