# 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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### Does anybody know what a low-degree Markov field is?

In the paper Fast Estimation of Sparse Quantum Noise I saw the following description: quantum devices approaching the fault-tolerant regime will have very few significant errors (and therefore are ...
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### Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
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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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### Why doesn't Z-gate change phase of |0⟩

Since the Pauli Z gate equate to a rotation around z axes of the Bloch sphere by $\pi$ radians, the phase of anything that lies on z axes is expected to change by $\pi$ by applying z-gate. As $|0⟩$ ...
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### what is Pauli twirling approximation?

In this video, Artur Ekert shows that for a single qubit, 4 Kraus operators can be chosen such that the action on state $\rho$ is given as $\rho \rightarrow \sum_m p_m A_m \rho A_m^\dagger$. We can ...
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### Efficient quantum algorithms to decompose Hessian matrices into sums of unitaries

Are there efficient quantum algorithms that given a d-sparse hessian $H \in \mathbb{C}^{N \times N}$ decompose it into a sum of unitaries (e.g. Pauli matrices)? $$H = \sum_i^q a_i U_i$$ If an ...
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### How to decompose a multi qubit Clifford unitary into a sequence of clifford gates

What are the algorithms that allow to decompose any given multi qubit Clifford unitary into elementary Clifford operations (e.g. Pauli+CNOT, with no T gate)?
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### In what contexts are different notations used for indicating measurement outcomes?

I have seen a few different notations for denoting measurement outcomes. Does anyone know of which notation is more widely used in various contexts? For instance, I like referring to this Wikipedia ...
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### Entanglement test in Pauli Representation

A 2-qubit state's density matrix can be written in Pauli representation as: $$\rho = \frac{1}{4} \sum_{i, j = 0}^{3} c_{i, j}\; \sigma_i \otimes \sigma_j$$ For a given $\rho$, to compute the ...
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### Why do we have only 3 Pauli gates X, Y and Z

This question is out of curiosity thus might not be of much importance. We have Pauli X, Y, Z gate which rotate the phase by π along X, Y and Z basis. Just wondering why not do we have these 3 gates ...
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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 ...
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### How does the Pauli Y gate act in the $|+\rangle, |-\rangle$ basis?

The X gate in the $|+\rangle$, $|-\rangle$ basis becomes the Z gate and vice versa. What is the Pauli Y gate as a matrix transformation in the $|+\rangle$, $|-\rangle$ basis?
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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 ...
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### 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, ...
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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 $X$ ...
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### 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 ...
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### 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 ...
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### Commutation rules between Pauli $X$ and controlled-Hadamard

Are there any known commutation rules between the $X$ gate and the $CH$ gate?
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### 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\...
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### 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 ...
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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 ...
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### 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 ...
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### 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 &...
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### 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 ...
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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 ...
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### 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 ...
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### 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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