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}}.

Filter by
Sorted by
Tagged with
3 votes
0 answers
30 views

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

Basically the title. If I have a $2^N\times 2^N$ Hamiltonian $H$ of random numbers (we can take the Hamiltonian as normalized if we want) and $N$ is an integer, is there an efficient way of writing $$ ...
  • 33
2 votes
1 answer
67 views

Algorithm for finding Pauli stabilizers of a code

Given the zero logical $ |0_L\rangle $ and one logical $ |1_L\rangle $ for an $ [[n,1,d]] $ code is there a well known/ efficient algorithm for determining which Pauli operators stabilize the code? ...
0 votes
1 answer
41 views

Expectation values of non-local operators in Qiskit

Is there a convenient way in Qiskit to calculate the expectation value for a non-local operator, i.e. I would like to calculate: $$ \langle \Psi|O|\Psi \rangle $$ More precisely, I would like to ...
1 vote
2 answers
50 views

Check if a Pauli string belongs to a stabilizer tableau

Given a Pauli string and a stabilizer tableau, how do I know that the Pauli string belongs to the tableau, i.e. can be written as a product of strings already in the tableau. Thanks.
  • 13
0 votes
0 answers
21 views

Equivalence check between rotational gates and Pauli gates

My question is highly related to this one. I am trying to understand the relationship between rotational gates $R_P(\theta)$, where $P \in \{X,Y,Z\}$. As stated here, $\exp(iPx)=\cos(x)I+i\sin(x)P$. ...
1 vote
1 answer
26 views

Decomposing a projector in the computational basis in terms of Pauli matrices

I have a $x \in \mathbb{N}$, and I would like to decompose it in terms of the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ and the identity. My first steps are as follows: $$ \begin{align} |x \...
0 votes
0 answers
30 views

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 ...
  • 359
3 votes
1 answer
91 views

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 ...
2 votes
1 answer
204 views

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 ...
3 votes
2 answers
96 views

How to show that a sum of Pauli operators is non-zero?

Suppose we have an $n$-qubit system. Let $Y_i$ and $Z_j$ denote the Pauli-Y and Pauli-Z operators acting on the $i$th and $j$th qubits, respectively. Suppose we have a finite set of tuples $E = \{(i,j)...
  • 1,161
1 vote
2 answers
52 views

What are the Pauli-Y eigenvectors?

The question should be pretty simple, but it turns out that there's more to it with respect to what I initially expected. Starting from the definition of the gate $Y = \begin{bmatrix} 0 & -i \\ i &...
  • 476
3 votes
2 answers
157 views

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 $|0⟩$ ...
1 vote
0 answers
120 views

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 ...
  • 123
2 votes
0 answers
26 views

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 ...
  • 189
1 vote
3 answers
106 views

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)?
4 votes
1 answer
89 views

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 ...
2 votes
1 answer
60 views

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 ...
  • 339
2 votes
1 answer
209 views

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 ...
1 vote
2 answers
191 views

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 ...
2 votes
1 answer
49 views

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?
4 votes
1 answer
127 views

Is the Pauli group isomorphic to the Heisenberg group over a finite field?

Let $ p $ be prime and let $ P_n(p) $ denote the Pauli group on $ n $ qudits each of size $ p $. Then $ P_n(p) $ and $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ are both extraspecial $ p $ groups of order $ ...
0 votes
1 answer
34 views

What is the actual probability of not losing information (in a depolarizing channel)

The probability that a depolarizing channel doesn't affect the information is usually assumed to be $1-3p$, while, for convenience, it is affected with same probability $p$ by any Pauli operator $X,Y,...
3 votes
0 answers
79 views

What is the correct name of this quantum gate? Possibly state control gate

Let $\vec v \in \mathbb{C}^2 $ be the following quantum state: $$ \vec v = \frac{1}{\sqrt{2}}\begin{bmatrix} v_{1} \\ v_{2} \\ \end{bmatrix},\space \lvert v_1 \rvert = 1,...
1 vote
1 answer
128 views

Calculate $\sqrt[4]{X}$ for the Pauli $X$ gate

I was trying to build a $cccx$ gate. According to this paper by Berenco et al., it requires a $\sqrt[4]{X}$ gate. Furthermore, I found another paper by Muradian and Frias with this formula: $$\sqrt A=\...
3 votes
1 answer
47 views

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 ...
3 votes
1 answer
117 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
2 answers
137 views

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 $X$ ...
6 votes
2 answers
182 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 ...
0 votes
1 answer
41 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 ...
  • 1,022
0 votes
0 answers
35 views

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 ...
  • 1,022
2 votes
2 answers
159 views

Commutation rules between Pauli $X$ and controlled-Hadamard

Are there any known commutation rules between the $X$ gate and the $CH$ gate?
3 votes
1 answer
270 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\...
  • 421
5 votes
1 answer
243 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
1 answer
60 views

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/...
  • 153
2 votes
2 answers
77 views

How to prove that the trace of n-qubit matrices satisfies ${\rm Tr}(XY)=2^n\sum_{M\in\{I,X,Y,Z\}^n} x_M y_M$?

It is known that for n-qubit matrices X, Y $\in \mathbb{C}^{2^{n}\times 2^{n}}$ (and Pauli matrices $I, X, Y, Z$) such that $$ X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n} $...
  • 153
1 vote
1 answer
125 views

How to express $n$-qubit Hermitian operator with Pauli matrices

How can we prove that all $n$-qubit Hermitian matrices can be written in terms of Pauli matrices $I$, $X$, $Y$, and $Z$ as $$ \sum_{W_k \in \{I, X, Y, Z\}} a_{W_1,\dots,W_n}W_{1}\otimes ... \otimes W_{...
  • 153
1 vote
1 answer
54 views

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
2 answers
63 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 ...
2 votes
1 answer
216 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 &...
  • 107
1 vote
1 answer
74 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 ...
  • 13
1 vote
1 answer
349 views

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 ...
  • 147
3 votes
1 answer
96 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?
4 votes
2 answers
468 views

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 ...
7 votes
1 answer
159 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 ...
3 votes
1 answer
75 views

Cliffords to Transform into Common Eigenbasis

Say I have the following Hamiltonian (given in terms of Pauli operators): \begin{equation} H=aX_1Z_2+bZ_1X_2. \end{equation} Both Pauli terms commute with each other. I want to make a measurement of $\...
3 votes
1 answer
136 views

Can you measure sums of Paulis in the stabilizer formalism?

Suppose we wanted to measure the observable $Z_{1} + Z_{2} + \cdots + Z_{N}$ in a stabilizer state. Is it possible to do this using only Clifford operations, and possibly adding some auxiliary qubits? ...
  • 215
1 vote
2 answers
147 views

Qiskit's PauliTrotterEvolution yields weird gates

I am trying to work with Qiskit's PauliTrotterEvolution() module, but the resulting circuits contain weird gates that I know nothing about. Here is a simple example: I want to implement the fermionic ...
6 votes
1 answer
543 views

Does the controlled Pauli Z gate cause entanglement?

I'm trying to understand the relationship between the factorability of a 2 qubit gate and that gate's ability to cause entanglement. I've begun by considering the controlled Pauli Z gate. After ...
  • 145
5 votes
2 answers
591 views

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,...
  • 147
5 votes
1 answer
136 views

What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?

Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as $$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
  • 338