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}}.
176
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Transform Pauli basis to other basis
The Pauli basis is
\begin{align}
I=\left[\begin{matrix}
1&0 \\
0&1
\end{matrix}
\right],
\end{align}
\begin{align}
X=\left[\begin{matrix}
0&1 \\
1&0
\end{matrix}
\right],
\...
- 175
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Pauli decomposed Hamiltonian as Diagonal U gate
While trying to implement a quantum circuit, I had to apply Hadamard gates to all qubits to achieve equal superposition. Done.
The next operation is decomposing the Hamiltonian into a sum of tensor ...
- 81
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votes
1
answer
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Express $e^{i\frac{\gamma}{2}Z\otimes Z}$ in terms of CNOT and rotations gate
I have the Hamiltonian evolution operator for a two qubit system :
\begin{equation}
\hat{U}= e^{-i\gamma H}\;,\quad H = \frac{1}{2}(I - Z\otimes Z)
\end{equation}
where the $Z$ gate is applied on both ...
- 1
4
votes
1
answer
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How many $ \sqrt{X} $ are there?
I was reading Square root of Pauli operators: is there a common convention to define them uniquely? and it got me thinking about square roots.
Recall the Pauli $ X $ gate
$$
X=\begin{bmatrix}
0 & ...
- 2,146
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votes
1
answer
56
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Tensor product of Pauli strings?
We define
\begin{equation}
\sum_{i=1}^{4^l} P_i \otimes P_i,
\sum_{i=1}^{4^m} Q_i \otimes Q_i,
\end{equation}
where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string.
Does ...
- 175
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2
answers
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Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as
$$
P=VDV^{-1}
$$
for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily ...
- 2,146
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vote
1
answer
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Given a unitary matrix, output the gate name using Qiskit
Given a unitary matrix of a unknown gate, can we write a program in Qiskit to output the name of the corresponding gate if that is a standard gate?
So suppose I have $U = \begin{pmatrix}
0 & 1\\
...
- 105
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votes
1
answer
48
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Propagation rules for the cross-resonance gate of IBM ($R_{zx}$)
The $R_{zx}$ is the fundamental two-qubit gate supported by IBM processors. I'd like to see how Pauli operators propagate over such a gate.
Other well-known cases are shown in the picture below.
- 1,166
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vote
2
answers
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Generating and executing large Pauli rotations in Python
I am interested in generating collective Pauli X, Y and Z spin operators for the purpose of rotating $2^N$ dimensional state vectors $|\psi\rangle$ (in the computational basis) for a quantum protocol. ...
- 747
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1
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99
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What does "commuting operators can be measured simultaneously" mean?
I want to understand better what it means by any commuting set of operators can be measured simultaneously.
Suppose I have an $n$-qubit arbitrary pure state $\rho = \lvert \psi \rangle \langle \psi \...
- 413
1
vote
2
answers
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Vanishing expectation value $|\langle Z_1Z_2...Z_N \rangle|$
I'm doing a research involving expectation values of different observables.
I've observed that, given a random Quantum Circuit $U$ with $N$ qubits acting on an inital state $|0\rangle$ in such a way ...
- 385
1
vote
2
answers
68
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How to find angle for Pauli-Z in U3
What are the angles (lambda, phi and theta) for Pauli-Z gate in U3 ?
2
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1
answer
128
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Tracking the signs of the inverse tableau
Given a tableau and its inverse, how can the signs of the inverse tableau be updated when an operation is appended?
For example, if a $S$ gate is applied on qubit $i$, then the sign of a given ...
user22310
2
votes
1
answer
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How VQE is scalable if the dimension of the Pauli basis of the given Hamiltonian grows exponentially with the number of qubits?
For a given Hamiltonian operator $H$, It's possible to approximate its smallest eigenvalue using VQE. Any Hamiltonian is a Hermitian operator. Therefore, for a system with $n$ qubits, the set $S$ of ...
- 1,574
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votes
1
answer
135
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What are the relations between the permutation group and the Clifford group?
I'm trying to understand the relation between the permutation group on all the $2^n$ bitstrings and the Clifford group. My question arises from the fact that the Toffoli gate (which can be thought of ...
- 1,700
6
votes
1
answer
417
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Is there a non-Clifford gate preserving both $X$ and $Z$ errors?
I would like to know if there exists an $n$-qubit (for $n \geq 2$) quantum gate $G_n$ that preserves both $X$ and $Z$ errors and that is additionnally non-Clifford.
In other words, I would like that $...
- 1,340
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vote
1
answer
61
views
Why use conjugate not transpose complex conjugate in superoperator？
For the n-qubit depolarizing noise, I want to know why it uses $\sigma_{0}^{i*}$ instead of $\sigma_{0}^{i}$ or $\sigma_{0}^{i\dagger}$.
- 437
1
vote
2
answers
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views
Does applying a random Pauli matrix to a density matrix result in the identity?
Nielsen and Chuang's textbook, Equation 8.101 (section 8.3.4 'Depolarizing Channel') shows that applying a random Pauli to a density matrix representing one qubit equals the identity (times one half):
...
- 1,291
3
votes
0
answers
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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
99
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?
- 2,146
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votes
1
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55
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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 ...
- 143
1
vote
2
answers
83
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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
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votes
0
answers
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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, $\exp(iPx)=\cos(x)I+i\sin(x)P$. ...
- 1,166
1
vote
1
answer
70
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 \...
- 327
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0
answers
31
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 ...
- 437
4
votes
1
answer
147
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 ...
- 41
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votes
1
answer
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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 ...
- 439
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votes
2
answers
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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)...
- 2,033
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vote
2
answers
156
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
217
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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⟩$ ...
- 439
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vote
0
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337
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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 ...
- 153
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0
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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$$
If an ...
- 321
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vote
3
answers
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views
How to decompose a multi qubit Clifford unitary into a sequence of clifford gates
What are the algorithms that allow to decompose any given multi qubit Clifford unitary into elementary Clifford operations (e.g. Pauli+CNOT, with no T gate)?
- 61
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votes
1
answer
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views
In what contexts are different notations used for indicating measurement outcomes?
I have seen a few different notations for denoting measurement outcomes. Does anyone know of which notation is more widely used in various contexts?
For instance, I like referring to this Wikipedia ...
- 1,291
2
votes
1
answer
112
views
Entanglement test in Pauli Representation
A 2-qubit state's density matrix can be written in Pauli representation as:
$$
\rho = \frac{1}{4} \sum_{i, j = 0}^{3} c_{i, j}\; \sigma_i \otimes \sigma_j
$$
For a given $\rho$, to compute the ...
- 823
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votes
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answer
301
views
Why do we have only 3 Pauli gates X, Y and Z
This question is out of curiosity thus might not be of much importance.
We have Pauli X, Y, Z gate which rotate the phase by π along X, Y and Z basis. Just wondering why not do we have these 3 gates ...
- 439
1
vote
2
answers
286
views
Expectation value of Pauli strings for VQE
I am studying VQE and have boiled it down to a matter of determining the expectation value of Pauli strings:
$$\langle H \rangle = \sum_i \alpha_i \langle\psi|\hat{P_i}|\psi\rangle.$$
I have been ...
- 11
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votes
1
answer
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How does the Pauli Y gate act in the $|+\rangle, |-\rangle$ basis?
The X gate in the $|+\rangle$, $|-\rangle$ basis becomes the Z gate and vice versa.
What is the Pauli Y gate as a matrix transformation in the $|+\rangle$, $|-\rangle$ basis?
4
votes
1
answer
190
views
Is the Pauli group isomorphic to the Heisenberg group over a finite field?
Let $ p $ be prime and let $ P_n(p) $ denote the Pauli group on $ n $ qudits each of size $ p $. Then $ P_n(p) $ and $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ are both extraspecial $ p $ groups of order $ ...
- 2,146
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votes
1
answer
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views
What is the actual probability of not losing information (in a depolarizing channel)
The probability that a depolarizing channel doesn't affect the information is usually assumed to be $1-3p$, while, for convenience, it is affected with same probability $p$ by any Pauli operator $X,Y,...
- 1,166
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votes
0
answers
84
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What is the correct name of this quantum gate? Possibly state control gate
Let $\vec v \in \mathbb{C}^2 $ be the following quantum state:
$$
\vec v = \frac{1}{\sqrt{2}}\begin{bmatrix}
v_{1} \\
v_{2} \\
\end{bmatrix},\space \lvert v_1 \rvert = 1,...
- 31
1
vote
1
answer
160
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=\...
- 356
3
votes
1
answer
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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 ...
- 129
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votes
1
answer
286
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, ...
- 275
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vote
2
answers
216
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 $X$ ...
8
votes
2
answers
306
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 ...
- 1,166
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votes
1
answer
53
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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 ...
- 1,124
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votes
0
answers
47
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 ...
- 1,124
3
votes
2
answers
395
views
Commutation rules between Pauli $X$ and controlled-Hadamard
Are there any known commutation rules between the $X$ gate and the $CH$ gate?
- 1,166
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votes
1
answer
381
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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\...
- 421