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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Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]
Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?
For example, I have a matrix 16x16 and want it to be decomposed into something ...
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How can I decompose a matrix in terms of Pauli matrices?
I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices.
I would prefer an option to do this in larger than 2 dimensions, ...
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Why can I apply $HS^\dagger$ and then measure in the computational basis to measure $Y$?
I come from a CS background
I was reading Neven and Farhi's paper ("Classification with Quantum Neural Networks on near Term Processors"), and I am trying to implement the subset parity problem using ...
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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How to construct matrix of regular and "flipped" 2-qubit CNOT?
When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning:
"If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix ...
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Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer?
Quantum annealers are single purpose machines allowing to solve quadratic unconstrained binary optimization (QUBO) problems. QUBO problems have following objective function:
$$
F=-\sum_{i<j}J_{ij}...
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2
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Qiskit CNOT-gate matrix mixup?
In the qiskit textbook chapter 1.3.1 "The CNOT-Gate" it says that the matrix representation on the right is the own corresponding to the circuit shown above, with q_0 being the control and ...
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Definition of the Pauli group and the Clifford group
There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as
\begin{align*}
\mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z,...
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Obtaining gate $e^{-i\Delta t Z}$ from elementary gates
I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't ...
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?
I see here in Olivia DeMatteo's notes, she states:
When we consider the action of the entire Clifford group on a single non-identity Pauli, it
maps that Pauli to each of the $d^2 − 1$ other possible ...
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What are the relations between the permutation group and the Clifford group?
I'm trying to understand the relation between the permutation group on all the $2^n$ bitstrings and the Clifford group. My question arises from the fact that the Toffoli gate (which can be thought of ...
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4
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Shorthand notation for the sign flip gate
I need to use the following matrix gate in a quantum circuit:
$$\text{Sign Flip}=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$$
$\text{Sign Flip}$ can be decomposed as (in terms ...
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4
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Simulate Hamiltonians with Pauli operations (controlled time evolution)
I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ...
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In Variational Quantum Eigensolvers, what does "grouping Pauli operators into tensor products requiring the same post-rotations" mean?
In this paper (nature version), the authors state
We group the Pauli operators into tensor product basis sets that require the same post-rotations.
As a result, they have the table S2 in the suppl. ...
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How to build a circuit for simulation of a simple Hamiltonian?
Consider very simple Hamiltonian $\mathcal{H} = Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$. It has eigenvalues 1 and -1 with coresponding eigenstates $|0\rangle$ and $|1\rangle$, ...
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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$ ...
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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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How to construct the two qubit gate generated by the Hamiltonian $H= X\otimes X + Y \otimes Y + Z \otimes Z $?
I know that the two qubit gate generated by $H=X\otimes X$ is $\exp\{-\text{i}\theta X\otimes X\}=\cos{\theta} \mathbb1 \otimes \mathbb1 - \text{i} \sin{\theta} X \otimes X$, where $X$ is the $\...
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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 $...
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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 ...
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How can I simulate Hamiltonians composed of Pauli matrices?
Suppose I want to perform the time-evolution simulation on the following Hamiltonians:
$$
H_{1} = X_1+ Y_2 + Z_1\otimes Z_2
\\
H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2
$$
Where $X,Y,Z$ are Pauli ...
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In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?
My question:
Is the center of $ \overline{{\rm Cl}_n} $ trivial?
Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group ...
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Commutation of $XX$ and $ZZ$ operators
It is known that the Pauli operators $XX$ and $ZZ$ commute. Consider the state $\vert{++}\rangle$ which is an eigenstate of $XX$. But we also know that
$$ZZ\vert{++}\rangle = \vert{--}\rangle$$
so ...
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2
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How does a single-qubit gate affect other qubits?
An instructional quantum computing article I'm reading (How the quantum search algorithm works) states that the following circuit takes $\vert x\rangle\vert 0\rangle$ to $−\vert x\rangle\vert 0\rangle$...
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2
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Expressing CNOT in the eigenbasis of $X$ (Preskill lecture notes eq. 7.6)
In chapter 7, equation 7.6 says CNOT works as follows:
CNOT: $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle )\otimes |x\rangle \rightarrow
\frac{1}{\sqrt{2}} (|0\rangle + (-1)^x |1\rangle ) \otimes |x\...
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Proof that $\frac{1}{d} | \text{Tr}(\mathcal{P}_i^\dagger \mathcal{E}(\mathcal{P}_j))| \leq 1$ for superoperators
I have two Pauli operators $\frac{1}{\sqrt{d}} \mathcal{P}_i$, $\frac{1}{\sqrt{d}} \mathcal{P}_j$, and an arbitrary quantum channel $\mathcal{E}$ (in the superoperator/Liouville representation) all ...
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Question Regarding Simulating Hamiltonian With Quantum Circuit
There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/...
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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 ...
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Why can any density operator be written this way? (quantum tomography)
From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product:
\begin{aligned}
\rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
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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=\...
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Showing that $e^{i \sigma_z \otimes \sigma_z t} = \text{CNOT}(I \otimes e^{i \sigma_zt})\text{CNOT}$
While working on circuit construction for Hamiltonian simulation using this answer as reference, I'm unable to see how the following equation is true:
$$
e^{i \sigma_z \otimes \sigma_z t} = \text{CNOT}...
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How does the stated Pauli decomposition for $\operatorname{CP\cdot A\cdot CP}$ arise?
I'm having a bit of trouble understand @DaftWullie's answer here.
I understood that the $4\times 4$ matrix $A$
$$ \frac{1}{4} \left[\begin{matrix}
15 & 9 & 5 & -3 \\
9 & 15 & 3 &...
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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.
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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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Cannot interpret transformations on the bloch sphere as matrix multiplications
I understand that X,Y and Z gates are rotations around the axes with the respective letters, but I cannot understand how can Y gate multiply the amplitude of 0 with unreal number and have it landing ...
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