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}}.
254 questions
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Coupling map in QISKIT transpile
I have a 3-qubit unitary represented by a circuit with the following dictionary: {'cx': 30, 'h': 22, 'rz': 15, 's': 4, 'sdg': 4}. I want to use this circuit on IBM ...
CommunityBot
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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 ...
CommunityBot
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Pauli Twirling not increasing Circuit depth?
This paper talks about the properties of Pauli twirling a circuit. Pauli twirling is a technique that converts arbitrary quantum noise into Pauli errors by applying random Pauli gates before and after ...
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How can I verify that the Pauli group is a group? And is it abelian? [duplicate]
So how can I verify that the Pauli group is a group? Furthermore, is it Abelian? And then to sum it up, what is the order of the group? Trying to do some research into the Pauli group but I can't find ...
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How to find the normalization factor of the eigenvectors of the $\sigma_x$ Pauli gate?
I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes ...
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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)
On page 26 of this paper (arXiv), it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
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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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Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
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Implementing an arbitrary complex valued single qubit unitary gate using a given matrix
I’m implementing an arbitrary single-qubit gate in Q# , using pauli decompositon. The goal is to match the behavior of a specified unitary matrix acting on a single qubit. Here’s the code I’ve written ...
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Deriving the number operator in a Cooper Pair Box (CPB)
The details of my question follows closely from this source (pg 10 - 12): https://clerkgroup.uchicago.edu/PDFfiles/0210247.pdf
The standard CPB hamiltonian in charge basis is written as
$$
H = 4E_{c}\...
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What class of matrices have efficient decompositions?
Suppose we have an arbitrary matrix $A \in \mathbb{C}^{N \times N}$ where $N=2^n$. Now suppose that we can decompose $A$ into tensor products of the Pauli basis given by $A=\sum_{l=0}^Lc_lS_l$ where $...
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Commutation relationship and measurement results
There are things I do not understand about the following circuit, and I would appreciate it if you could explain.
...
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What's the earliest reference noting that a NOT gate is the same as a Pauli X operator?
Coming from an electrical/computer engineering background, I knew what a NOT gate is well before I learned about the Pauli matrices. But the NOT gate is operationally the same as a Pauli X matrix, ...
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Trying to prove Theorem 4.1 from Nielsen and Chuang algebraically
Background
Theorem 4.1 of Nielsen and Chuang (10th Anniversary Edition) states how a universal single-qubit unitary can be constructed from Y and Z rotations.
Suppose $U$ is a unitary operation on a ...
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Clarification on the Pauli Stabilizer Condition in Aaronson & Gottesman’s Efficient Simulation Paper
In Aaronson's & Gottesman's paper about the efficient simulation of a stabilizer circuit (arXiv),
I have a problem with finding the reason why the following statement holds.
Restating the ...
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Is decomposing high-dimensional states in terms of Pauli matrices impossible?
I've been trying to decompose a $3\times3$ density matrix with 3-dimensional Pauli matrices but it doesn't work for all matrices.
For example, the density matrix of the state $|0\rangle + |1\rangle + |...
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Forward propogation of rotated clifford gates in lattice surgery
In A Game of Surface Codes (arXiv), I have been trying to understand the process that is described for forward propogating rotated clifford gates through other rotated clifford gates, and either I am ...
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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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CX and CZ commutation
Suppose I have control qubits $1$ and $2$ and target qubit $3$. I have the circuit element
$$E = CX_{1-> 3}CZ_{2-> 3}$$
I would like to swap the order and have
$$E' = CZ_{2-> 3}CX_{1-> 3}$$...
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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
$$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?
I see here in Olivia DeMatteo's notes, she states:
When we consider the action of the entire Clifford group on a single non-identity Pauli, it
maps that Pauli to each of the $d^2 − 1$ other possible ...
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If $g \in S$, does this necessitate that $g^{\dagger} \in S$?
In Lecture 5 of Aram Harrow's lecture series "Quantum Information Science II", he sets about proving that $\Pi_{S} = \frac{1}{|S|} \sum_{g \in S} g$ can be written as $\Pi_{S}= \prod_{i=1}^{...
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How to express $n$-qubit Hermitian operator with Pauli matrices
How can we prove that all $n$-qubit Hermitian matrices can be written in terms of Pauli matrices $I$, $X$, $Y$, and $Z$ as
$$
\sum_{W_k \in \{I, X, Y, Z\}} a_{W_1,\dots,W_n}W_{1}\otimes ... \otimes W_{...
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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
$$
X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n}
$...
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In the Clifford group, is the center of $ \overline{\text{Cl}_n} \equiv\text{Cl}_n/U(1)$ trivial?
My question:
Is the center of $ \overline{{\rm Cl}_n} $ trivial?
Recall that the algebra generated by the Pauli group is the full matrix algebra. So any matrix that commutes with the Pauli group ...
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Why no $Z$'s in the $\operatorname{F} (\sum_{j=0}^{n-1} 2^j Z_j) \operatorname{F}^\dagger$ operator?
An interesting numerical observation is that an operator defined as $\phi=\sum_{j=0}^{n-1} 2^j Z_j$ upon a QFT is rotated into an operator $\pi=\operatorname{F} \phi \operatorname{F}^\dagger$ which ...
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Coding a hamiltonian in qiskit
I have a hamiltonian of the form:
$H=\sum_{i=1}^N Z_i Z_{i+1}-Z_NZ_1$
And another one as:
$H=-\sum_{i=1}^N X_i$
I need it to it for N terms.
I am a bit lost can anybody help. I tried looking for ...
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Is the Pauli group isomorphic to the Heisenberg group over a finite field?
Let $ p $ be prime and let $ P_n(p) $ denote the Pauli group on $ n $ qudits each of size $ p $. Then $ P_n(p) $ and $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ are both extraspecial $ p $ groups of order $ ...
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What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?
I'm looking for an algorithm to write an arbitrarily sized tridiagonal matrix as a linear combination of Pauli matrices. The tridiagonal matrix has the form, for example,
\begin{pmatrix}
2 & -1 &...
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Spectral theorem for Pauli matrices
Let $ P $ be a Pauli matrix, hence normal. So by the spectral theorem $ P $ can be written as
$$
P=VDV^{-1}
$$
for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily diagonalizable). ...
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How many $ \sqrt{X} $ are there?
I was reading this post and it got me thinking about square roots.
Recall the Pauli $ X $ gate
$$
X=\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix};
$$
How many unitary matrices $ U $ are there ...
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How is the definition of $n$-qubit Pauli group derived?
The authors give the following definition for the Pauli group in the paper Averaged circuit eigenvalue sampling.
The n-qubit Pauli group $P_n$ consists of n-fold tensor products of single-qubit Pauli ...
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Eigenvalues of Pauli Gate and connection to measurement
Suppose I measure a qubit in the $Z$ basis. If I measure and obtain the outcome $+1$, I get the post-measurement state $\vert 0\rangle\langle 0\vert$ and if I measure and obtain the outcome $-1$, I ...
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How does one identify Pauli logicals in a hyperbolic surface code?
I understand how the Pauli X and Z logicals on the planar surface code work - they are stringlike operators between either the rough boundaries (in case of the Z logical) or between smooth boundaries (...
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What is the connection between an observable and a gate?
I am reading some introductory quantum mechanics and I don't understand the connection between an observable and a gate. I thought a gate just applies a rotation to a state while a measurement gives ...
glS♦
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How can I measure qutrits in the X basis using cirq?
I attempted to create a custom measurement class which, in my case, allows us to go from the z basis to x basis using a hadamard gate transformation, and then we measure wrt that new basis. However, ...
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Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?
As stated in the question, I have seen this claim made that a pure state can be written as an eigenstate of $aX + bY + cZ$ for some $a,b,c$ where $X,Y,Z$ are Pauli matrices. Why is this true and what ...
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QuTiP tutorial: How to derive the analytical solution to the expectation value of an operator for a system evolving by Lindbladian
I am following the simple tutorial below:
(https://nbviewer.org/urls/qutip.org/qutip-tutorials/tutorials-v5/time-evolution/003_qubit-dynamics.ipynb)
In this they look at single qubit with Hamiltonian $...
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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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Changing the Basis
I am attempting to use a VQE algorithm to find the ground state of a deuterium nucleus by applying a constructed hamiltonian to an ansatz state with one parameter created by a circuit. While I am ...
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Dimension reduction to subsystem in Pauli-Liouville basis: How to implement partial trace of Pauli-Transfer Matrices?
I have a complementary question to my previous (thankfully answered, but I can't verify for larger systems) question:
Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-...
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Calculating number of CNOT gates in Pauli evolution gate
How to calculate the number of CNOT gates for a Pauli exponentiation for given time?
I am performing Trotterization which involves performing Pauli evolution ...
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Moving pauli product rotations past measurements
I'm trying to understand how the clifford + T compiler works in "A Game of Surface codes".
How do I move a pauli product rotation block past a pauli product measurement block? More ...
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Why is the Pauli Y gate eigenstate so hard to create?
In a lot of quantum computing formalism, it is relatively easy to create $\vert 0\rangle$, $\vert 1\rangle$, $\vert +\rangle$ and $\vert -\rangle$. However, it is hard to create $\vert i\rangle$. Why ...
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How to modify the quantum circuit to do superdense coding with the state $|00\rangle-|11\rangle$?
Suppose I want to perform the superdense coding protocol, but instead of using the state $\beta_{00}=\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$, I have to initialize it with the state $\beta_{10}=\frac{...
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Efficient way to calculate trace of product of Pauli string and matrix?
Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
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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$$
so ...
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2
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Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?
Basically the title. If I have a $2^N\times 2^N$ Hamiltonian $H$ of random numbers (we can take the Hamiltonian as normalized if we want) and $N$ is an integer, is there an efficient way of writing
$$
...
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In context of stabilizer codes, are logical gates and Pauli operators the same?
I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
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1
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Connection between a Pauli measurement and the corresponding Pauli gate?
Suppose I have a qubit and the ability to act a Pauli $Z$ gate on it. This is a black box that does the phase flip and I don't know how it works on the inside. Can I use this black box to implement a ...
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