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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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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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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,...
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What does it mean that a qubit is a triple $(H,X,Z)$ with $H$ Hilbert space and $X,Z$ Pauli operators?

In this paper, http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it gives the definition of a qubit as follows: A qubit is a triple $(H, X, Z)$ consisting of a separable Hilbert space H and ...
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How does Spin Measurement correspond to quantum NOT gate?

Newbie in quantum computing (and stack overflow) here. I am confused regarding the relation between spin measurement in quantum mechanics and the quantum NOT gate. I have a Bloch sphere picture of a ...
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Trying to prove Theorem 4.1 from Neilsen and Chuang algebraically

Background Theorem 4.1 of Neilsen 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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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 ...
consthatza's user avatar
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Interested in software helping with projecting multi-qubit states onto irreducible components

My interest in QC comes from a problem in geometry called the Atiyah problem on configurations of points. In short, there is a nice one-to-one correspondence between quantum states of a single qubit ...
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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 ...
R.G.J's user avatar
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Is decomposing high-dimensional states in terms of Pauli matrices impossible?

I've been trying to decompose a 3x3 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 + |2\...
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Measurement on a specific basis and proof of circuit output

I am trying to understand a proof from Practical optimization for hybrid quantum-classical algorithms. In particular, I need clarifications on how do you perform the measurement on a different basis ...
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Understanding the EPR argument with a simple description using Pauli matrices

Can someone explain the EPR argument with a simple description using Pauli matrices? Two non-commuting physical quantity are being discussed philosophically whether there is an element of reality ...
Eara Shahirah's user avatar
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Measuring vs gate - what is the relation?

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 ...
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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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Action of below circuit using heisenberg representation

Can someone please explain how the above gate affects logical operators? My understanding is that the circle indicates that we are measuring the second qubit? My initial guess is that it is equivalent ...
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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-...
Onur Danaci's user avatar
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scaling of error of sum of Pauli strings with number of shots

I have a question which I suppose is quite basic. Let's say I want to measure the average of an obersvable which is the sum of non-commuting Pauli strings on $N_q$ qubits: $$ \langle O\rangle =\sum_i^{...
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Measuring a single-qubit PauliZ using Qiskit's EstimatorQNN

I am currently working with the EstimatorQNN from Qiskit to construct a Quantum Neural Network using a custom Parametrized Quantum Circuit. But I want to change the ...
yeray142's user avatar
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How is Pauli twirling so powerful?

So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to: $\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$ where $\sigma^0 = \mathbb{I}, \...
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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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Correctability of X, Y, and Z Errors in Quantum Surface Codes and Color Codes

In surface codes and color codes, when the code distance is $d$, you can correct up to $[(d-1)/2]$ Pauli errors. I would like to know what this $[(d-1)/2]$ Pauli errors means for $X$, $Y$, and $Z$. ...
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Conjugating pairs of Paulis to each other with a non-entangling Clifford

This a follow-up question to Conjugating pairs of Paulis to each other with a Clifford We call a Clifford gate local if it is a tensor product of single qubit Clifford gates. We call a Clifford gate ...
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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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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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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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The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)

On page 26 of arXiv:quant-ph/0504218, 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 \...
kong's user avatar
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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. ...
david's user avatar
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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 ...
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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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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 ...
Ron Cohen's user avatar
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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 ...
Tareq Hamarneh's user avatar