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Results tagged with pauli-gates
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user 14239
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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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Paul...
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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In what sense are Pauli matrices measurement operators?
Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle …
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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): …
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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 pa …
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sig...
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 $\sigma$ is $\mathrm …
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Proof for Cardinality of the Clifford Group
A Clifford $C_n$, defined by how it maps each of $X_i$ and $Z_i$ for $1 \leq i \leq n$, via the functions $g_i(\sigma_i)$ where $$\sigma_i = \{\pm I_i, \pm X_i, \pm Y_i, \pm Z_i\},$$ can be seen as th …
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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 u …
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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{1}{\sqrt{2}} …
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Proof for Cardinality of the Clifford Group
In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how t …
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