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Accepted

### What unitary commutes with all local Pauli operators?

TL;DR: The only $U$ that commutes with all $\sigma_{X,i}$ and all $\sigma_{Z,i}$ is a scalar multiple of identity. This follows from the Schur's lemma, but can also be shown using elementary linear ...
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• 1,818
Accepted

### What do the cosets of the group $E/Z(E)$ look like? (E is the quantum error group and Z(E) is the centre of E)

The elements of the n-fold Pauli group are either commute or anti-commute. In the quotient over the center they will all commute, since $-1$ wouldn't matter. Note that a general element of the Pauli ...
• 7,342
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• 1,818
1 vote
Accepted

### When is a block diagonal matrix a tensor product of Pauli matrices?

Assume $U$ equals a Pauli matrix and has the form $|0\rangle \langle 0| \otimes U_1 + |1 \rangle \langle 1| \otimes U_2$. The set of Pauli matrices $\mathcal{P}_N := \{I, X, Y, Z\}^{\otimes N}$ is an ...
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1 vote

### Efficient way to calculate trace of product of Pauli string and matrix?

Any Pauli string has exactly one non-zero element in each row and column. Moreover, if you switch between $I$ and $Z$, and between $X$ and $Y$ in a Pauli string then the pattern of zeros will be the ...
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1 vote
Accepted

### Efficiently finding an explicit presentation for $N(S)/S$, for any stabilizer group $S$

What you are looking to construct are the logical operators of the code. These operators are the representatives of each coset in $N(S)/S$. There exists a process to do so, which was presented in ...
• 1,818
1 vote

### Efficiently finding an explicit presentation for $N(S)/S$, for any stabilizer group $S$

Yes there is. You put the code in "standard" form. See Chapter 9 of https://www.amazon.com/Quantum-Information-Processing-Error-Correction/dp/0123854911 The process uses gaussian elimination ...
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1 vote
Accepted

### How grouping of Pauli strings is handled in Qiskit when running VQE?

The Estimator primitive may group the operator paulis internally when computing the observable value of the operator and some offer control over that: There are ...
• 1,488
1 vote

### Tensor product of Pauli strings?

It doesn't hold. You can consider the simple example where $l=1$ and $m=1$ as a counter-example. $$\sum_i P_i \otimes P_i = II + XX + YY + ZZ$$ $$\sum_i Q_i \otimes Q_i = II + XX + YY + ZZ$$ Then, \...

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