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8 votes
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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 ...
Adam Zalcman's user avatar
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6 votes
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Proof that for an $[\![n,k,d]\!]$ code we have $N(S)/S\simeq{\cal G}_k$ with $S$ stabilizer

TL;DR: This can be seen by decoding the $k$ logical qubits out of the code subspace into the first $k$ physical qubits while keeping track of the stabilizer group. Code subspace Consider an $[\![n,k]\!...
Adam Zalcman's user avatar
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4 votes
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Moments of Pauli coefficients of Haar-random states

Let us compute the value for $\alpha=4$, averaged over Haar-random states. We have the following identity: $$ \sum_{P\in\mathcal{P}_n} \mathrm{tr}(\rho P)^4= \sum_{P\in\mathcal{P}_n} \mathrm{tr}(\rho^...
Markus Heinrich's user avatar
4 votes
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Under what conditions are two sets of Pauli operators Clifford-equivalent?

What I would do is to extend S and T into full Clifford tableaus $C_S$ and $C_T$ by discarding linearly dependent products and filling in generators for the unspanned $n$-qubit space. The Clifford $...
Craig Gidney's user avatar
  • 37.8k
3 votes
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In context of stabilizer codes, are logical gates and Pauli operators the same?

Let $\mathcal{S}$ be the set of stabilizers. It is not true that $N(\mathcal{S}) = C(\mathcal{S})$. There can exist an operator $U$ such that for all $S\in \mathcal{S}$, $USU^\dagger = S'$ for some $S'...
rnva's user avatar
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3 votes
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What are the elements of quotienting the Pauli group $\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$, and how to do calculations with it?

It seems like the answer you are looking for may be found at the beginning of section 2 of this reference: https://arxiv.org/pdf/1909.08123.pdf. Succinctly, what you call the $n$-qubit Pauli group $\...
fcrp's user avatar
  • 156
3 votes

Simulating stabilizer groups

Stim doesn't have native functionality for tracking partial knowledge of the state. Stim does have generic tools for working with stabilizers, that you could use to make it easier to build this ...
Craig Gidney's user avatar
  • 37.8k
3 votes
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What does Pauli's $Y$ matrix represent?

$Y = iXZ = -iZX$, so it can be thought of as both a bit flip and a phase flip, plus an overall $i$ phase.
Abdullah Khalid's user avatar
3 votes

Minimum-weight presentation for stabilizer group $S$ and logical Pauli group $N(S)/S$

The standard form of the generator matrix is $$ H = \left(\begin{array}{ccc|ccc} I & A_1 & A_2 & B & 0 & C_2 \\ 0 & 0 & 0 & D & I & E_2 \end{array}\right)....
Abdullah Khalid's user avatar
2 votes
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A question on the structure of the Clifford group

Yes, if a Clifford operation $C$ conjugates every Pauli product $P$ into $\pm P$, then $C$ is a Pauli product. First, show that $PAP^\dagger = \pm A$ for any Pauli products $A$ and $P$. In other words,...
Craig Gidney's user avatar
  • 37.8k
2 votes

Simulating stabilizer groups

The QuantumClifford.jl has pretty great support for mixed stabilizer states and a lot of the algebra related to them (various decompositions, canonicalizations, ...
Krastanov's user avatar
  • 181
2 votes

Simulating stabilizer groups

PyClifford is a python based clifford circuit simulation package which not only offers the fast simulation but also supports analytical level manipulation of pauli operators and stabilizer states. ...
Abdullah Khalid's user avatar
2 votes
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How to find the order of the error group $E$

Note that $2^{2n+2} = 4^{n+1}$. Now it is easy to solve the combinatorial problem. There are $n$ elements in the tensor product. Each element can be one of four operators $I, X, Y, Z$. That gives us $...
Abdullah Khalid's user avatar
2 votes
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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 ...
Danylo Y's user avatar
  • 7,342
2 votes
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Shortcutting Clifford circuit computations using relations between stabilizers

Circuits with measurements still have "stabilizer flows" $A \rightarrow B$ where $A$ and $B$ are Pauli products. You can interpret $A \rightarrow B$ as saying "if you knew $\langle A \...
Craig Gidney's user avatar
  • 37.8k
1 vote

What unitary commutes with all local Pauli operators?

If $P$ commutes with $U$, that means $U$ conjugates $P$ into $P$. $$ \begin{aligned} &([P, U] = 0) \\\equiv& (P U = U P) \\\equiv& (U^\dagger P U = P) \end{aligned}$$ In the stabilizer ...
Craig Gidney's user avatar
  • 37.8k
1 vote

Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$

Perhaps my understanding is wrong, I had previously thought that $X(a)$ was the Pauli gate $X$ acting on the vector $a$ and $Z(b)$ was the Pauli gate $Z$ acting on the vector $b$. Your understanding ...
Peter-Jan's user avatar
  • 1,549
1 vote
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Does the fact that the elements of the normalizer group commute with elements of the stabilizer group imply that the normalizer is abelian?

would this not instead imply that S′ is the abelian group No. It does not. We have the following definitions. (1) Every element of $S'$ commutes with every element of $S$. (2) $S$ is a subgroup of $S'...
Abdullah Khalid's user avatar
1 vote
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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 ...
forky40's user avatar
  • 7,113
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 ...
Danylo Y's user avatar
  • 7,342
1 vote
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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 ...
Abdullah Khalid's user avatar
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 ...
unknown's user avatar
  • 2,197
1 vote
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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 ...
Steve Wood's user avatar
  • 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, \...
Alejandro G's user avatar

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