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Quantum Computing Stack Exchange

Questions tagged [pauli-group]

Questions about or related to the Pauli group.

The Pauli group on $n$ qubits can be defined as $$ G_n = P_n \equiv \{ \pm1 ,\pm i \} \times \{ \sigma_1^{\alpha_1} \otimes \cdots \otimes \sigma_n^{\alpha_n} \}_{\alpha_j \in \{0,x,y,z \}}\,.$$

https://en.wikipedia.org/wiki/Pauli_group

43 questions
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Is the Pauli norm of an observable minimized in its eigenbasis?

Consider an observable $O = \sum_i \lambda_i P_i$ decomposed into Paulistrings $P_i$ and a unitary $U$ each acting on $n$ qubits. The Pauli-norm of $O$ is defined as the 1-norm of the Pauli vector, i.e. $$\|O\|_P = \sum_i |\lambda_i|.$$ I can write…
Refik Mansuroglu
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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 does not have any Pauli $Z$'s in its expansion. Is…
mavzolej
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Minimum-weight presentation for stabilizer group $S$ and logical Pauli group $N(S)/S$

Given some stabilizer group $S$ with presentation $\langle s_1, \dots, s_r \rangle$, what is known about finding a minimal-weight presentation for it? By this, I mean a new presentation $\langle s_1', \dots, s_r' \rangle$ such that $\sum_{j=1}^{r}…
Alex Townsend-Teague
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How many elements are present in $N(S)$ where $S$ is a stabilizer group?

Question Suppose I have $n$ qubits and $n-k$ stabilizer generators. Let the set of stabilizers be $S$. I then have $k$ logical qubits. Define $N(S)$ to be $$N(S) = \{p \in P_n\ |\ sp = ps\ \forall s\in S\},$$ where $P_n$ is an $n$-qubit Pauli. How…
guest01
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What unitary commutes with all local Pauli operators?

I was thinking about this problem of identifying a set of unitary operations (other than the identity operation) that commute with local pauli $\sigma_X$ and $\sigma_Z$ matrices, i.e. find $U$ such that $[U, \sigma_{X, i}] = [U, \sigma_{Z, i}] = 0 ~…
Mohan
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Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is well known that when $\alpha=2$, the quantity…
Feng Pan
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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 in polynomial time (or better) the…
Physics Penguin
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Simulating stabilizer groups

Can any existing software be used (either directly or with a bit of persuading) to work with general stabilizer groups? From what I can see, tableau-based options like Stim and Qiskit can be used to work with stabilizer groups over $n$ qubits with…
Alex Townsend-Teague
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The weight distribution of uniformly random Clifford conjugation

For an $n$ qubit system, I fix a non-identity Pauli $P$ and perform the following experiment $N$ times: Sample a Clifford gate $U_i$ uniformly at random from the Clifford group (iid). Compute $w_i$, the weight of $U_i^\dagger P U_i$ Is there an…
forky40
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Proof that for an $[\![n,k,d]\!]$ code we have $N(S)/S\simeq{\cal G}_k$ with $S$ stabilizer

I am reading about Quantum Error Correction and more specifically about the stabilizer formalism. Nielsen's textbook introduces the selection of logical Pauli as a kind of "ad hoc" process which, although intuitive, I think lacks rigor. Later, they…
George Giapitzakis
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Clifford gates constructed from CNOT, H and S gates

Trying to prove that all Clifford gates can be constructed with CNOT, H and S gates, I'm following the classical path by induction (Nielsen and Chuang, Quantum Computation and Quantum Information -- Exercice 10.40). Assuming that $U (Z_1 \otimes…
JMark
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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?

Let $\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$ together with all the phases $\{\pm 1, \pm i\}$ the regular Pauli group, and $N = \langle \pm i I\rangle $. I would like to understand the following group-theoretic…
R.W
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"Twisted" traces of Pauli strings

Consider a system of $L$ qubits. A generic string of Pauli operators acting on such a system can be written as (neglecting phases for simplicity) $$\mathcal{O}(\vec{v},\vec{w}) = \bigotimes_{j=1}^{L}Z^{v_j}X^{w_j}$$ where $\vec{v}, \vec{w}$ are…
miggle
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Why are there equally many commuting and non-commuting Pauli operators with respect to a given Pauli operator?

I'm working with $\mathbf{P}_n$, the set of all $n$-fold tensor products of Pauli matrices, including the identity. For a given Pauli operator $Q \in \mathbf{P}_n$ (excluding the identity), I want to understand why there are an equal number of Pauli…
tare_
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Proof for how logical operators generated systematically will satisfy Pauli commutation

Let me assume a CSS code. Suppose I have $n$ qubits and have $(n-k)$ stabilizers which we label by the set $S = \{S_1, S_2, .. S_{n-k}\}$. Let me then find $2k$ Pauli operators $L_i$ that are Not products of the stabilizers Not products of each…
JRT
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