Pauli Subgroup Computations

[dsvandet]

This set of requirements covers the various group computations for subgroups of the $n$-qubit Pauli Groups $\mathcal{P}_n$.

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Problem: Given a subgroup $A$ of $\mathcal{P}_n$ find a minimal generating set for $A$.

Note: As $\mathcal{P}_n$ is a $p$-group it follows by Burnside's basis theorem that all minimal
generating sets (by including) have the samecardinality, hence there is not ambiguity in the question.

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Problem: Given a subgroup $A$ of $\mathcal{P}_n$ find a hyperbolic generating set for $A$.
An optional argument should return a normalized version.

A set $\mathcal{Q}=\{A_1,A_2,...,A_k,A_{k+1},B_{k+1},...,A_{k+t},B_{k+t}\}\subseteq\mathcal{P}_n$ is a hyperbolic generating
set if

• $\mathcal{Q}$ is a minimal generating set for $\langle \mathcal{Q}\rangle$
• $Z(\mathcal{Q} )=\langle A_1,A_2,...,A_k \rangle$
• $A_i$ commutes with all generators except for $B_i$.
• $B_i$ commutes with all generators except for $A_i$.

Thus the standard generating set $\{iI,X_1,Z_1,X_2,Z_2,...,X_n,Z_n\}$ is a hyperbolic generating set for
$\mathcal{P}_n$.

A hyperbolic generating set $\mathcal{Q}=\{A_1,A_2,...,A_k,A_{k+1},B_{k+1},...,A_{k+t},B_{k+t}\}\subseteq\mathcal{P}_n$
is said to be normal if all generators have order $2$ except possibly for one element of the center.

A hyperbolic generating set can be found using a modified Gram-Schmidt process. See for example
-T. Brun, Min-Hsiu Hsieh https://arxiv.org/abs/1610.04013

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Problem: Let $A$ be a subgroup of $\mathcal{P}_n$ with hyperbolic generating set $\mathcal{Q}=\{A_1,A_2,...,A_k,A_{k+1},B_{k+1},...,A_{k+t},B_{k+t}\}$. Extend $\mathcal{Q}$ to a hyperbolic generating set
for $\mathcal{P}_n$

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Problem: Given a subset $X$ of $\mathcal{P}_n$ find its Center, Normalizer, Centralizer, relative to $\mathcal{P}_n$.

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Problem: Given a Gauge group find a minimal generating set for its bare logical operators

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Problem: Let $A$ be a subgroup of $\mathcal{P}_n$. 1) Find all maximal subgroups of $A$. 2) Find all maximal gauge groups of $A$, find all maximal stabilizer subgroups of $A$. Construct generating sets (hyperbolic, or normal hyperbolic is required) for all maximal subgroups found.

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Problem: Let $A\in\mathcal{P}_n$ and $\mathcal{G}$ a Pauli subgroup. Determine if $A$ is in $G$. (IsElementOf)