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As quantum devices make steady progress towards intermediate scale and fault-tolerant quantum computing, it is essential to develop rigorous and efficient measurement protocols that account for known sources of noise. Most existing quantum characterization protocols such as gate set tomography and randomized benchmarking assume the noise acting on the qubits is Markovian. However, this assumption is often not valid, as for the case of 1/f charge noise or hyperfine nuclear spin noise. Here, we present a general framework for quantum process tomography (QPT) in the presence of time-correlated noise. We further introduce fidelity benchmarks that quantify the relative strength of different sources of Markovian and non-Markovian noise. As an application of our method, we perform a comparative theoretical and experimental analysis of silicon spin qubits. We first develop a detailed noise model that accounts for the dominant sources of noise and validate the model against experimental data. Applying our framework for time-correlated QPT, we find that the number of independent parameters needed to characterize one and two-qubit gates can be compressed by 10x and 100x, respectively, when compared to the fully generic case. These compressions reduce the amount of tomographic measurements needed in experiment, while also significantly speeding up numerical simulations of noisy quantum circuit dynamics compared to time-dependent Hamiltonian simulation. Using this compressed noise model, we find good agreement between our theoretically predicted process fidelities and two qubit interleaved randomized benchmarking fidelities of 99.8% measured in recent experiments on silicon spin qubits. More broadly, our formalism can be directly extended to develop efficient and scalable tuning protocols for high-fidelity control of large-arrays of quantum devices with non-Markovian noise., Comment: 11 pages, 3 figures, comments welcome

We investigate the occurrence of elements of order $p$ in the upper central series of a finite $p$-group., Comment: 11 pages - In this version we have added some indecomposable examples

Letourmy and Vendramin have recently introduced a concept of isoclinism for skew braces. We show that for a skew brace the properties of being bi-skew, $\lambda$-homomorphic, and inner are invariant under isoclinism., Comment: 7 pages

Let $p, q$ be distinct primes, with $p > 2$. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order $p^2q$, for which the Sylow $p$-subgroups of the multiplicative group is cyclic. In this paper we complete the classification by dealing with the case when the Sylow $p$-subgroups of the Galois group are elementary abelian. According to Greither and Pareigis, and Byott, we will do this by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementary abelian. We rely on the use of certain gamma functions $\gamma:G\to \operatorname{Aut}(G)$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual., Comment: 95 pages

We consider the quotient group $T(G)$ of the multiple holomorph by the holomorph of a finite $p$-group $G$ of class two for an odd prime $p$. By work of the first-named author, we know that $T(G)$ contains a cyclic subgroup of order $p^{r-1}(p-1)$, where $p^r$ is the exponent of the quotient of $G$ by its center. In this paper, we shall exhibit examples of $G$ (with $r = 1$) such that $T(G)$ has order exactly $p-1$, which is as small as possible., Comment: Fixed some typos and a small mistake in the introduction

Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd prime $p$, in which case $T(G)$ may be studied using certain bilinear forms. For any $n\geq 4$, we exhibit examples of $G$ of order $p^{n+{n\choose 2}}$ such that $T(G)$ contains a subgroup isomorphic to \begin{equation*} \operatorname{GL}_n(\mathbb{F}_p) \times \operatorname{GL}_{\binom{n}{2}-n}(\mathbb{F}_p).\end{equation*} For finite $p$-groups $G$, the prime factors of the order of $T(G)$ which are known so far all came from $p(p-1)$. Our examples show that the order of $T(G)$ can have other prime factors as well. In fact, we can embed any finite group into $T(G)$ for a suitable choice of $G$., Comment: 20 pages, added Definition 2.2 and a few small edits

Rota-Baxter operators for groups were recently introduced by L. Guo, H. Lang, and Y. Sheng. V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one can associate a skew brace. Skew braces on a group $G$ can be characterised in terms of certain gamma functions from $G$ to its automorphism group $\operatorname{Aut}(G)$, that are defined by a functional equation. For the skew braces obtained from a Rota-Baxter operator the corresponding gamma functions take values in the inner automorphism group $\operatorname{Inn}(G)$ of $G$. In this paper, we give a characterisation of the gamma functions on a group $G$, with values in $\operatorname{Inn}(G)$, that come from a Rota-Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota--Baxter operator. For gamma functions that can be obtained from a Rota-Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits., Comment: 14 pages; accepted for publication, Annali di Matematica Pura e Applicata

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families of group operations on a set $G$ such that any two of them induce a skew brace structure on $G$. We construct these operations by using bilinear maps and liftings of endomorphisms of quotient groups with respect to a central subgroup. We provide several examples of the construction, showing that there are brace blocks which consist of distinct operations of any given cardinality. One of the examples we give yields an answer to a question of Cornelius Greither. This example exhibits a sequence of distinct operations on the $p$-adic Heisenberg group $(G, \cdot)$ such that any two operations give a skew brace structure on $G$ and the sequence of operations converges to the original operation "$\cdot$"., Comment: 19 pages

In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and multiplicative group. This means that in this case the isomorphism type of $(G,\circ)$ determines the isomorphism type of $(G,\cdot)$. In this paper we complement Bachiller's result in two directions. In Theorem 2.2 we prove that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$, then $(G,\cdot)$ has small rank (i.e. $< p-1$) if and only if $(G,\circ)$ has small rank. We also provide examples of groups of rank $p-1$ in which elements of arbitrarily large order in the additive group become of prime order in the multiplicative group. When the rank is larger, orders may increase.

The interplay between set-theoretic solutions of the Yang--Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism of a group $(G, \cdot)$ whose image is abelian. In this paper, we characterise the endomorphisms of a group $(G, \cdot)$ for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang--Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf--Galois structures., Comment: v1 of this manuscript has been split in two. The first part has appeared in J. Algebra 587 (2021), 462-487, and carries a slightly revised title - it is v2 here (19 pages). The second part is now arXiv:2110.11028

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie algebras as $F$-subalgebras of Lie algebras $M$ of maximal class over $E$. We characterise the thin Lie $F$-subalgebras of $M$ generated in degree $1$. Moreover we show that every thin Lie algebra $L$ whose ring of graded endomorphisms of degree zero of $L^3$ is a quadratic extension of $F$ can be obtained in this Lie algebra of maximal class over $E$ which are ideally $r$-constrained for a positive integer $r$., Comment: 10 pages

L. Childs has defined a skew brace $(G, \cdot, \circ)$ to be a bi-skew brace if $(G, \circ, \cdot)$ is also a skew brace, and has given applications of this concept to the equivalent theory of Hopf-Galois structures. The goal of this paper is to deal with bi-skew braces $(G, \cdot, \circ)$ from the yet equivalent point of view of regular subgroups of the holomorph of $(G, \cdot)$. In particular, we find that certain groups studied by T. Kohl, F. Dalla Volta and the author, and C. Tsang all yield examples of bi-skew braces. Building on a construction of Childs, we also give various methods for constructing further examples of bi-skew braces., Comment: 16 pages - accepted for publication in the Journal of Algebra

$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $\Gamma$ are groups of order $p^{2} q$ with non-isomorphic Sylow $p$-subgroups, then there are no regular subgroups of the holomorph of $G$ which are isomorphic to $\Gamma$. Equivalently, a Galois extension with Galois group $\Gamma$ has no Hopf-Galois structures of type $G$. Our method relies on the alternate brace operation $\circ$ on $G$, which we use mainly indirectly, that is, in terms of the functions $\gamma : G \to \Aut(G)$ defined by $g \mapsto (x \mapsto (x \circ g) \cdot g^{-1})$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual., Comment: 43 pages

We record for reference a detailed description of the automorphism groups of the groups of order $p^{2} q$, where $p$ and $q$ are distinct primes., Comment: 9 pages

S. S. Magliveras et al. have described symmetric and public key cryptosystems based on logarithmic signatures (also known as group bases) for finite permutation groups. In this paper we show that if $G$ is a nontrivial finite group which is not cyclic of order a prime, or the square of a prime, then the round (or encryption) functions of these systems, that are the permutations of $G$ induced by the exact-transversal logarithmic signatures (also known as transversal group bases), generate the full symmetric group on $G$. This answers a question of S. S. Magliveras, D.R. Stinson and Tran van Trung., Comment: This is an old paper, published in 2006

$\DeclareMathOperator{\Hol}{Hol}$$\DeclareMathOperator{\Aut}{Aut}$$\newcommand{\Gp}[0]{\mathcal{G}(p)}$$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$Let $G$ be a group, and $S(G)$ be the group of permutations on the set $G$. The (abstract) holomorph of $G$ is the natural semidirect product $\Aut(G) G$. We will write $\Hol(G)$ for the normalizer of the image in $S(G)$ of the right regular representation of $G$, \begin{equation*} \Hol(G) = N_{S (G)}(\rho(G)) = \Aut(G) \rho(G) \cong \Aut(G) G, \end{equation*} and also refer to it as the holomorph of $G$. More generally, if $N$ is any regular subgroup of $S(G)$, then $N_{S(G)}(N)$ is isomorphic to the holomorph of $N$. G.A.~Miller has shown that the group \begin{equation*} T(G) = N_{S(G)}(\Hol(G))/\Hol(G) \end{equation*} acts regularly on the set of the regular subgroups $N$ of $S(G)$ which are isomorphic to $G$, and have the same holomorph as $G$, in the sense that $N_{S(G)}(N) = \Hol(G)$. If $G$ is non-abelian, inversion on $G$ yields an involution in $T(G)$. Other non-abelian regular subgroups $N$ of $S(G)$ having the same holomorph as $G$ yield (other) involutions in $T(G)$. In the cases studied in the literature, $T(G)$ turns out to be a finite $2$-group, which is often elementary abelian. In this paper we exhibit an example of a finite $p$-group $\Gp$ of class $2$, for $p > 2$ a prime, which is the smallest $p$-group such that $T(\Gp)$ is non-abelian, and not a $2$-group. Moreover, $T(\Gp)$ is not generated by involutions when $p > 3$. More generally, we develop some aspects of a theory of $T(G)$ for $G$ a finite $p$-group of class $2$, for $p > 2$. In particular, we show that for such a group $G$ there is an element of order $p-1$ in $T(G)$, and exhibit examples where $\Size{T(G)} = p - 1$, and others where $T(G)$ contains a large elementary abelian $p$-subgroup., Comment: 19 pages This version fixes some misprints

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated by a couple of examples that might be of independent interest., Comment: 19 pages - minor revisions

W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(\rho(G))$, where $\rho$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(\rho(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G, +)$, what are the commutative ring structures $(G, +, \cdot)$ such that all automorphism of $G$ as a group are also automorphisms of $G$ as a ring?, Comment: 18 pages. Accepted for publication, Journal of Algebra, March 2017

We define a cipher that is an extension of GOST, and study the permutation group generated by its round functions. We show that, under minimal assumptions on the components of the cipher, this group is the alternating group on the plaintext space. This we do by first showing that the group is primitive, and then applying the O'Nan-Scott classification of primitive groups., Comment: 20 pages. Fixed mistake with parity of permutations, shortened argument in final section

According to Li, Nicholson and Zan, a group $G$ is said to be morphic if, for every pair $N_{1}, N_{2}$ of normal subgroups, each of the conditions $G/N_{1} \cong N_{2}$ and $G/N_{2} \cong N_{1}$ implies the other. Finite, homocyclic $p$-groups are morphic, and so is the nonabelian group of order $p^{3}$ and exponent $p$, for $p$ an odd prime. It follows from results of An, Ding and Zhan on self dual groups that these are the only examples of finite, morphic $p$-groups. In this paper we obtain the same result under a weaker hypotesis., Comment: 7 pages. Critical reference added, and manuscript revised accordingly