The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Search and test algorithms for Triple Product Property triples

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare their runtime. Furthermore we show that the search for subgroup TPP triples of nontrivial size in a nonabelian group can be restricted to the set of all nonnormal subgroups of that group. Finally we describe brute-force search algorithms for maximal subgroup and subset TPP triples. In addition we present the results of the subset brute-force search for all groups of order less than 25 and selected results of the subgroup brute-force search for 2-groups, $SL(n,q)$ and $PSL(2,q)$.Comment: 14 pages, 2 figures, 4 tables; ISSN (Online) 1869-6104, ISSN (Print) 1867-114

Further research problems and theorems on prime power groups

Below we state a great number of research problems concerning finite p-groups. This list is a continuation of the six lists in [1, 2, 3, 4, 5, 6]. Below we also stated some new theorems with proofs. For explanation of notation see the beginning of the above volumes

Orientably-Regular π\pi-Maps and Regular π\pi-Maps

Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable} if the group $G^+$ of all orientation-preserving automorphisms (resp. the group $G$ of automorphisms) is solvable; and called {\it normal} if $G^+$ (resp. $G$) contains a normal $\pi$-Hall subgroup. In this paper, it will be proved that orientably-regular $\pi$-maps are solvable and normal if $2\notin \pi$ and regular $\pi$-maps are solvable if $2\notin \pi$ and $G$ has no sections isomorphic to ${\rm PSL}(2,q)$ for some prime power $q$. In particular, it's shown that a regular $\pi$-map with $2\notin \pi$ is normal if and only if $G/O_{2^{'}}(G)$ is isomorphic to a Sylow $2$-group of $G$. Moreover, nonnormal $\pi$-maps will be characterized and some properties and constructions of normal $\pi$-maps will be given in respective sections.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2201.0430

Inducing π\pi-partial characters with a given vertex

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup of a subgroup $V$. Suppose \phi \in \ibr G. If either $|G|$ is odd or $p = 2$, we prove that the number of Brauer characters of $H$ inducing $\phi$ with vertex $Q$ is at most |\norm GQ: \norm VQ|

Borcherds symmetries in M-theory

It is well known but rather mysterious that root spaces of the $E_k$ Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on $T^k$ corresponds to blow-up of $k$ points in general position with respect to each other. All theories of the Magic triangle that reduce to the $E_n$ sigma model in three dimensions correspond to singular del Pezzo surfaces with $A_{8-n}$ (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively aris

Characterization of finite abelian and minimal nonabelian groups

In this note we present the following characterizations of finite abelian and minimal nonabelian groups: (i) A group G is abelian if and only if G\u27 = Φ(G)\u27. (ii) A group G is either abelian or minimal nonabelian if and only if Φ(G)\u27 = H\u27 for all maximal subgroups H of G. We also prove a number of related results