A generalization of the Davenport constant over abelian groups

Magic squares on Abelian groups

Point group symmetry has been exploited for designing compact ansätze in the Variational Quantum Eigensolver (VQE), thereby facilitating the computation of molecular energy levels on current Noisy Intermediate-Scale Quantum (NISQ) devices. However, the widely used Symmetry-reduced Unitary Coupled Cluster Singles and Doubles (SymUCCSD) is restricted to molecules with Abelian point groups and often yields deficient ansätze for non-Abelian molecular systems. In this paper, we propose Hamiltonian-informed UCCSD (HiUCCSD), a novel shallow ansatz engineered based on the intrinsic information encoded in the molecular Hamiltonian. We theoretically prove the effectiveness of HiUCCSD for molecules belonging to Abelian point groups. Furthermore, numerical results for 10 molecular systems with distinct symmetries demonstrate that HiUCCSD may also be applicable to non-Abelian point group systems. Compared with the standard UCCSD ansatz, HiUCCSD reduces the parameter count and circuit size of VQE by 18-83% and 26-83%, respectively, and shrinks the size of the excitation operator pool for Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT)-VQE by 27-84% across the studied molecules. Given its superior performance and broad applicability, we expect that HiUCCSD will facilitate the realization of large-scale molecular VQE implementations.

Quantum-chemical calculations often make use of point-grouptheoryto exploit molecular symmetry, resulting in a reduction of the computationalcost and in insights into the electronic structure. This exploitationis often limited to subgroups of D2h that are Abelian with real characters. Here, we extend thesymmetry exploitation to Abelian point groups with complex characters.Such point groups are often encountered in calculations that involvefinite magnetic fields, though their occurrence is not limited tothese cases alone. We present the evaluation of integrals over symmetry-adaptedorbitals using the double-coset decomposition, as well as the useof these symmetries in the contractions needed within post-Hartree-Fockcalculations in the context of block tensors. Efficiency gains arediscussed for four simple hydrocarbons that exhibit a complex Abelianpoint group in the presence of a magnetic field.

Abstract We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups Z 3 2 , Z 2 p × Z 3 2 for p = 1 , 2 , 3 , and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg–Wyler’s color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.

In this paper, we study the structure of square-free (summand-square-free) modules, that is the modules that do not contain two nonzero isomorphic disjoint submodules (direct summands) and the modules that are dual to square-free modules over the ring of integers. The class of square-free abelian groups is completely characterized. We determine the structure of both summand-square-free and dual-square-free torsion and finitely generated abelian groups. An abelian group is summand-square-free if its torsion subgroup and the factor group by the torsion subgroup are summand-square-free. The rank of torsion-free dual-square-free groups does not exceed 2. Finally, we establish several results concerning dual-square-free and simply-distinct right modules and provide generalizations of some known results.

Let A be an infinitely generated free abelian group. For every natural number m ⩾ 1 , we provide two generating sets for the congruence group Γ A ( m ) of level m of the group Γ = Aut ( A ) , and then use one of these generating sets to describe the automorphisms of the group Γ A ( m ) .

Given polynomials f 0 , f 1 , …, f k the Ideal Membership Problem, IMP for short, asks if f 0 belongs to the ideal generated by f 1 , …, f k . In the search version of this problem, the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications. For instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA’19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP(Γ). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP(Γ) where Γ is a Boolean constraint language, while Bulatov and Rafiey [STOC’22] advanced these results to several cases of CSPs over finite domains. In this article, we consider IMPs arising from CSPs over “affine” constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS’21] for linear equations modulo 2, and by Bulatov and Rafiey [STOC’22] for systems of linear equations over GF ( p ), p prime. Here, we prove that if Γ is an affine constraint language then IMP(Γ) is solvable in polynomial time assuming the input polynomial has bounded degree.

Separating polynomial invariants over non-closed fields of finite abelian groups

Orientable quadratic equations in wreath products of abelian groups

Let G be a finite group acting on a finite dimensional complex vector space V via linear transformations. Let C [ V ] G be the algebra of polynomials that are invariant under the induced G-action on the polynomial ring C [ V ] . A subset S ⊆ C [ V ] G is a separating set if it separates the orbits of the group action. If G is abelian, then there exist finite separating sets consisting of monomials. In this paper we investigate properties of separating sets from three different points of view, including the minimal size of separating sets consisting of monomials, the exact value of the separating Noether number β sep ( G ) of abelian groups of rank 4, and an inverse problem of β sep ( G ) for abelian groups of rank 2.

In this paper, the authors study the concept of domination in the negation of Cayley signed graphs over finite abelian groups. They establish sharp bounds for the domination numbers and characterize the groups that attain these bounds. The study reveals that, for certain Cayley sets, the sum of the domination numbers of a Cayley signed graph and its negation exceeds the group’s order. The results are illustrated with several examples.

Abstract An Abelian group 𝐺 is called strongly not divisible if n ⁢ G ≠ G nG\neq G for every integer n ≠ ± 1 n\neq\pm 1 . In J. A. Lewallen and N. Sagullo, A note on OI torsion abelian groups, Missouri J. Math. Sci. 27 (2015), 1, 33–36, a characterization of the torsion strongly not divisible groups (referred to there as OI-groups) was obtained. In this note, we provide characterizations for broad classes of torsion-free and mixed strongly not divisible groups.

Abstract We define the concept of slenderness for arbitrary semigroups containing at least one idempotent element, and give necessary and sufficient conditions for semigroups to be slender. We describe the semigroup of all homomorphisms of the direct product of a nonempty family of groups into a slender rectangular abelian group.

Quantum error correction in Kitaev’s quantum double model for Abelian groups

Abstract We propose a family of quantum algorithms for estimating Gowers uniformity norms $$ U^k $$ U k over finite abelian groups, extending earlier quantum methods for the $$ U^2 $$ U 2 -norm to arbitrary prime fields and higher-order uniformity norms. Our algorithms prepare quantum states encoding higher-order finite differences and apply Fourier sampling together with amplitude estimation to obtain estimates of Gowers norms. As a central application, we study algebraic property testing problems of distinguishing whether a bounded function $$ f: \mathbb {F}_p^n \rightarrow \mathbb {C} $$ f : F p n → C is a low-degree phase polynomial or is far from any such structure. We show that whenever an inverse theorem for the $$ U^{d+1} $$ U d + 1 -norm is available, our quantum framework yields a corresponding structure-testing algorithm whose query and measurement complexity depends explicitly on the quantitative bounds of that inverse theorem. In particular, using the recent quasipolynomial inverse theorem for the $$ U^4 $$ U 4 -norm over $$ \mathbb {F}_p^n $$ F p n , we obtain quasipolynomial-time quantum algorithms for detecting cubic phase polynomials. For higher-order norms such as $$ U^5 $$ U 5 and $$ U^6 $$ U 6 over $$ \mathbb {F}_2^n $$ F 2 n , the best known inverse theorems provide only tower-type quantitative bounds; accordingly, our detection algorithms remain correct but inherit complexity corresponding to tower-type quantitative bounds. We also present a quantum method for estimating the number of 3-term arithmetic progressions in Boolean functions via the $$ U^2 $$ U 2 -norm. Although not query-optimal compared to Grover-style counting, this approach is sensitive to additive structure and naturally aligned with tools from higher-order Fourier analysis. Finally, we observe that Gowers norms are invariant under certain classes of shift-type noise, implying that our algorithms retain robustness under natural quantum noise models. This suggests that Gowers norm-based quantum procedures may serve as stable primitives for quantum property testing, learning theory, and the analysis of pseudorandomness in the NISQ regime.

Abstract A perfect code in a graph $$\Gamma = (V, E)$$ Γ = ( V , E ) is a subset C of V such that no two vertices in C are adjacent and every vertex in $$V \setminus C$$ V \ C is adjacent to exactly one vertex in C . A total perfect code in $$\Gamma $$ Γ is a subset C of V such that every vertex of $$\Gamma $$ Γ is adjacent to exactly one vertex in C . In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

In this work we study the self-similarity of infinite products of the group of the integers. In [A. C. Dantas, T. M. G. Santos and S. N. Sidki, Self-similar abelian groups and their centralizers, Groups Geom. Dyn. 17 (2023) 1–23], the authors ask if there is a self-similar free abelian group of uncountable rank. On the other hand, in [L. Bartholdi and S. N. Sidki, Self-similar products of groups, Groups Geom. Dyn. 14(1) (2020) 107–115] is asked if the Baer–Specker group [Formula: see text] is self-similar (by a result of R. Baer, this group is not free abelian). We answer both questions positively.

This paper classifies all left-invariant Hermitian and hyper-Hermitian structures on Lie groups whose commutator groups are one-dimensional. Furthermore, for each natural number n = 2k and n = 4k (k > 1 for the case of hyper-Hermitian structures), we construct an n-dimensional Lie group with a one-dimensional commutator group that possesses left-invariant Hermitian and hyper-Hermitian structures, respectively.

Our main result is a combinatorial characterization of when a horospherical variety has (at worst) quotient singularities. Using this characterization, we show that every quasiprojective horospherical variety with quotient singularities is globally the quotient of a smooth variety by a finite abelian group.