arXiv subjects

Boris Tsvelikhovskiy

Publications and source records attributed to Boris Tsvelikhovskiy.

16 recordsLinked to original sources

arXiv2026

Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles

For an almost simple complex algebraic group $G$ of semisimple rank $m$, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most $m+2$ members does. We determine the resulting Helly number $(h(G))$ for almost simple groups of types $A$, $B$, $C$, $G_2$, $F_4$, $E_7$, and $E_8$, and obtain bounds differing by one in types $D$ and $E_6$. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal $G$-bundle splits equivariantly. In particular, for $d \ge 2$, this splitting property holds on $\mathbb{P}^d$ if and only if $d \ge h(G)$

math.AG↗

arXiv2026

The Expressive Power of Constrained QAOA: What You Might Have MISsed

Feasibility-preserving mixer Hamiltonians offer an attractive alternative to penalty encodings for constrained quantum optimization, yet how enforcing constraints alters the state space reachable by the Quantum Approximate Optimization Algorithm (QAOA) remains largely unexplored. We address this question within the context of the Maximum Independent Set (MIS) problem. Alongside the standard controlled bit-flip mixer, we analyze the alternative $\mathit{Flip\text{-}or\text{-}Stay}$ mixer, whose restriction to the feasible subspace $W_{\mathcal{F}}$ corresponds precisely to the shifted negative Laplacian of the independent-set reconfiguration graph. Although both mixers induce identical transitions between independent sets, the diagonal modification changes the spectral structure and can substantially enlarge the set of reachable quantum states. For every connected input graph with at least two vertices, we prove that independently parameterized local controls associated with either mixer generate the full unitary Lie algebra on the feasible subspace. Standard QAOA exhibits a more nuanced algebraic structure. We prove that the dynamical Lie algebra associated with MIS-QAOA employing the conventional mixer embeds into its Flip-or-Stay counterpart. Furthermore, we construct an infinite family of graphs for which the corresponding vacuum-state dynamical group orbits exhibit strictly distinct dimensions, establishing a qualitative expressivity separation between the two architectures. For either mixer, we show that standard QAOA initialized in the empty set can prepare a state supported entirely on maximum independent sets at finite depth. For the free architectures, we also derive an exact loss-variance formula in the unitary-design limit, expressed through the number of feasible independent sets and the variance of their cardinalities.

quant-ph↗

arXiv2026

Efficient Estimation of Reduced QAOA Expressibility on Acyclic Graphs

Classically equivalent formulations of an optimization problem need not lead to equivalent quantum algorithms. For MaxCut, fixing the value of a single vertex removes a simple global symmetry without changing the underlying optimization problem, yet it can substantially alter the quantum dynamics generated by the Quantum Approximate Optimization Algorithm (QAOA). These dynamics are captured by the circuit's dynamical Lie algebra (DLA), whose direct construction can become exponentially expensive. Here we show that, for symmetry-reduced QAOA on trees, substantial information about the reduced DLA can instead be obtained efficiently from the graph alone. We introduce a polynomial classical algorithm that recursively distinguishes vertices using shortest path structure and degree parity. When all vertices are resolved individually, the method determines the complete reduced DLA and, under the corresponding graph conditions, certifies maximal expressibility of the reduced QAOA ansatz. Even when full resolution is not achieved, the algorithm identifies embedded subalgebras, provides rigorous lower bounds on DLA dimension, and certifies controllable subsystems. Our theoretical and experimental results show how classical graph structure can be used to diagnose and potentially guide the quantum dynamics of variational algorithms before running them on quantum hardware.

quant-ph↗

arXiv2026

Reductions of QAOA Induced by Classical Symmetries: Theoretical Insights and Practical Implications

The performance of the Quantum Approximate Optimization Algorithm (QAOA) is closely tied to the structure of the dynamical Lie algebra (DLA) generated by its Hamiltonians, which determines both its expressivity and trainability. In this work, we show that classical symmetries can be systematically exploited as a design principle for QAOA. Focusing on the MaxCut problem with global bit-flip symmetry, we analyze reduced QAOA instances obtained by fixing a single variable and study how this choice affects the associated DLAs. We show that the structure of the DLAs can change dramatically depending on which variable is held fixed. In particular, we construct explicit examples where the dimension collapses from exponential to quadratic, uncovering phenomena that do not appear in the original formulation. Numerical experiments on asymmetric graphs indicate that such reductions often produce DLAs of much smaller dimension, suggesting improved trainability. We also prove that any graph can be embedded into a slightly larger one (requiring only quadratic overhead) such that the standard reduced DLA coincides with the free reduced DLA, in most cases implying exponential dimension and irreducibility on the Hilbert space for reduced QAOA instances. These results establish symmetry-aware reduction as a principled tool for designing expressive and potentially trainable QAOA circuits.

quant-ph↗

arXiv2025

Notes from a family of smooth $G$-Hilbert schemes

Let $ G = \mathbb{Z}/r\mathbb{Z}$ be the cyclic group of order $r$, and let $\varpi = e^{2πi / r}$ denote a primitive $r$ th root of unity. Consider the action of $G$ on $\mathbb{C}^n$ via the embedding $$ φ: G \hookrightarrow GL_n(\mathbb{C}), \qquad φ(1) = \mathrm{diag}\!\bigl( \underbrace{\varpi, \dots, \varpi}_{s},\, \underbrace{\varpi^{-1}, \dots, \varpi^{-1}}_{n - s} \bigr), $$ where $0 < s < n $. Denote the corresponding GIT quotient by $$ \mathcal{X}_{s,n,r} = \mathrm{Spec}\bigl((\mathbb{C}[z_1,\dots,z_n])^G\bigr). $$ Then the varieties $\mathcal{X}_{s,n,r}$ is a cyclic quotient singularity of type $\tfrac{1}{r}\bigl(\underbrace{1,\dots,1}_{s}, \underbrace{-1,\dots,-1}_{n-s}\bigr)$. We show that the associated $G$-Hilbert schemes $\mathcal{Y}_{s,n,r}$ are smooth, connected, and irreducible. The natural morphism $$ ρ_{s,n,r}:\mathcal{Y}_{s,n,r}\longrightarrow\mathcal{X}_{s,n,r} $$ is a projective resolution of $\mathcal{X}_{s,n,r}$, discrepant for $n \ge 3$. We establish that the irreducible components of the central fiber $ρ_{s,n,r}^{-1}(0)$ are in bijection with the nontrivial characters of $G$, thereby realizing the classical McKay correspondence in this family of examples. Finally, we describe a canonical choice of this bijection via the Fourier--Mukai type functor $$ Ψ: D^b(\mathrm{Coh}_G(\mathbb{C}^n)) \longrightarrow D^b(\mathrm{Coh}(\mathcal{Y}_{s,n,r})), $$ by showing that, for each nontrivial irreducible representation of $G$, the corresponding skyscraper sheaf is mapped to a complex whose $0^{\text{th}}$ cohomology is supported on a unique irreducible component of the central fiber $ρ_{s,n,r}^{-1}(0)$.

math.AG↗

arXiv2025

Provable avoidance of barren plateaus for the Quantum Approximate Optimization Algorithm with Grover mixers

We analyze the dynamical Lie algebras (DLAs) associated with the Grover-mixer variant of the Quantum Approximate Optimization Algorithm (GM-QAOA). When the initial state is the uniform superposition of computational basis states, we show that the corresponding DLA is isomorphic to $\mathfrak{su}(d) \oplus \mathfrak{u}(1)\oplus \mathfrak{u}(1)$, where $d$ denotes the number of distinct values of the objective function. We also establish an analogous result for other choices of initial states and Grover-type mixers. Furthermore, we prove that the DLA of GM-QAOA has the largest possible commutant among all QAOA variants initialized with the same state, corresponding physically to the maximal set of conserved quantities. We derive an explicit formula for the variance of the GM-QAOA loss function in terms of the objective function values, and we show that for a broad class of optimization problems, GM-QAOA with sufficiently many layers avoids barren plateaus.

quant-ph↗

arXiv2025

On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence

Let $G \subset SL_3(\mathbb{C})$ be a finite abelian subgroup, and let $Y = G\operatorname{-}\mathrm{Hilb}(\mathbb{C}^3)$ be the corresponding $G$-Hilbert scheme. Denote by $Ψ: D^b_G(\mathrm{Coh}(\mathbb{C}^3)) \to D^b(\mathrm{Coh}(Y))$ the Bridgeland--King--Reid derived equivalence. For a nontrivial character $χ$ of $G$, let $χ^!$ be the corresponding skyscraper sheaf supported at the origin. It is known that $Ψ(χ^!)$ is always a pure sheaf supported either in degree $0$ or in degree $-1$. We prove that the proportion of characters $χ$ for which $Ψ(χ^!)$ is supported in degree $0$ is a rational number lying between $0.25$ and $1$, with both bounds being sharp. Moreover, we exhibit families of resolutions for which these proportions attain certain explicit values within this range.

math.AG↗

arXiv2025

Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $Φ$ from the fan of $\mathfrak{X}$ to the (extended) building of $GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles.

math.AG↗

arXiv2024

Equivariant QAOA and the Duel of the Mixers

Constructing an optimal mixer for Quantum Approximate Optimization Algorithm (QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving combinatorial optimization problems. We present a systematic methodology for constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the inherent symmetries of classical optimization problem objectives. The key to our approach is to identify an operator that commutes with the action of the group of symmetries on the QAOA underlying Hilbert space and meets the essential technical criteria for effective mixer Hamiltonian functionality. We offer a construction method specifically tailored to the symmetric group $S_d$, prevalent in a variety of combinatorial optimization problems. By rigorously validating the required properties, providing a concrete formula and corresponding quantum circuit for implementation, we establish the viability of the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical mixer $B$ commutes only with a subgroup of $S_d$ of significantly smaller order than the group itself, enhancing the efficiency of the proposed approach. To evaluate the effectiveness of our methodology, we compare two QAOA variants utilizing different mixer Hamiltonians: conventional $B=\sum X_i$ and the newly proposed $H_M$ in edge coloring and graph partitioning problems across various graphs. We observe statistically significant differences in mean values, with the new variant consistently demonstrating superior performance across multiple independent simulations. Additionally, we analyze the phenomenon of poor performance in alternative warm-start QAOA variants, providing a conceptual explanation supported by recent literature findings.

quant-ph↗

arXiv2024

Symmetries and Dimension Reduction in Quantum Approximate Optimization Algorithm

In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on the generalized formulation of optimization problems defined on the sets of $n$-element $d$-ary strings. Our main contribution encompasses dimension reductions for the originally proposed QAOA. These reductions retain the same problem Hamiltonian as the original QAOA but differ in terms of their mixer Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of exponential scaling in $n$, where certain reduced QAOA spaces exhibit dimensions governed by polynomial functions. This phenomenon is illustrated in this paper, by providing partitions corresponding to polynomial dimensions of the corresponding subspaces. As a result, each reduced QAOA partition encapsulates unique classical solutions absent in others, allowing us to establish a lower bound on the number of solutions to the initial optimization problem. Our novel approach opens promising practical advantages in accelerating the algorithm. Restricting the algorithm to Hilbert spaces of smaller dimension may lead to significant acceleration of both quantum and classical simulation of circuits and serve as a tool to cope with barren plateaus problem.

quant-ph↗

arXiv2022

The universe inside Hall algebras of coherent sheaves on toric resolutions

Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given $\mathfrak{n}_-$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_3(\mathbb{C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname{-}\mbox{Hilb}(\mathbb{C}^3)$ and $Ψ: D^b_G(Coh(\mathbb{C}^3))\rightarrow D^b(Coh(Y))$ the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in $Coh_G(\mathbb{C}^3)$, s.t. the Hall algebra generated by their images under $Ψ$ is isomorphic to $U(\mathfrak{n}_-)$. In case the field $\Bbbk$ (in place of $\mathbb{C}$) is finite and $\mbox{char}(\Bbbk)$ is coprime with the order of $G$, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras $U_v(\mathfrak{n}_-)$ at $v=\sqrt{|\Bbbk|}$.

math.RT↗

arXiv2020

Manipulable outcomes within the class of scoring voting rules

Coalitional manipulation in voting is considered to be any scenario in which a group of voters decide to misrepresent their vote in order to secure an outcome they all prefer to the first outcome of the election when they vote honestly. The present paper is devoted to study coalitional manipulability within the class of scoring voting rules. For any such rule and any number of alternatives, we introduce a new approach allowing to characterize all the outcomes that can be manipulable by a coalition of voters. This gives us the possibility to find the probability of manipulable outcomes for some well-studied scoring voting rules in case of small number of alternatives and large electorates under a well-known assumption on individual preference profiles.

econ.TH↗

arXiv2019

On Poincare polynomials of shuffle algebra representations

In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-λ$ for both generic $q_{1},q_{2}$ and satisfying the relation $q_{1}^{a}q_{2}^{b}=1$ with $a, b \in \mathbb{N}$.

math.RT↗

arXiv2019

On categories $\mathcal{O}$ of quiver varieties overlying the bouquet graphs

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations $\overline{\mathcal{A}}_λ(n, \ell)$ if dim $V=n$ is greater than $1$ and so is the number of loops $\ell$. We find that there is a Hamiltonian torus action with finitely many fixed points in case $n\leq 3$, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ and compute the multiplicities of simples in standards for $n=2$ in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localisation theorem and find the values of parameters, for which the quantizations have infinite homological dimension.

math.RT↗

arXiv2018

Ring Theoretic Aspects of Quandles

We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group $Inn(X)$ acts orbit $2$-transitively on $X$, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles $R_n$ is given. It is also shown that if for two quandles $X$ and $Y$ the inner automorphism groups act $2$-transitively and $\mathbf{k}[X]$ is isomorphic to $\mathbf{k}[Y]$, then the quandles are of the same partition type. However, we provide examples when the quandle rings $\mathbf{k}[X]$ and $\mathbf{k}[Y]$ are isomorphic, but the quandles $X$ and $Y$ are not isomorphic. These examples answer some open problems in [3].

math.RA↗

arXiv2018

Nontrivial Topological Quandles

We show that there are infinitely many nonisomorphic quandle structures on any topogical space $X$ of positive dimension. In particular, we disprove the conjecture, asserting that there are no nontrivial quandle structures on the closed unit interval $[0,1]$.

math.GT↗