Science
Below are my scientific articles (and other scientific projects), in order from recent to older. See also my ResearchGate profile, Google Scholar profile.
Pointwise Bounds on the Number of Rubik's Snake Shapes (2026)
Abstract. Let S(n) count the collision-free configurations of an oriented n-wedge Rubik's Snake. For every n ≥ 1, we prove (3.45056)^(n-1) ≤ S(n) ≤ 2.95621(3.66179)^(n-1). The proof is formalized in Lean.
Links: paper, mirror, github, OEIS.
CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks (2026)
Abstract.
This is the fourth paper in the CayleyPy project, which applies AI methods to the exploration of large graphs. In this work, we suggest the existence of a new discrete version of holographic string dualities for this setup, and discuss their relevance to AI systems and mathematics.
Many modern AI tasks -- such as those addressed by GPT-style language models or RL systems -- can be viewed as direct analogues of predicting particle trajectories on graphs. We investigate this problem for a large family of Cayley graphs, for which we show that surprisingly it admits a dual description in terms of discrete strings. We hypothesize that such dualities may extend to a range of AI systems where they can lead to more efficient computational approaches. In particular, string holographic images of states are proposed as natural candidates for data embeddings, motivated by the "complexity = volume" principle in AdS/CFT.
For Cayley graphs of the symmetric group S_n, our results indicate that the corresponding dual objects are flat, planar polygons. The diameter of the graph is equal to the number of integer points inside the polygon scaled by n. Vertices of the graph can be mapped holographically to paths inside the polygon, and the usual graph distances correspond to the area under the paths, thus directly realising the "complexity = volume" paradigm. We also find evidence for continuous CFTs and dual strings in the large n limit. We confirm this picture and other aspects of the duality in a large initial set of examples. We also present new datasets (obtained by a combination of ML and conventional tools) which should be instrumental in establishing the duality for more general cases.
Link: arxiv.
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (2025)
Abstract.
This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs. Compared with systems such as GAP and Sage, CayleyPy handles much larger graphs and performs several orders of magnitude faster.
Using CayleyPy we obtained about 200 new conjectures on Cayley and Schreier graphs, focused on diameters and growth. For many Cayley graphs of symmetric groups Sn we observe quasi polynomial diameter formulas: a small set of quadratic or linear polynomials indexed by n mod s. We conjecture that this is a general phenomenon, giving efficient diameter computation despite the problem being NP hard. We propose a refinement of the Babai type conjecture on diameters of Sn: n^2/2 + 4n upper bounds in the undirected case, compared to previous O(n^2) bounds. We also provide explicit generator families, related to involutions in a square with whiskers pattern, conjectured to maximize the diameter; search confirms this for all n up to 15. We further conjecture an answer to a question posed by V M Glushkov in 1968 on directed Cayley graphs generated by a cyclic shift and a transposition.
For nilpotent groups we conjecture an improvement of J S Ellenberg's results on upper unitriangular matrices over Z/pZ, showing linear dependence of diameter on p.
Some conjectures are LLM friendly, naturally stated as sorting problems verifiable by algorithms or Python code. To benchmark path finding we created more than 10 Kaggle datasets. CayleyPy works with arbitrary permutation or matrix groups and includes over 100 predefined generators. Our growth computation code outperforms GAP and Sage up to 1000 times in speed and size.
Co-authored with Alexander Chervov, Elena Konstantinova and many others.
Published: The 5th Workshop on Mathematical Reasoning and AI at NeurIPS 2025, San Diego (link).
Quantum Arithmetic Algorithms: Implementation, Resource Estimation, and Comparison (2025)
Abstract. As quantum computing technology advances, the need for optimized arithmetic circuits continues to grow. This paper presents the implementation and resource estimation of a library of quantum arithmetic algorithms, including addition, multiplication, division, and modular exponentiation. Using the Azure Quantum Resource Estimator, we evaluate runtime, qubit usage, and space-time trade-offs and identify the best-performing algorithm for each arithmetic operation. We explore the design space for division, optimize windowed modular exponentiation, and identify the tipping point between multipliers, demonstrating effective applications of resource estimation in quantum research. Additionally, we highlight the impact of parallelization, reset operations, and uncomputation techniques on implementation and resource estimation. Our findings provide both a practical library and a valuable knowledge base for selecting and optimizing quantum arithmetic algorithms in real-world applications.
Co-authored with Yingrong Chen and Brian Goldsmith.
Published: 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), Albuquerque, NM, USA, 2025.
This project was done as part of the QOSF Quantum Computing Mentorship Program. Presented on 2025-09-04 (slides).
Links: IEEE Xplore, arxiv, github.
New Circuit for Quantum Adder by Constant (2025)
Abstract. We propose a new circuit for in-place addition of a classical n-bit constant to a quantum n-qubit integer modulo 2n. Our circuit uses n-3 ancilla qubits and has a T-count of 4n-5. We also propose a controlled version of this circuit that uses n-2 ancillas and has a T-count of 11n-15. We implement these circuits in Q#.
Link: arxiv.
Counting Rubik's Snake Shapes (2024)
In this project I numerically computed the number of n-wedge Rubik's Snake shapes for n up to 26. I also computed some other related sequences representing the number of shapes under some symmetries and/or restrictions and analyzed their asymptotic behaviour.
Binary Tree Elimination algorithm for computing marginal probabilities in graphical models (2022)
Abstract. We propose a method to efficiently compute marginal probabilities for a probabilistic graphical model. This method can be built on top of any elimination algorithm for computing the partition function and is significantly more efficient and accurate than the trivial reduction. We show how to apply this technique to three elimination algorithms: Bucket Elimination, Mini-Bucket Elimination and Mini-Bucket Renormalization.
Link: paper.
Quantum Circuits for Elementary Cellular Automata (2021)
Abstract. In this paper we identify the full list of Elementary Cellular Automata rules which can be simulated using a quantum circuit (there are 22 such rules). For every such rule we present a quantum circuit implementing it with O(N) gates.
Decomposition of unitary matrix into quantum gates (2019)
Abstract. An algorithm is proposed to convert an arbitrary unitary matrix to a sequence of X gates and fully controlled Ry, Rz and R1 gates. This algorithm is used to generate a Q# implementation for an arbitrary unitary matrix. Some optimizations are considered and the complexity of the result is analyzed.
This was a course project for the online course "8.06x - Application of Quantum Mechanics" at MIT. Published in 8.06x Physical Review.
Introduction to Probability Theory (2018, seminar notes)
These are notes (in Russian) for one seminar at MIPT on the course "Algorithms and Computational Models", for second-year students who haven't studied Probability Theory yet. I tried to give a minimal introduction sufficient for further study of this course (e.g. probabilistic algorithms), and yet to be maximally rigorous. They also include 15 problems.
Link: pdf (in Russian).
Ising Model simulation (2017, Course project)
This is a course project for the course "Introduction to Scientific Computing", attended by me at Skoltech. In this work I numerically obtain the dependency of energy and magnetization on temperature for the two-dimensional Ising model in zero outer field and compare it with theoretical results. Also I do modelling with CUDA and analyze how modelling error decreases with an increase in lattice size. Report (html), Code+Data.
Topic models interactive visualization technology (2017, B. Sc. Thesis)
Abstract. In this work we study methods of topic models visualization. We consider several ways of visualizing topic models (including temporal and hierarchical). We set a problem of building a topic model spectrum - a permutation of its topics, in which semantically close topics are located nearby. We suggest several algorithms to solve this problem and design a method to assess their quality. The task is generalized for the case of hierarchical topic models. Also we describe an information system, VisARTM, which is designed for the automated creation and visualization of topic models.
Advisor: Konstantin Vorontsov. Defended on 2017-06-29 at Dorodnitsyn Computing Centre.
Links: thesis (in Russian), mirror, slides, code.
Hierarchic topic models visualization (2016)
Course work on the course "Creating of exploited models of Machine Learning". Supervisor V. V. Strijov.
Abstract. Hierarchic topic models are a good tool for representing a large number of text documents. However, displaying such models on screen is a difficult problem itself. This paper discusses the problem of hierarchic topic model visualization. It introduces the concept of tree visualization with polygons. Also it considers problems of quality measuring and representation of additional information. This article also describes the implementation of visualization algorithms using the BigARTM topic modeling library.
Links: paper, presentation, code, sysdoc, video.Mixtures of models of vector autoregression in the problem of time series forecasting (2016)
Abstract. In this paper we investigate the problem of short-term time series forecasting. We consider the time series of different scales, which are interconnected and have the property of periodicity. The forecasting problem is reduced to the problem of regression, which is solved by using a linear model. To improve its accuracy we propose to use a composition of models. The compositions are built using bagging, the random subspace method and the boosting algorithm AdaBoost. We also propose a heuristic iterative algorithm of model composition based on the idea of the clustering algorithm K-means. Using the proposed methods, we forecast energy consumption in Turkey and Poland and spot prices for electricity in Germany.
Co-authored with Vadim Strijov and Radoslav Neychev.
Links: paper (in Russian), mirror1, mirror2, slides, code.
Computer modelling in application to city traffic (2013)
Abstract. A model of traffic in a big city is developed. We consider: 1) one- and bidirectional movement, 2) intersections, 3) basic requirements of traffic safety. We have created a decision support system for traffic optimization. For computer realization of the model we have created a project using object-oriented programming.
Links: paper (in Ukrainian), program.
Percolation phenomenon in cellular automata (2012)
Abstract. In this paper, we investigate synchronous cellular automata, similar to the 'Game of Life'. We study the phenomenon of percolation — abrupt filling of all cells during evolution from random initial state when a critical concentration is reached. We give a theoretical explanation for this phenomenon. Also we study the dependency of borders and the critical index of percolation on the form of the grid and parameters of the automata.
Link: paper (in Ukrainian).
Modelling and research of the Game of Life and its modifications on triangular, square and hexagonal grids (2011)
Abstract. Algorithms and a program are created for analysis of the evolution of counter colonies on two-dimensional lattices filled with rectilinear triangles and hexagons under different initial configurations and survival conditions.
Published in "Mathematics. Informatics. Physics." (the scientific magazine of Dnipropetrovsk Lyceum of Informational Technology), Vol. 2.