arXiv subjects

Youming Qiao

Publications and source records attributed to Youming Qiao.

At least 19 recordsLinked to original sources

arXiv2026

From almost tight $p$-group enumeration to $2$-group domination among finite groups

Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. The pioneering works of Higman (\emph{Proc. Lond. Math. Soc.}, 1960) and Sims (\emph{Proc. Lond. Math. Soc.}, 1965), with improvement by Newman and Seeley, establish that \[ \frac{2}{27}m^3-\frac{4}{9}m^2+Ω(m)\leq \log_p f(p^m)\leq \frac{2}{27}m^3+O(m^{5/2}). \] In the same paper in 1965, Sims speculated on improving the error term ($O(m^{8/3})$ at that time) in the upper bound. To show that $f(p^m)\leq \frac{2}{27}m^3+O(m^2)$ is now known as Sims' conjecture. Our first main result proves Sims' conjecture in a strong form as follows: \[ \log_p f(p^m)\leq \frac{2}{27}m^3-\frac{4}{9}m^2+O(m). \] Techniques developed to prove the above lead to progress on some well-known open questions in group enumeration in affirmative, including: (1) 2-group domination in finite group enumeration (Pyber, \emph{Eur. Congr. Math.}, 1996): Let $F(n)$ be the number of groups of order $\leq n$, and $F_2(n)$ be the number of $2$-groups of order $\leq n$. Then $$ \lim_{n\to\infty}\frac{F_2(n)}{F(n)} =1. $$ (2) A question attributed to Higman and Erdős (see Blackburn--Neumann--Venkataraman, \emph{Cambridge Tracts in Maths.}, 2007) when $m$ is large enough: $$ \text{for } 1<n<2^m, f(n)<f(2^m). $$ Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalisation by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a $p^{-Ω(m^2)}$ fraction of groups of order $p^m$ have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser--Li--Qiao--Rogovskyy (\emph{arXiv}, 2025) for $p$-groups of Frattini class $2$ yields the linear error term.

math.GR↗

arXiv2026

Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2

The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos--Mendoza--Qiao--Sun--Zhang, FOCS '24; Grochow--Qiao--Stange--Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.

cs.DS↗

arXiv2026

Verifiable quantum advantage based on polynomials with planted structures

A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around $10^{-6}$.

quant-ph↗

arXiv2026

The planted tensor problem over finite fields: algorithms and cryptography

Inspired by the planted clique problem for random graphs, we introduce the planted totally-isotropic space problem for random tensors as follows. Let $U\cong \mathbb{F}_q^n$ and $W\cong \mathbb{F}_q^m$ be finite-dimensional vector spaces over a finite field $\mathbb{F}_q$. Given $d\in \mathbb{N}$, choose a random $d$-dimensional subspace $V\leq U$, and construct a random alternating bilinear map $ϕ:U\times U\to W$ subject to the constraint $ϕ(V,V)=0$. Such a $V$ is known as a totally-isotropic space of $ϕ$, and the goal is to recover $V$. Building on the recent probabilistic analysis of random tensors (Pham--Qiao--Wigderson--Wigderson, \emph{in progress}), we initiate the study of the algorithmic hardness of this problem. Setting $m=\lceil n/\log n\rceil$, we show that this problem admits an average-case polynomial-time algorithm for $d\geq n/2$, by leveraging recent advances on the non-commutative rank problem. We also show that this problem admits a $q^{O(n\log n)}$-time algorithm. We carry out algorithmic experiments using polynomial-system solving. From these results, we conjecture that the planted totally-isotropic space problem for $d=\lceil n/C\rceil$ with some constant $C\geq 3$ is exponentially hard. Based on this evidence of computational hardness, we explore cryptographic applications of the planted totally-isotropic space problem and related planted tensor problems. We present private simultaneous messages and secret sharing protocols based on planted tensor problems, following the protocols based on planted subgraphs in (Abram--Beimel--Ishai--Kushilevitz--Narayanan, \emph{TCC}'23). At the same security level, the public information size of protocols based on planted subgraphs is (moderately) exponential in that of protocols based on planted tensors, while the communication costs of these protocols are polynomially related.

cs.DS↗

arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗

arXiv2026

Sharp Norms from Finite Structure: Graph Matrices and Structured Chaoses

Graph matrices encode dependencies in random matrices built from shared random variables and arise in spectral algorithms, sum-of-squares (SoS), and high-dimensional statistics. We determine how finite graph structure controls their sharp spectral growth. For every fixed simple graph shape in the dense Rademacher model, including overlapping or empty matrix boundaries, we prove $\mathbb E\|M_α\|=Θ_α(n^{(v+h-s)/2}(\log n)^{a_*/2})$, where $v$ counts vertices, $h$ isolated summation vertices, $s$ the minimum boundary-separator size, and $a_*$ maximizes an active-component count over minimum separators. Thus two finite cut optimizations determine both the polynomial and logarithmic exponents. The formula closes the polylogarithmic gap in separator bounds, and an infinite family with identical coarse parameters but different norms shows that the logarithmic exponent records genuinely new structure. The proof controls all defect layers in growing trace moments by converting label loss into separator excess; conditional flattening and synchronized fluctuations yield matching lower bounds. We extend the analysis to specified independent-factor chaoses, local weights, unequal dimensions, bounded asymmetric noise, Gaussian inputs, and fixed-degree Hermite inputs. Applications include degree-four clique SoS feasibility for $9\le k\le c\sqrt n$ without an asymptotic logarithmic loss, and Gaussian random tensor networks: deviation thresholds, sharp expected scales, entropy estimates, and, for connected loopless equal-dimensional networks, convergence of the rescaled largest output eigenvalue to the exact right edge of the limiting law. These results connect finite structure to sharp growth scales, and additional algebraic and spectral structure to full feasibility and exact limiting constants.

math.PR↗

arXiv2026

Synchronization Strings over the Optimal Alphabet

Synchronization strings provide deterministic position labels for recovering coordinates after insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 215/216. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with a positive interval conditional min-entropy rate supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into an exponentially small probability of a near-complete common subsequence between adjacent intervals, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, synchronization circles, and synchronization within a class of extremal square-free words. Adding constraints on distant intervals gives a Las Vegas construction in expected O(n^2 log^3(n+2)) time.

cs.IT↗

arXiv2026

A q-analogue of graph independence polynomials with a group-theoretic interpretation

We define totally-isotropic polynomials of alternating matrix spaces over finite fields, by analogy with independence polynomials of graphs. Our main result shows that totally-isotropic polynomials of graphical alternating matrix spaces give rise to a natural q-analogue of graph independence polynomials. For p-groups of class 2 and exponent p, this family of polynomials over fields of order p can be naturally interpreted as enumerating their abelian subgroups containing the commutator subgroup according to the orders. With this interpretation, our main result has implications to graphical groups over finite fields, in the same spirit as the results in (Bull. Lond. Math. Soc., 2022) by Rossmann, who studied enumerating conjugacy classes of graphical groups over finite fields.

math.CO↗

arXiv2026

On the average-case complexity landscape for Tensor-Isomorphism-complete problems over finite fields

In Grochow and Qiao (SIAM J. Comput., 2021), the complexity class Tensor Isomorphism (TI) was introduced and isomorphism problems for groups, algebras, and polynomials were shown to be TI-complete. In this paper, we study average-case algorithms for several TI-complete problems over finite fields, including algebra isomorphism, matrix code conjugacy, and $4$-tensor isomorphism. Our main results are as follows. Over the finite field of order $q$, we devise (1) average-case polynomial-time algorithms for algebra isomorphism and matrix code conjugacy that succeed in a $1/Θ(q)$ fraction of inputs and (2) an average-case polynomial-time algorithm for the $4$-tensor isomorphism that succeeds in a $1/q^{Θ(1)}$ fraction of inputs. Prior to our work, algorithms for algebra isomorphism with rigorous average-case analyses ran in exponential time, albeit succeeding on a larger fraction of inputs (Li--Qiao, FOCS'17; Brooksbank--Li--Qiao--Wilson, ESA'20; Grochow--Qiao--Tang, STACS'21). These results reveal a finer landscape of the average-case complexities of TI-complete problems, providing guidance for cryptographic systems based on isomorphism problems. Our main technical contribution is to introduce the spectral properties of random matrices into algorithms for TI-complete problems. This leads to not only new algorithms but also new questions in random matrix theory over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (J. London Math. Soc., 1998) and the characteristic sum method of Gorodetsky and Rodgers (Trans. Amer. Math. Soc., 2021).

cs.CC↗

arXiv2026

Random tensor isomorphism under orthogonal and unitary actions

We study the problem of testing whether two tensors in $\mathbb{R}^\ell\otimes \mathbb{R}^m\otimes \mathbb{R}^n$ are isomorphic under the natural action of orthogonal groups $\textbf{O}(\ell, \mathbb{R})\times\textbf{O}(m, \mathbb{R})\times\textbf{O}(n, \mathbb{R})$, as well as the corresponding question over $\mathbb{C}$ and unitary groups. These problems naturally arise in several areas, including graph and tensor isomorphism (Grochow--Qiao, SIAM J. Comp. '21), scaling algorithms for orbit closure intersections (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), and quantum information (Liu--Li--Li--Qiao, Phys. Rev. Lett. '12). We study average-case algorithms for orthogonal and unitary tensor isomorphism, with one random tensor where each entry is sampled uniformly independently from a sub-Gaussian distribution, and the other arbitrary. For the algorithm design, we develop algorithmic ideas from the higher-order singular value approach into polynomial-time exact (algebraic) and approximate (numerical) algorithms with rigorous average-case analyses. Following (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), we present an algorithm for a gapped version of the orbit distance approximation problem. For the average-case analysis, we work from recent progress in random matrix theory on eigenvalue repulsion of sub-Gaussian Wishart matrices (Christoffersen--Luh--O'Rourke--Shearer and Han, arXiv '25) by extending their results from side lengths of Wishart matrices linearly related to polynomially related.

cs.CC↗

arXiv2025

On average orders of automorphism groups of bilinear maps over finite fields

Let $φ:V\times V\to W$ be a bilinear map of finite vector spaces $V$ and $W$ over a finite field $\mathbb{F}_q$. We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of $\mathrm{GL}(V)$ and $\mathrm{GL}(W)$, when $\dim(V)$ and $\dim(W)$ are linearly related. As motivations and applications of the results, we present almost tight upper bounds on the number of $p$-groups of Frattini class $2$ as first studied by Higman (Proc. Lond. Math. Soc., 1960). Such bounds lead to answers for some open questions by Blackburn, Neumann, and Venkataraman (Cambridge Tracts in Mathematics, 2007). Further applications include sampling matrix spaces with the trivial automorphism group, and asymptotic bounds on the number of isomorphism classes of finite cube-zero commutative algebras.

math.CO↗

arXiv2024

Canonical forms for matrix tuples in polynomial time

Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we present polynomial-time algorithms for computing canonical forms of matrix tuples over a finite field under these actions. Our algorithm builds upon new structural insights for matrix tuples, which can be viewed as a generalization of Schur's lemma for irreducible representations to general representations.

cs.DS↗

arXiv2024

On the complexity of isomorphism problems for tensors, groups, and polynomials III: actions by classical groups

We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. Such problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the decision problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problem over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d>3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

cs.CC↗

arXiv2024

On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications

Many isomorphism problems for tensors, groups, algebras, and polynomials were recently shown to be equivalent to one another under polynomial-time reductions, prompting the introduction of the complexity class TI (Grochow & Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow & Qiao (CCC '21) then gave moderately exponential-time search- and counting-to-decision reductions for a class of $p$-groups. A significant issue was that the reductions usually incurred a quadratic increase in the length of the tensors involved. When the tensors represent $p$-groups, this corresponds to an increase in the order of the group of the form $|G|^{Θ(\log |G|)}$, negating any asymptotic gains in the Cayley table model. In this paper, we present a new kind of tensor gadget that allows us to replace those quadratic-length reductions with linear-length ones, yielding the following consequences: 1. If Graph Isomorphism is in P, then testing equivalence of cubic forms in $n$ variables over $F_q$, and testing isomorphism of $n$-dimensional algebras over $F_q$, can both be solved in time $q^{O(n)}$, improving from the brute-force upper bound $q^{O(n^2)}$ for both of these. 2. Combined with the $|G|^{O((\log |G|)^{5/6})}$-time isomorphism-test for $p$-groups of class 2 and exponent $p$ (Sun, STOC '23), our reductions extend this runtime to $p$-groups of class $c$ and exponent $p$ where $c<p$, and yield algorithms in time $q^{O(n^{1.8}\cdot \log q)}$ for cubic form equivalence and algebra isomorphism. 3. Polynomial-time search- and counting-to-decision reduction for testing isomorphism of $p$-groups of class $2$ and exponent $p$ when Cayley tables are given. This answers questions of Arvind and Tóran (Bull. EATCS, 2005) for this group class, thought to be one of the hardest cases of Group Isomorphism.

cs.CC↗

arXiv2023

Turán and Ramsey problems for alternating multilinear maps

Guided by the connections between hypergraphs and exterior algebras, we study Turán and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lovász (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992). Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.

math.CO↗

arXiv2023

On linear-algebraic notions of expansion

A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of graph edge expansion. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.

math.CO↗

arXiv2022

Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

cs.DS↗

arXiv2022

Connections between graphs and matrix spaces

Given a bipartite graph $G$, the graphical matrix space $\mathcal{S}_G$ consists of matrices whose non-zero entries can only be at those positions corresponding to edges in $G$. Tutte (J. London Math. Soc., 1947), Edmonds (J. Res. Nat. Bur. Standards Sect. B, 1967) and Lovász (FCT, 1979) observed connections between perfect matchings in $G$ and full-rank matrices in $\mathcal{S}_G$. Dieudonné ({Arch. Math., 1948) proved a tight upper bound on the dimensions of those matrix spaces containing only singular matrices. The starting point of this paper is a simultaneous generalization of these two classical results: we show that the largest dimension over subspaces of $\mathcal{S}_G$ containing only singular matrices is equal to the maximum size over subgraphs of $G$ without perfect matchings, based on Meshulam's proof of Dieudonné's result (Quart. J. Math., 1985). Starting from this result, we go on to establish more connections between properties of graphs and matrix spaces. For example, we establish connections between acyclicity and nilpotency, between strong connectivity and irreducibility, and between isomorphism and conjugacy/congruence. For each connection, we study three types of correspondences, namely the basic correspondence, the inherited correspondence (for subgraphs and subspaces), and the induced correspondence (for induced subgraphs and restrictions). Some correspondences lead to intriguing generalizations of classical results, such as for Dieudonné's result mentioned above, and for a celebrated theorem of Gerstenhaber regarding the largest dimension of nil matrix spaces (Amer. J. Math., 1958). Finally, we show some implications of our results to quantum information and present open problems in computational complexity motivated by these results.

math.CO↗