arXiv subjects

Yota Maeda

Publications and source records attributed to Yota Maeda.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Algorithms and Hardness for Point-Count Approximation over Finite Fields

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

quant-ph↗

arXiv2026

Random Access Codes: Explicit Constructions, Optimality, and Classical-Quantum Gaps

A random access code (RAC) encodes an $L$-bit string into a $k$-bit message, $L>k$, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses $k$ qubits instead. We give a geometric characterization of optimal classical $(L,k)$-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing $2^k$ representatives in $\{0,1\}^L$, while the worst-case criterion is reduced to a minimax problem over $2^k$ points in $[0,1]^L$ with a distance-like objective. This framework proves optimality for several parameter families, with many optimal constructions arising from standard infinite families of binary linear codes. It also yields two explicit classical--quantum separations. First, for every $L>1$, we construct a $(L,1)$-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family $(2^k-1,k)$, we prove worst-case optimality of a classical RAC and construct a QRAC with strictly larger worst-case success probability. For the family $(L,L-1)$, the framework identifies a classical RAC that is average-case optimal and, under a stated conjecture, also worst-case optimal. The same viewpoint further recovers explicit $(L,L-1)$-QRACs attaining a previously conjectured upper-bound value.

quant-ph↗

arXiv2025

The Kodaira dimension of even-dimensional ball quotients

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for an odd discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\mathrm{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.

math.AG↗

arXiv2025

Remarks on two problems by Hassett

One of the ultimate goals of the Hassett-Keel program is the determination of the log canonical models of the moduli spaces of pointed rational curves $\overline{M}_{0,n}$. In this paper, we study log canonical models of $\overline{M}_{0,5}$ with \textit{asymmetric} boundary divisors. Our results generalize previous work by Alexeev-Swinarski, Fedorchuk-Smyth, Kiem-Moon and Simpson for the first non-trivial case, namely $n=5$. We prove that all moduli spaces of weighted pointed rational curves $\overline{M}_{0,A}$ arise as log canonical models of $\overline{M}_{0,5}$ for suitable choices of boundary coefficients, thereby also recovering a theorem of Fedorchuk and Moon. In addition, we relate these moduli spaces to Deligne-Mostow ball quotients. We further study log canonical models of the moduli spaces $\overline{M}_{0,n\cdot (1/k)}$ with symmetric weight, which differ from $\overline{M}_{0,n}$. The case $n=5$ can be viewed as an explicit guiding example in a very general program and the paper can thus also serve as an expository introduction.

math.AG↗

arXiv2025

Finiteness of Free Algebras of Modular Forms on Unitary Groups

Classical results on the classification of reflections in an arithmetic subgroup $\Gamma$ imply that if the graded algebra of modular forms $M_*(\Gamma)$ is freely generated, then $\Gamma$ must be an arithmetic subgroup of either the orthogonal group $\operatorname{O}^+(2,n)$ or the unitary group $\operatorname{U}(1,n)$. Vinberg and Schwarzman showed that in the orthogonal case, if $n>10$, then it is never free. In this paper, we investigate the remaining unitary case and prove that, up to scaling, there are only finitely many isometry classes of Hermitian lattices of signature $(1, n)$ with $n > 2$ over imaginary quadratic fields with odd discriminant that admit a free algebra of modular forms. In particular, when $n>99$ (except over $\mathbb{Q}(\sqrt{-3})$, where we require $n > 154$), the graded algebra $M_*(\Gamma)$ is never free for any arithmetic subgroup $\Gamma<\operatorname{U}(1,n)$, thereby partially confirming a conjecture by Wang and Williams. As a byproduct, we also establish a finiteness result for reflective modular forms. In the course of this proof, we derive a formula for the covolume of an arithmetic subgroup of a special unitary group, presented as the stabiliser of a Hermitian lattice, which generalises Prasad's volume formula for principal arithmetic subgroups in the case of special unitary groups.

math.NT↗

arXiv2025

The Universe of Deligne-Mostow Varieties

Deligne and Mostow investigated period maps on the configuration spaces $M_{0,n}$ of $n$ ordered points on $\mathbb{P}^1$. The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered $n$-tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log $K$-equivalent to the unique toroidal compactification.

math.AG↗

arXiv2025

Geometric Generality of Transformer-Based Gr\"obner Basis Computation

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr\"obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr\"obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr\"obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

cs.LG↗

arXiv2025

Quantum-enhanced causal discovery for a small number of samples

The discovery of causal relations from observed data has attracted significant interest from disciplines such as economics, social sciences, and biology. In practical applications, considerable knowledge of the underlying systems is often unavailable, and real data are usually associated with nonlinear causal structures, which makes the direct use of most conventional causality analysis methods difficult. This study proposes a novel quantum Peter-Clark (qPC) algorithm for causal discovery that does not require any assumptions about the underlying model structures. Based on conditional independence tests in a class of reproducing kernel Hilbert spaces characterized by quantum circuits, the proposed algorithm can explore causal relations from the observed data drawn from arbitrary distributions. We conducted systematic experiments on fundamental graphs of causal structures, demonstrating that the qPC algorithm exhibits better performance, particularly with smaller sample sizes compared to its classical counterpart. Furthermore, we proposed a novel optimization approach based on Kernel Target Alignment (KTA) for determining hyperparameters of quantum kernels. This method effectively reduced the risk of false positives in causal discovery, enabling more reliable inference. Our theoretical and experimental results demonstrate that the quantum algorithm can empower classical algorithms for accurate inference in causal discovery, supporting them in regimes where classical algorithms typically fail. In addition, the effectiveness of this method was validated using the datasets on Boston housing prices, heart disease, and biological signaling systems as real-world applications. These findings highlight the potential of quantum-based causal discovery methods in addressing practical challenges, particularly in small-sample scenarios, where traditional approaches have shown significant limitations.

quant-ph↗

arXiv2024

Statistical inference for quantum singular models

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.

quant-ph↗

arXiv2024

MambaPEFT: Exploring Parameter-Efficient Fine-Tuning for Mamba

An ecosystem of Transformer-based models has been established by building large models with extensive data. Parameter-efficient fine-tuning (PEFT) is a crucial technology for deploying these models to downstream tasks with minimal cost while achieving effective performance. Recently, Mamba, a State Space Model (SSM)-based model, has attracted attention as a potential alternative to Transformers. While many large-scale Mamba-based models have been proposed, efficiently adapting pre-trained Mamba-based models to downstream tasks remains unexplored. In this paper, we conduct an exploratory analysis of PEFT methods for Mamba. We investigate the effectiveness of existing PEFT methods for Transformers when applied to Mamba. We also modify these methods to better align with the Mamba architecture. Additionally, we propose new Mamba-specific PEFT methods that leverage the distinctive structure of Mamba. Our experiments indicate that PEFT performs more effectively for Mamba than Transformers. Lastly, we demonstrate how to effectively combine multiple PEFT methods and provide a framework that outperforms previous works. To ensure reproducibility, we will release the code after publication.

cs.CL↗

arXiv2024

Compactifications of the Eisenstein ancestral Deligne-Mostow variety

All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In a previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers. This variety is related to the moduli space of unordered 12 points on $\mathbb{P}^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza.

math.AG↗

arXiv2023

Estimation of mutual information via quantum kernel method

Recently, the importance of analysing data and collecting valuable insight efficiently has been increasing in various fields. Estimating mutual information (MI) plays a critical role to investigate the relationship among multiple random variables with a nonlinear correlation. Particularly, the task to determine whether they are independent or not is called the independence test, whose core subroutine is estimating MI from given data. It is a fundamental tool in statistics and data analysis that can be applied in a wide range of application such as hypothesis testing, causal discovery and more. In this paper, we propose a method for estimating mutual information using the quantum kernel. We investigate the performance under various problem settings, such as different sample size or the shape of the probability distribution. As a result, the quantum kernel method showed higher performance than the classical one under the situation that the number of samples is small, the variance is large or the variables posses highly non-linear relationships. We discuss this behavior in terms of the central limit theorem and the structure of the corresponding quantum reproducing kernel Hilbert space.

quant-ph↗

arXiv2022

Revisiting the moduli space of 8 points on $\mathbb{P}^1$

The moduli space of $8$ points on $\mathbb{P}^1$, a so-called ancestral Deligne-Mostow space, is, by work of Kond\={o}, also a moduli space of K3 surfaces. We prove that the Deligne-Mostow isomorphism does not lift to a morphism between the Kirwan blow-up of the GIT quotient and the unique toroidal compactification of the corresponding ball quotient. Moreover, we show that these spaces are not $K$-equivalent, even though they are natural blow-ups at the unique cusps and have the same cohomology. This is analogous to the work of Casalaina-Martin-Grushevsky-Hulek-Laza on the moduli space of cubic surfaces. The moduli spaces of ordinary stable maps, that is, the Fulton-MacPherson compactification of the configuration space of points on $\mathbb{P}^1$, play an important role in the proof. We further relate our computations to new developments in the minimal model program and recent work of Odaka. We briefly discuss other cases of moduli space of points on $\mathbb{P}^1$ where a similar behaviour can be observed, hinting at a more general, but not yet fully understood phenomenon.

math.AG↗

arXiv2022

The singularities and Kodaira dimension of unitary Shimura varieties

The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli spaces of polarized K3 surfaces. They proved that they have canonical singularities and are of general type if the polarization degree is sufficiently large. In this paper, we work on similar problems for unitary Shimura varieties. We show that they have canonical singularities if $n > 4$. As an application, we show that certain unitary Shimura varieties associated with Hermitian forms over the rings of integers of $\mathbb{Q}(\sqrt{-1})$, $\mathbb{Q}(\sqrt{-3})$ are of general type. We use modular forms of low weight vanishing on ramification divisors, which are the restrictions of the quasi-pullbacks of the Borcherds form $Φ_{12}$.

math.NT↗

arXiv2022

Reflective obstructions of unitary modular varieties

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than $138$. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula.

math.AG↗

arXiv2021

Irregular cusps of ball quotients

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.

math.AG↗

arXiv2021

Fano Shimura varieties with mostly branched cusps

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.

math.AG↗

arXiv2021

Modularity of special cycles on unitary Shimura varieties over CM-fields

We study the modularity of the generating series of special cycles on unitary Shimura varieties over CM-fields of degree $2d$ associated with a Hermitian form in $n+1$ variables whose signature is $(n,1)$ at $e$ real places and $(n+1,0)$ at the remaining $d-e$ real places for $1\leq e 1$, we prove that the generating series of special cycles of codimension $er$ in the Chow group is a Hermitian modular form of weight $n+1$ and genus $r$, assuming the Beilinson-Bloch conjecture with respect to orthogonal Shimura varieties. Our result is a generalization of $\textit{Kudla's modularity conjecture}$, solved by Liu unconditionally when $e=1$.

math.NT↗