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Ching-Jui Lai

Publications and source records attributed to Ching-Jui Lai.

15 recordsLinked to original sources
arXiv2026

On slope unstable Fano varieties

For Fano varieties, significant progress has been made recently in the study of $K$-stability, while the understanding of the weaker but more algebraic concept of $(-K)$-slope stability remains intricate. For instance, a conjecture attributed to Iskovskikh states that the tangent bundle of a Picard rank one Fano manifold is slope stable. Peternell-Wi\'sniewski and Hwang proved this conjecture up to dimension five in 1998, but Kanemitsu later disproved it in 2021. To address this gap in understanding, we present a method that aims to characterize the geometry associated with the maximal destabilizing sheaf of the tangent sheaf of a Fano variety. This approach utilizes modern advancements in the foliated minimal model program. In dimension two, our approach leads to a complete classification of $(-K)$-slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that $\mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{F}_1$ are the only $(-K)$-slope unstable nonsingular del Pezzo surfaces, recovering a classical result of Fahlaoui in 1989. We also uncover a phenomenon that does not occur for Fano manifolds: there exists a del Pezzo surface with type A singularities admitting a weak K\"ahler-Einstein metric, yet whose tangent sheaf is slope unstable.

math.AG↗
arXiv2025

On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two

We show that for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two ($\mathbb{Q}$-factorial with at worst terminal singularities), the anticanonical volume satisfies $-K_X^3\leq72$ except in one case, and the equality holds only if $X=\mathbb{P} (\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The approach in this article can serve as a general strategy to establish the optimal upper bound of $-K_X^3$ for any canonical Fano threefolds, where the described main result serves as the first step.

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arXiv2024

Hybrid Quantum-Classical Clustering for Preparing a Prior Distribution of Eigenspectrum

Determining the energy gap in a quantum many-body system is critical to understanding its behavior and is important in quantum chemistry and condensed matter physics. The challenge of determining the energy gap requires identifying both the excited and ground states of a system. In this work, we consider preparing the prior distribution and circuits for the eigenspectrum of time-independent Hamiltonians, which can benefit both classical and quantum algorithms for solving eigenvalue problems. The proposed algorithm unfolds in three strategic steps: Hamiltonian transformation, parameter representation, and classical clustering. These steps are underpinned by two key insights: the use of quantum circuits to approximate the ground state of transformed Hamiltonians and the analysis of parameter representation to distinguish between eigenvectors. The algorithm is showcased through applications to the 1D Heisenberg system and the LiH molecular system, highlighting its potential for both near-term quantum devices and fault-tolerant quantum devices. The paper also explores the scalability of the method and its performance across various settings, setting the stage for more resource-efficient quantum computations that are both accurate and fast. The findings presented here mark a new insight into hybrid algorithms, offering a pathway to overcoming current computational challenges.

quant-ph↗
arXiv2024

Molecular Ground State Simulation by Subspace Restriction and Hund's Rule

Simulation of molecular ground states on near-term quantum hardware is constrained by qubit availability and the cost of variational optimization. To address these challenges, the Subspace Restriction Scheme (SRS) is introduced as a mathematical framework that projects the molecular Hamiltonian onto a selected Fock subspace prior to qubit encoding. By enforcing molecular multiplicity and a generalized Hund's rule, the Multi-Hund Subspace (MHS) is constructed. This physically motivated restriction significantly reduces the effective Fock-space dimension, asymptotically saving $N$ qubits for a Hamiltonian of $M$ spatial orbitals and $N$ electrons. As a result, we successfully overcome classical memory bottlenecks and enable simulations of large systems, such as the $H_{22}$ chain, which requires 44 qubits under standard Jordan-Wigner (JW) encoding. While the strict pairing structure may limit accuracy in strongly correlated dissociation regimes, MHS effectively captures the essential low-energy physics of closed-shell molecules near equilibrium. In Variational Quantum Eigensolver (VQE) benchmarks, MHS enhances optimization behaviour and achieves high accuracy with a shallow ansatz. These findings demonstrate that physically motivated subspace restriction offers an effective approach to more resource-efficient quantum-chemistry simulations.

quant-ph↗
arXiv2023

The movable cone of Calabi--Yau threefolds in ruled Fano manifolds

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi--Yau threefold in a non-split $(n + 4)$-dimensional $\mathbb{P}^{n}$-ruled Fano manifold of index $n + 1$ and Picard number two. Moreover, all birational minimal models of such Calabi--Yau threefolds are found whose number is finite.

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arXiv2023

Implementation of Trained Factorization Machine Recommendation System on Quantum Annealer

Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. It requires a run-time of $O((N_m \log N_m)^2)$, where $N_m$ is the number of items in the dataset. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a faster than quadratic speedup in finding good user suggestions. We then demonstrate the aforementioned computational advantage on current NISQ hardware by experimenting with a real example on a D-Wave annealer.

quant-ph↗
arXiv2023

On character table of Clifford groups

Based on a presentation of $\mathcal{C}_n$ and the help of [GAP], we construct the character table of the Clifford group $\mathcal{C}_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in [HWW, WF] and reveal some new phenomena. We prove that when $n \geq 3$, (1) the trivial character is the only linear character for $\mathcal{C}_n$ and hence $\mathcal{C}_n$ equals to its commutator subgroup, (2) the $n$-qubit Pauli group $\mathcal{P}_n$ is the only proper non-trivial normal subgroup of $\mathcal{C}_n$, (3) the matrix representation $\mathcal{M}_{2^n}$ is a faithful representation for $\mathcal{C}_n$. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.

math.RT↗
arXiv2021

Exceptional collection of objects on some fake projective planes

The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of Galkin-Katzarkov-Mellit-Shinder in 2015. Our method shows that 30 fake projective planes support such a sequence, most of which are new. In particular, this provides many new H-phantom categories.

math.AG↗
arXiv2021

The movable cone of certain Calabi-Yau threefolds of Picard number two

We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism.

math.AG↗
arXiv2021

Examples of surfaces with canonical maps of maximal degree

It was shown by A. Beauville that if the canonical map $φ_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then ${\rm deg}(φ_{|K_M|})\leq 36$. The first example of a surface with canonical degree 36 was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group ${\rm Aut}(X)=C_7:C_3$ (the unique non-abelian group of order 21), the base locus of the canonical map is finite, and we verify that 35 of these surfaces have maximal canonical degree 36. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of C. Hacon and J.-X. Cai.

math.AG↗
arXiv2015

Bogomolov-Gieseker Type Inequality on Calabi-Yau and Fano 3-folds

We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langer's estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.

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arXiv2012

Bounding the volumes of singular Fano threefolds

Let $(X,Δ)$ be an $n$-dimensional $ε$-klt log $\QQ$-Fano pair. We give an upper bound for the volume ${\rm Vol}(-(K_X+Δ))=(-(K_X+Δ))^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $ρ(X)=1$. This bound is essentially sharp for $n=2$. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of $ε$-klt log $\QQ$-Fano varieties of a given dimension $n$.

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arXiv2010

Varieties fibered by good minimal models

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

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