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Zhi Yin

Publications and source records attributed to Zhi Yin.

At least 19 recordsLinked to original sources
arXiv2026

Well-posedness and stationary distribution of free stochastic differential equations

This paper studies free stochastic differential equations driven by free Brownian motion. Under local operator Lipschitz and Lyapunov-type conditions on the coefficients, we prove the global well-posedness of solutions in the noncommutative probability setting using free It\^o calculus. We further establish the existence and uniqueness of stationary solutions under appropriate dissipativity conditions. Our results extend classical theory to the free probability framework.

math.PR↗
arXiv2026

An improvement bound on a problem of Picasarri-Arrieta and Rambaud

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $C(k,\ell)$ is a digraph consisting of two internally vertex disjoint directed paths of lengths $k$ and $\ell$ with the same initial vertex and terminal vertex. Picasarri-Arrieta and Rambaud (European J. Combin., 2024) proved that for any $k\geq 2$, every digraph of minimum out-degree at least two and girth at least $8k-6$ contains a subdivision of $C(k,k)$. They also construct a family of digraphs showing that the girth cannot be reduced to $k-1$, and posed the problem of determining the minimum girth such that every digraph of minimum out-degree at least two contains a subdivision of $C(k,k)$. In this paper, we improve the lower bound on the girth from $8k-6$ to $4k+2$, and construct a family of digraphs in which every member has minimum out-degree two and girth $k$ but contains no subdivision of $C(k,k)$. Thus our results show that the girth in question lies between $k+1$ and $4k+2$.

math.CO↗
arXiv2026

Monochromatic triangle-tilings in dense graphs without large independent sets

Given two graphs $H$ and $G$, an $H$-tiling is a family of vertex-disjoint copies of $H$ in $G$. A perfect $H$-tiling covers all vertices of $G$. The Corradi-Hajnal theorem (1963) states that an $n$-vertex graph $G$ with minimum degree $\delta(G)\ge 2n/3$ contains a perfect triangle-tiling. For an $n$-vertex graph $G$ with independence number $\alpha(G)=o(n)$, Balogh, Molla and Sharifzadeh (Random Structures & Algorithms, 2016) showed that a minimum degree of $(\frac12+o(1))n$ forces a perfect triangle-tiling. In a 2-edge-colored graph, Balogh, Freschi, Treglown (European J. Combin. 2026) determined the (asymptotic) minimum degree threshold for forcing a strong or weak monochromatic triangle-tiling covering a prescribed proportion of the vertices: a strong tiling requires all triangles to be in the same color class, while a weak tiling only requires each triangle to be monochromatic. In this paper, we combine the conditions from these two lines of work and prove that every $2$-edge-colored $n$-vertex graph $G$ with $\alpha(G)=o(n)$ contains a weak monochromatic triangle-tiling $\Gamma$ of size \[ |\Gamma|\ge \begin{cases} 2\delta(G)-n-o(n), & \text{if }\frac12 n\le \delta(G)\le \frac35 n,\\[2mm] \delta(G)/3-o(n), & \text{if }\delta(G)>\frac35 n. \end{cases} \] Both bounds are asymptotically optimal. We use the degree form regularity lemma in our proof.

math.CO↗
arXiv2025

Benchmarking the Impact of Active Space Selection on the VQE Pipeline for Quantum Drug Discovery

Quantum computers promise scalable treatments of electronic structure, yet applying variational quantum eigensolvers (VQE) on realistic drug-like molecules remains constrained by the performance limitations of near-term quantum hardwares. A key strategy for addressing this challenge which effectively leverages current Noisy Intermediate-Scale Quantum (NISQ) hardwares yet remains under-benchmarked is active space selection. We introduce a benchmark that heuristically proposes criteria based on chemically grounded metrics to classify the suitability of a molecule for using quantum computing and then quantifies the impact of active space choices across the VQE pipeline for quantum drug discovery. The suite covers several representative drug-like molecules (e.g., lovastatin, oseltamivir, morphine) and uses chemically motivated active spaces. Our VQE evaluations employ both simulation and quantum processing unit (QPU) execution using unitary coupled-cluster with singles and doubles (UCCSD) and hardware-efficient ansatz (HEA). We adopt a more comprehensive evaluation, including chemistry metrics and architecture-centric metrics. For accuracy, we compare them with classical quantum chemistry methods. This work establishes the first systematic benchmark for active space driven VQE and lays the groundwork for future hardware-algorithm co-design studies in quantum drug discovery.

physics.chem-ph↗
arXiv2025

Global Fluctuations of Gaussian Elliptic Matrices

We introduce a spoke-arc decomposition of non-crossing annular pair partitions $NC_2(p,q)$ that records spoke type and orientation, isolates spoke-level contributions, and factorizes the dependence on the ellipticity parameter $\gamma$ into a spoke factor and arc weights. This yields closed-form descriptions of the limiting covariance of Gaussian elliptic matrices. As a corollary, we show that an independent family of Gaussian elliptic random matrices is asymptotically second-order free.

math.PR↗
arXiv2025

SMILES-Inspired Transfer Learning for Quantum Operators in Generative Quantum Eigensolver

Given the inherent limitations of traditional Variational Quantum Eigensolver(VQE) algorithms, the integration of deep generative models into hybrid quantum-classical frameworks, specifically the Generative Quantum Eigensolver(GQE), represents a promising innovative approach. However, taking the Unitary Coupled Cluster with Singles and Doubles(UCCSD) ansatz which is widely used in quantum chemistry as an example, different molecular systems require constructions of distinct quantum operators. Considering the similarity of different molecules, the construction of quantum operators utilizing the similarity can reduce the computational cost significantly. Inspired by the SMILES representation method in computational chemistry, we developed a text-based representation approach for UCCSD quantum operators by leveraging the inherent representational similarities between different molecular systems. This framework explores text pattern similarities in quantum operators and employs text similarity metrics to establish a transfer learning framework. Our approach with a naive baseline setting demonstrates knowledge transfer between different molecular systems for ground-state energy calculations within the GQE paradigm. This discovery offers significant benefits for hybrid quantum-classical computation of molecular ground-state energies, substantially reducing computational resource requirements.

physics.chem-ph↗
arXiv2025

Outlier eigenvalues for full rank deformed single ring random matrices

Let $A_n$ be an $n \times n$ deterministic matrix and $Σ_n$ be a deterministic non-negative matrix such that $A_n$ and $Σ_n$ converge in $*$-moments to operators $a$ and $Σ$ respectively in some $W^*$-probability space. We consider the full rank deformed model $A_n + U_n Σ_n V_n,$ where $U_n$ and $V_n$ are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of $A_n + U_nΣ_n V_n$ in two domains that are outside the support of the Brown measure of $a +u Σ$. We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When $A_n$ has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).

math.PR↗
arXiv2025

Stochastic theta methods for free stochastic differential equations

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assume that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz, we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $θ\in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler-Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler-Maruyama method.

math.NA↗
arXiv2025

Dirac-type condition for Hamilton-generated graphs

The cycle space $\mathcal{C}(G)$ of a graph $G$ is defined as the linear space spanned by all cycles in $G$. For an integer $k\ge 3$, let $\mathcal{C}_k (G)$ denote the subspace of $\mathcal{C}(G)$ generated by the cycles of length exactly $k$. A graph $G$ on $n$ vertices is called Hamilton-generated if $\mathcal{C}_n (G) = \mathcal{C}(G)$, meaning every cycle in $G$ is a symmetric difference of some Hamilton cycles of $G$. %A necessary condition for this property is that $n$ must be odd. Heinig (European J. Combin., 2014) showed that for any $σ>0$ and sufficiently large odd $n$, every $n$-vertex graph with minimum degree $(1+ σ)n/2$ is Hamilton-generated. He further posed the question that whether the minimum degree requirement could be lowered to the Dirac threshold $n/2$. Recent progress by Christoph, Nenadov, and Petrova~(arXiv:2402.01447) reduced the minimum degree condition to $n/2 + C$ for some large constant $C$. In this paper, we resolve Heinig's problem completely by proving that for sufficiently large odd $n$, every Hamilton-connected graph $G$ on $n$ vertices with minimum degree at least $(n-1)/2$ is Hamilton-generated. Moreover, this result is tight for the minimum degree and the Hamilton-connected condition. The proof relies on the parity-switcher technique introduced by Christoph, et al in their recent work, as well as a classification lemma that strengthens a previous result by Krivelevich, Lee, and Sudakov~(Trans. Amer. Math. Soc., 2014).

math.CO↗
arXiv2024

A Hybrid Quantum Computing Pipeline for Real World Drug Discovery

Quantum computing, with its superior computational capabilities compared to classical approaches, holds the potential to revolutionize numerous scientific domains, including pharmaceuticals. However, the application of quantum computing for drug discovery has primarily been limited to proof-of-concept studies, which often fail to capture the intricacies of real-world drug development challenges. In this study, we diverge from conventional investigations by developing \rev{a hybrid} quantum computing pipeline tailored to address genuine drug design problems. Our approach underscores the application of quantum computation in drug discovery and propels it towards more scalable system. We specifically construct our versatile quantum computing pipeline to address two critical tasks in drug discovery: the precise determination of Gibbs free energy profiles for prodrug activation involving covalent bond cleavage, and the accurate simulation of covalent bond interactions. This work serves as a pioneering effort in benchmarking quantum computing against veritable scenarios encountered in drug design, especially the covalent bonding issue present in both of the case studies, thereby transitioning from theoretical models to tangible applications. Our results demonstrate the potential of a quantum computing pipeline for integration into real world drug design workflows.

physics.chem-ph↗
arXiv2024

The Brown measure of a sum of two free random variables, one of which is triangular elliptic

The triangular elliptic operators are natural extensions of the elliptic deformation of circular operators. We obtain a Brown measure formula for the sum of a triangular elliptic operator $g_{_{α, β, γ}}$ with a random variable $x_0$, which is $*$-free from $g_{_{α, β, γ}}$ with amalgamation over certain unital subalgebra. Let $c_t$ be a circular operator. We prove that the Brown measure of $x_0 + g_{_{α, β, γ}}$ is the push-forward measure of the Brown measure of $x_0 + c_t$ by an explicitly defined map on $\mathbb{C}$ for some suitable $t$. We show that the Brown measure of $x_0+c_t$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{C}$ and its density is bounded by $1/(π{t})$. This work generalizes earlier results on the addition with a circular operator, semicircular operator, or elliptic operator to a larger class of operators. We extend operator-valued subordination functions, due to Biane and Voiculescu, to certain unbounded operators. This allows us to extend our results to unbounded operators.

math.OA↗
arXiv2023

Spectrum of Laplacian matrices associated with large random elliptic matrices

A Laplacian matrix is a square matrix whose row sums are zero. We study the limiting eigenvalue distribution of a Laplacian matrix formed by taking a random elliptic matrix and subtracting the diagonal matrix containing its row sums. Under some mild assumptions, we show that the empirical spectral distribution of the Laplacian matrix converges to a deterministic probability distribution as the size of the matrix tends to infinity. The limiting measure can be interpreted as the Brown measure of the sum of an elliptic operator and a freely independent normal operator with a Gaussian distribution.

math.PR↗
arXiv2023

The degree threshold for covering with all the connected $3$-graphs with $3$ edges

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with $δ_{i}(H)>c_i(n,F)$ has an $F$-covering. Falgas-Ravry, Markstöm and Zhao (Combin. Probab. Comput., 2021) asymptotically determined $c_1(n,K_{4}^{(3)-})$, where $K_{4}^{(3)-}$ is obtained by deleting an edge from the complete $3$-graph on $4$ vertices. Later, Tang, Ma and Hou (arXiv, 2022) asymptotically determined $c_1(n,C_{6}^{(3)})$, where $C_{6}^{(3)}$ is the linear triangle, i.e. $C_{6}^{(3)}=([6],\{123,345,561\})$. In this paper, we determine $c_1(n,F_5)$ asymptotically, where $F_5$ is the generalized triangle, i.e. $F_5=([5],\{123,124,345\})$. We also determine the exact values of $c_1(n,F)$, where $F$ is any connected $3$-graphs with $3$ edges and $F\notin\{K_4^{(3)-}, C_{6}^{(3)}, F_5\}$.

math.CO↗
arXiv2023

Limit distribution of partial transposition of block random matrices

It is well known that, under some assumptions, the limit distribution of random block matrices and their partial transposition converges to the distributions of random variables in some noncommutative probability space. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we are able to derive a new family of co-completely positive and k-positive maps by using the Wishart ensemble.

quant-ph↗
arXiv2022

The spectrum of local random Hamiltonians

The spectrum of a local random Hamiltonian can be represented generically by the so-called $ε$-free convolution of its local terms' probability distributions. We establish an isomorphism between the set of $ε$-noncrossing partitions and permutations to study its spectrum. Moreover, we derive some lower and upper bounds for the largest eigenvalue of the Hamiltonian.

math-ph↗
arXiv2021

Rainbow independent sets in graphs with maximum degree two

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $\mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019) conjectured that (i) $f_G(n,n-1)=n-1$ for all graphs $G\in\mathcal{D}(2)$ and (ii) $f_{C_t}(n,n)=n$ for $t\ge 2n+1$. Lv and Lu (2020) showed that the conjecture (ii) holds when $t=2n+1$. In this article, we show that the conjecture (ii) holds for $t\ge\frac{1}{3}n^2+\frac{44}{9}n$. Let $C_t$ be a cycle of length $t$ with vertices being arranged in a clockwise order. An ordered set $I=(a_1,a_2,\ldots,a_n)$ on $C_t$ is called a $2$-jump independent $n$-set of $C_t$ if $a_{i+1}-a_i=2\pmod{t}$ for any $1\le i\le n-1$. We also show that a collection of 2-jump independent $n$-sets $\mathcal{F}$ of $C_t$ with $|\mathcal{F}|=n$ admits a rainbow independent $n$-set, i.e. (ii) holds if we restrict $\mathcal{F}$ on the family of 2-jump independent $n$-sets. Moreover, we prove that if the conjecture (ii) holds, then (i) holds for all graphs $G\in\mathcal{D}(2)$ with $c_e(G)\le 4$, where $c_e(G)$ is the number of components of $G$ isomorphic to cycles of even lengths.

math.CO↗
arXiv2020

Finite free convolutions via Weingarten calculus

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain \emph{quadrature property}, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.

math.CO↗