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Chenghua Liu

Publications and source records attributed to Chenghua Liu.

18 recordsLinked to original sources
arXiv2026

Hidden Circuits and Exact Counting in Ordered Graphs

We prove that counting perfect matchings is $\#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and Müller (JACM 2017), complementing their rapid-mixing theorem. Inspired by quantum circuits, our reductions implement a circuit simulation using globally coupled matching-transfer operators. The key construction is an exact projection, implemented by a polynomial-length sequence of normalized transfers, that restores tensor-product locality and makes encoded gates composable. Interpolation-based cancellation then reduces circuit evaluation to unweighted perfect-matching counts in all three classes. We also place Dyer and Müller's class QChains within the distance-hereditary graphs and give an $O(n^2)$-arithmetic-operation counting algorithm for the latter, improving the $O(n^4)$ bound obtainable from Curticapean and Marx (SODA 2016). Together with prior results, these advances complete the exact-counting classification of the graph classes in Dyer and Müller's diagram (SIDMA 2019).

cs.CC↗
arXiv2026

Quantum Behaviors Are Not Semialgebraic

The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.

quant-ph↗
arXiv2026

Flow-Matched Motion Priors: Online Optimal-Transport Rewards for Imitation Learning

Learning a motion prior requires a reward that guides a policy from its current behavior toward demonstrated motion. Adversarial Motion Priors (AMP) provide such a reward with a discriminator. However, adversarial objectives can become uninformative when policy and expert supports are far apart. A naive use of optimal transport (OT) averages matched expert successors into a barycentric target. Averaging across gait phases can weaken the target's joint motion. We introduce Flow-Matched Motion Priors (FMP), an online scalar reward learned from paths connecting current rollout histories to an expert motion bank. Entropic OT supplies the coupling. Before each policy update, we train a neural potential with flow matching (FM) along the rollout-to-expert paths, endpoint-gradient supervision, and relative-value calibration. The actor receives only physical observations and the reward remains a scalar, as in AMP. Controlled reward-model experiments show substantially better generalization beyond the fitting rollout than value-only or endpoint-only fitting. On Unitree G1, matched 50-million-transition experiments compare FMP with AMP, a barycentric OT reward, and nested ablations under demonstration and fixed-pose initialization. FMP produces stable forward walking at 0.727 m/s from demonstration resets and 0.338 m/s from a fixed default pose. In the fixed-pose condition, it incurs 129 falls versus 243 for the endpoint-only control. Against a static score-gradient teacher, dynamic FM reduces score-increment error at interpolation fractions 0.25 and 0.50 while using 29% less offline fitting time.

cs.RO↗
arXiv2026

Optimal Covariance Inflation under Gaussian Tilts

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $μ_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(μ_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=Θ(n^{2/5})$, closing the gap between the known $Ω(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $Ω(n^{2/5})$.

math.PR↗
arXiv2026

Bounded Relative Boundary Implies Narrow DNF Approximation

Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing that, for $0<p\leq 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$. A separate high-bias argument completes the proof for all $p\in(0,1)$. Our proof builds on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)). Tracking Hatami's construction isolates an adaptive representation with increasing local activations and dimension-free arity and multiplicity-counted load bounds. Our main new ingredient is a bias-matched randomized shifting procedure that converts the pseudo-junta approximator into an increasing function while retaining exact measurability with respect to a controlled forced refinement of its adaptive representation. From the resulting monotone adaptive representation, we extract positive certificates and truncate them to obtain the required narrow DNF.

cs.CC↗
arXiv2026

Lower Bounds for Domination-Type Problems Parameterized by Rank-Width

For graphs of rank-width \(w\), the algorithms of Bui-Xuan, Telle, and Vatshelle (\emph{Theor. Comput. Sci.}, 2013) for fixed finite/cofinite \((σ,ρ)\)-problems and of Bergougnoux and Kanté (\emph{SIAM J. Discrete Math.}, 2021) for Connected Dominating Set run in \(2^{O(w^2)}n^{O(1)}\) time. Bergougnoux, Korhonen, and Nederlof (STACS 2023) proved a matching lower bound under the Exponential Time Hypothesis (ETH) for \emph{Weighted} Dominating Set, but left the unweighted problem open. We prove that, unless ETH fails, Dominating Set admits no \(2^{o(w^2)}n^{O(1)}\)-time algorithm, even on split graphs and, separately, on bipartite graphs of diameter at most four, and even with a rank-decomposition or witnessing vertex order supplied. The proof replaces the earlier weights by a two-guard gadget and uses a low-rank equality gadget to carry \(k^2\) assignment bits through cuts of rank \(O(k)\). The construction also gives the same lower bound for Independent, Connected, and Total Dominating Set on restricted graph classes and applies to a broad family of \((σ,ρ)\)-set problems. This family includes cases in which \(σ\) is neither finite nor cofinite and contains the entire nontrivial cofinite--cofinite minimization regime. Every solution within the target budget has target size and corresponds bijectively to a satisfying assignment. Under the counting Exponential Time Hypothesis (\(\#\mathrm{ETH}\)), the same bounds therefore hold for counting solutions of size at most or exactly the target. Together with the known algorithms, our results show that the quadratic dependence on the rank-width \(w\) is optimal up to constant factors in the exponent for the classical problems above and throughout the covered finite/cofinite regime.

cs.CC↗
arXiv2026

From Block Orthogonality to Decidability in Complex-Weighted Counting CSP

In a landmark JACM paper recognized with the 2021 G{ö}del Prize, Cai and Chen established a complete complexity dichotomy for counting CSPs over arbitrary finite domains with algebraic complex weights. Its polynomial-time side is characterized by three conditions---Block Orthogonality, Type Partition, and preservation by a common Mal'tsev operation---quantified over the countably infinite family $W_{\mathcal{F}}$ generated from arbitrary $\#\mathrm{CSP}(\mathcal{F})$ instances by partial summation. They asked whether these infinitary conditions are decidable from the finite language $\mathcal{F}$ alone---equivalently, whether the polynomial-time side of this complete fixed-language classification is uniformly recognizable. We settle this problem by giving, for every nonempty finite domain $D$ and every finite exactly encoded algebraic-complex language $\mathcal{F}$, a total exact algorithm that decides all three conditions on the full unbounded family $W_{\mathcal{F}}$. Beyond decidability, we prove that Block Orthogonality alone forces both Type Partition and the existence of a single Mal'tsev operation preserving all generated support and row-equivalence relations. Thus the three-condition characterization collapses to Block Orthogonality, and the finite input $(D,\mathcal{F})$ determines which side of the dichotomy applies. The same framework decides the corresponding conditions in the dichotomy theorem for degree-multiple counting CSP proved by Lin.

cs.CC↗
arXiv2026

Quantum Speedups for Log-Concave Sampling from Local Structure

For a convex function $f \colon \mathbb{R}^d \to \mathbb{R}$, the problem of sampling from a distribution proportional to $e^{-f(x)}$ is called log-concave sampling. In many practical scenarios, the function $f(x)$ turns out to admit a local decomposition $f(x) = \sum_{a=1}^R ψ_a(x_{S_a})$. In this paper, we consider log-concave sampling using local queries, i.e., evaluation and gradient queries to each clause $ψ_a(\cdot)$, which can be computationally much cheaper than the queries to $f(x)$ itself. We show that if each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using $\widetilde{O}(\sqrtκd)$ local queries, where $κ$ is the condition number. This improves the prior best classical result $\widetilde{O}(κd)$ due to Ascolani, Lavenant, and Zanella (Ann. Probab. 2026) and the quantum result $\widetilde{O}(\sqrtκ d^2)$ implied by Childs et al. (NeurIPS 2022). Our quantum sampler applies to a broad class of locally structured models from statistical computing and machine learning, with representative examples including Gaussian Markov random fields, finite-element latent Gaussian models, and sparse generalized linear models. These results demonstrate that local structure is not merely an implementation detail, but a quantum algorithmic resource for high-dimensional sampling.

quant-ph↗
arXiv2026

A Correlation-Gap Bound for Nonlinear Gaussian PCA

Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Loève (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.

cs.DS↗
arXiv2026

Rank-Independent Spectral Hypergraph Sparsification via Global-Dictionary Chaining

We show that every weighted hypergraph on $n$ vertices admits a spectral $\varepsilon$-sparsifier with $O(n\log n/\varepsilon^2)$ hyperedges, strengthening the independent STOC 2023 works of Lee and Jambulapati--Liu--Sidford by removing their rank dependence and answering Lee's open question on whether this loss is inherent. The key idea is global-dictionary chaining: after choosing clique edge weights with balanced effective resistances, every hyperedge seminorm is Lipschitz with respect to the same global-dictionary norm generated by normalized vertex-pair directions; the local rank complexity is thereby replaced by the Gaussian width of this common dictionary. Since these STOC 2023 works have become standard analytic primitives across a broad subsequent literature on spectral hypergraph sparsification and its variants, our rank-independent theorem sharpens many later guarantees that inherit their sampling bounds.

cs.DS↗
arXiv2026

Quantum Communication Lower Bounds for Search Problems via Matrix Discrepancy

We study one-way quantum communication lower bounds for search problems. Unlike decision problems, search problems can have many valid outputs, which pose a fundamental barrier to standard quantum lower-bound techniques. We overcome this by developing a novel method based on matrix discrepancy, which allows us to bound the output measurements of a quantum protocol jointly. As applications of our method, we establish the first tight quantum lower bounds for two fundamental search problems in some natural parameter regimes: collision finding and triangle finding. For collision finding, we prove a tight $Ω(N^{1/4})$ one-way quantum communication lower bound. Previously, the best-known quantum communication lower bound for collision finding was $Ω(N^{1/12})$ due to Göös and Jain (RANDOM 2022), and no stronger bound was known even under the one-way restriction. For triangle finding in graph streams, we prove a one-pass quantum streaming space lower bound of $Ω\left(\sqrt{Δ_V}\right)$ for graphs with $m$ edges, $Θ(m)$ triangles, and constant $Δ_E$, where $Δ_V$ and $Δ_E$ denote the maximum number of triangles sharing a common vertex and edge, respectively, under the condition that $1\le Δ_V\le m^{2/3}$. This constitutes the first nontrivial quantum space lower bound in this regime, matching the classical upper bound of Jayaram and Kallaugher (RANDOM 2021) up to logarithmic factors. Notably, our method also recovers the classical lower bound of Kallaugher and Price (SODA 2017) through an entirely different argument, avoiding their Boolean-Hidden-Matching reduction that breaks down for quantum protocols.

quant-ph↗
arXiv2026

Lévy-Khintchine Structure Enables Fast-Forwardable Lindbladian Simulation

Simulation of open quantum systems is an area of active research in quantum algorithms. In this work, we revisit the connection between Markovian open-system dynamics and averages of Hamiltonian real-time evolutions, which we refer to as Hamiltonian twirling channels. By applying the Lévy-Khintchine representation theorem, we clarify when and how a dissipative dynamics can be realized using Hamiltonian twirling channels. Guided by the general theory, we explore Hamiltonian twirling with Gaussian, compound Poisson and symmetric stable distributions and their algorithmic implications. These give wide classes of Lindbladians that can be simulated in $Θ(t^{1/α})$ Hamiltonian simulation time without any extra ancilla or other quantum gates for $1\le α\le 2$. Moreover, we prove that these time complexities are asymptotically optimal using an information theoretic approach, which, to the best of our knowledge, is the first result of lower bounds on fast-forwarding simulation algorithms.

quant-ph↗
arXiv2025

Accelerating Regression Tasks with Quantum Algorithms

Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks -- including linear and multiple regression, Lasso, Ridge, Huber, $\ell_p$-, and $δ_p$-type regressions -- achieving up to a quadratic improvement in the number of samples $m$ over the best classical algorithms. This speedup is achieved by extending the recent classical breakthrough of Jambulapati et al. (STOC'24) using several quantum techniques, including quantum leverage score approximation (Apers &Gribling, 2024) and the preparation of many copies of a quantum state (Hamoudi, 2022). For problems of dimension $n$, sparsity $r < n$, and error parameter $ε$, our algorithm solves the problem in $\widetilde{O}(r\sqrt{mn}/ε+ \mathrm{poly}(n,1/ε))$ quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.

quant-ph↗
arXiv2025

Quantum Speedup for Hypergraph Sparsification

Graph sparsification serves as a foundation for many algorithms, such as approximation algorithms for graph cuts and Laplacian system solvers. As its natural generalization, hypergraph sparsification has recently gained increasing attention, with broad applications in graph machine learning and other areas. In this work, we propose the first quantum algorithm for hypergraph sparsification, addressing an open problem proposed by Apers and de Wolf (FOCS'20). For a weighted hypergraph with $n$ vertices, $m$ hyperedges, and rank $r$, our algorithm outputs a near-linear size $\varepsilon$-spectral sparsifier in time $\widetilde O(r\sqrt{mn}/\varepsilon)$. This algorithm matches the quantum lower bound for constant $r$ and demonstrates quantum speedup when compared with the state-of-the-art $\widetilde O(mr)$-time classical algorithm. As applications, our algorithm implies quantum speedups for computing hypergraph cut sparsifiers, approximating hypergraph mincuts and hypergraph $s$-$t$ mincuts.

quant-ph↗
arXiv2025

Quantum Speedup for Sampling Random Spanning Trees

We present a quantum algorithm for sampling random spanning trees from a weighted graph in $\widetilde{O}(\sqrt{mn})$ time, where $n$ and $m$ denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in $\widetilde{O}(m)$ time. The approach carefully combines, on one hand, a classical method based on ``large-step'' random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi's multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.

quant-ph↗
arXiv2023

Framework for Quality Evaluation of Smart Roadside Infrastructure Sensors for Automated Driving Applications

The use of smart roadside infrastructure sensors is highly relevant for future applications of connected and automated vehicles. External sensor technology in the form of intelligent transportation system stations (ITS-Ss) can provide safety-critical real-time information about road users in the form of a digital twin. The choice of sensor setups has a major influence on the downstream function as well as the data quality. To date, there is insufficient research on which sensor setups result in which levels of ITS-S data quality. We present a novel approach to perform detailed quality assessment for smart roadside infrastructure sensors. Our framework is multimodal across different sensor types and is evaluated on the DAIR-V2X dataset. We analyze the composition of different lidar and camera sensors and assess them in terms of accuracy, latency, and reliability. The evaluations show that the framework can be used reliably for several future ITS-S applications.

cs.CV↗
arXiv2021

Stationary Diffusion State Neural Estimation for Multiview Clustering

Although many graph-based clustering methods attempt to model the stationary diffusion state in their objectives, their performance limits to using a predefined graph. We argue that the estimation of the stationary diffusion state can be achieved by gradient descent over neural networks. We specifically design the Stationary Diffusion State Neural Estimation (SDSNE) to exploit multiview structural graph information for co-supervised learning. We explore how to design a graph neural network specially for unsupervised multiview learning and integrate multiple graphs into a unified consensus graph by a shared self-attentional module. The view-shared self-attentional module utilizes the graph structure to learn a view-consistent global graph. Meanwhile, instead of using auto-encoder in most unsupervised learning graph neural networks, SDSNE uses a co-supervised strategy with structure information to supervise the model learning. The co-supervised strategy as the loss function guides SDSNE in achieving the stationary state. With the help of the loss and the self-attentional module, we learn to obtain a graph in which nodes in each connected component fully connect by the same weight. Experiments on several multiview datasets demonstrate effectiveness of SDSNE in terms of six clustering evaluation metrics.

cs.LG↗