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The claw problem is central in the fields of theoretical computer science as well as cryptography. The optimal quantum query complexity of the problem is known to be $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ for input functions $f\colon [F]\to Z$ and $g\colon [G]\to Z$. However, the lower bound was proved when the range $Z$ is sufficiently large (i.e., $|{Z}|=\Omega(FG)$). The current paper proves the lower bound holds even for every smaller range $Z$ with $|{Z}|\ge F+G$. This implies that $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ is tight for every such range. In addition, the lower bound $\Omega\left(\sqrt{G}+F^{1/3}G^{1/6}M^{1/6}\right)$ is provided for even smaller range $Z=[M]$ with every $M\in [2,F+G]$ by reducing the claw problem for $|{Z}|= F+G$. The proof technique is general enough to apply to any $k$-symmetric property (e.g., the $k$-claw problem), i.e., the Boolean function $\Phi$ on the set of $k$ functions with different-size domains and a common range such that $\Phi$ is invariant under the permutations over each domain and the permutations over the range. More concretely, it generalizes Ambainis's argument [Theory of Computing, 1(1):37-46] to the multiple-function case by using the notion of multisymmetric polynomials., Comment: Title page + 24 pages

We tackle the challenge of rapidly adapting an agent's behavior to solve spatiotemporally continuous problems in novel settings. Animals exhibit extraordinary abilities to adapt to new contexts, a capacity unmatched by artificial systems. Instead of focusing on generalization through deep reinforcement learning, we propose viewing behavior as the physical manifestation of a search procedure, where robust problem-solving emerges from an exhaustive search across all possible behaviors. Surprisingly, this can be done efficiently using online modification of a cognitive graph that guides action, challenging the predominant view that exhaustive search in continuous spaces is impractical. We describe an algorithm that implicitly enumerates behaviors by regulating the tight feedback loop between execution of behaviors and mutation of the graph, and provide a neural implementation based on Hebbian learning and a novel high-dimensional harmonic representation inspired by entorhinal cortex. By framing behavior as search, we provide a mathematically simple and biologically plausible model for real-time behavioral adaptation, successfully solving a variety of continuous state-space navigation problems. This framework not only offers a flexible neural substrate for other applications but also presents a powerful paradigm for understanding adaptive behavior. Our results suggest potential advancements in developmental learning and unsupervised skill acquisition, paving the way for autonomous robots to master complex skills in data-sparse environments demanding flexibility., Comment: 26 pages, 5 figures

We develop an Effective Field Theory approach for jet observables in heavy-ion collisions, where the jet is treated as an open quantum system interacting with a hot and dense QCD medium. Within this framework, we derive a novel factorization formula for inclusive jet production, expressed as a series expansion with an increasing number of radiating subjet functions that encode forward scattering with the Quark-Gluon Plasma, convolved with perturbative matching coefficients. This work provides a systematic framework for computing jet observables at higher order and understanding their non-perturbative aspects, paving the way for future applications in heavy-ion phenomenology., Comment: 8 pages, 2 figures

We investigate the geometrical structure of multiply enhanced codimension-two singularities in the $SU(5)$ model of six-dimensional F-theory, where the rank of the singularity increases by two or more. We perform blow-up processes for the enhancement $SU(5)\rightarrow G'$, where $G'=E_6$, $E_7$ or $E_8$, to examine whether a sufficient set of exceptional curves emerge that can explain the charged matter generation predicted from anomaly cancelation. We first apply one of the six Esole-Yau small resolutions to the multiply enhanced singularities, but it turns out that the proper transform of the threefold equation does not reflect changes in the singularity or how the generic codimension-two singularities gather there. We then use a(n) (apparently) different way of small resolutions than Esole-Yau to find that, except for the cases of $G'= E_6$ and special cases of $E_7$, either (1) the resolution only yields exceptional curves that are insufficient to cancel the anomaly, or (2) there arises a type of singularity that is neither a conifold nor a generalized conifold singularity. Finally, we revisit the Esole-Yau small resolution and show that the change of the way of small resolutions amounts to simply exchanging the proper transform and the constraint condition, and under this exchange the two ways of small resolutions are completely equivalent., Comment: 43 pages, 7 figures

Tau leptons serve as an important tool for studying the production of Higgs and electroweak bosons, both within and beyond the Standard Model of particle physics. Accurate reconstruction and identification of hadronically decaying tau leptons is a crucial task for current and future high energy physics experiments. Given the advances in jet tagging, we demonstrate how tau lepton reconstruction can be decomposed into tau identification, kinematic reconstruction, and decay mode classification in a multi-task machine learning setup. Based on an electron-positron collision dataset with full detector simulation and reconstruction, we show that common jet tagging architectures can be effectively used for these sub-tasks. We achieve comparable momentum resolutions of 2-3% with all the tested models, while the precision of reconstructing individual decay modes is between 80-95%. We find ParticleTransformer to be the best-performing approach, significantly outperforming the heuristic baseline. This paper also serves as an introduction to a new publicly available $\mathtt{Fu}\tau\mathtt{ure}$ dataset for the development of tau reconstruction algorithms. This allows to further study the resilience of ML models to domain shifts and the efficient use of foundation models for such tasks.

The study of interplay between the geometric nature of Bloch electrons and transverse responses under strong field offers new opportunities for optoelectronic applications. Here, we present a comprehensive study of the strong-field response of Weyl Dirac nodes in bilayer T'-WTe2 using time-dependent first-principles formalism. The electron dynamics is explored focusing on the mid-infrared frequency, ranging from the perturbative to nonperturbative regime. In the nonperturbative regime, the high-harmonic generation (HHG) spectra under a strong field clearly exhibit a plateau and energy cutoffs for both longitudinal and anomalous Hall (transverse) currents, with the latter being due to the large interband Berry curvature of the Weyl-Dirac semimetal. For the longitudinal harmonics, the intraband contributions increase with intensity, resulting in a complex interplay between interband polarization and intraband motions. Remarkably, if we take a comprehensive all-band perspective enabled by time-dependent density functional calculations, the anomalous Hall responses are purely attributed to the interband processes, even in the nonperturbative regime, thus Hall HHG can be crucial to understand the carrier dynamics. Our findings suggest that HHG associated with the ultrafast strong-field driven electron dynamics holds immense potential for exploring the nonlinear high Hall responses in Weyl semimetal., Comment: 19 pages, 5 figures

In Reinforcement Learning (RL), artificial agents are trained to maximize numerical rewards by performing tasks. Exploration is essential in RL because agents must discover information before exploiting it. Two rewards encouraging efficient exploration are the entropy of action policy and curiosity for information gain. Entropy is well-established in literature, promoting randomized action selection. Curiosity is defined in a broad variety of ways in literature, promoting discovery of novel experiences. One example, prediction error curiosity, rewards agents for discovering observations they cannot accurately predict. However, such agents may be distracted by unpredictable observational noises known as curiosity traps. Based on the Free Energy Principle (FEP), this paper proposes hidden state curiosity, which rewards agents by the KL divergence between the predictive prior and posterior probabilities of latent variables. We trained six types of agents to navigate mazes: baseline agents without rewards for entropy or curiosity, and agents rewarded for entropy and/or either prediction error curiosity or hidden state curiosity. We find entropy and curiosity result in efficient exploration, especially both employed together. Notably, agents with hidden state curiosity demonstrate resilience against curiosity traps, which hinder agents with prediction error curiosity. This suggests implementing the FEP may enhance the robustness and generalization of RL models, potentially aligning the learning processes of artificial and biological agents., Comment: 54 pages, 11 figures, to be published in Neural Computation

We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a non-negative vector cost $c_e \in \mathbb{R}^{\ell}_{\ge 0}$. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the $\ell_p$-norm of the obtained cost vector (we assume that $p \ge 1$ is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.

The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid $\Bbbk$-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous $\Bbbk$-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous $\Bbbk$-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous $\Bbbk$-algebra., Comment: 14 pages, 6 figures

Humans excel at applying learned behavior to unlearned situations. A crucial component of this generalization behavior is our ability to compose/decompose a whole into reusable parts, an attribute known as compositionality. One of the fundamental questions in robotics concerns this characteristic. "How can linguistic compositionality be developed concomitantly with sensorimotor skills through associative learning, particularly when individuals only learn partial linguistic compositions and their corresponding sensorimotor patterns?" To address this question, we propose a brain-inspired neural network model that integrates vision, proprioception, and language into a framework of predictive coding and active inference, based on the free-energy principle. The effectiveness and capabilities of this model were assessed through various simulation experiments conducted with a robot arm. Our results show that generalization in learning to unlearned verb-noun compositions, is significantly enhanced when training variations of task composition are increased. We attribute this to self-organized compositional structures in linguistic latent state space being influenced significantly by sensorimotor learning. Ablation studies show that visual attention and working memory are essential to accurately generate visuo-motor sequences to achieve linguistically represented goals. These insights advance our understanding of mechanisms underlying development of compositionality through interactions of linguistic and sensorimotor experience., Comment: 64 pages, 6 figures, 10 supplementary figures

The development of algorithms to accurately decode of neural information is a long-standing effort in the field of neuroscience. Brain decoding is typically employed by training machine learning models to map neural data onto a preestablished vector representation of stimulus features. These vectors are usually derived from image- and/or text-based feature spaces. Nonetheless, the intrinsic characteristics of these vectors might be fundamentally different than those encoded by the brain, limiting the ability of algorithms to accurately learn this mapping. To address this issue, here, we propose a representation learning framework, called brain-aligning of semantic vectors, that fine-tunes pretrained feature vectors to better align with the structure of neural representations of visual stimuli in the human brain. We trained this model with functional magnetic resonance imaging (fMRI) data representing 150 visual stimulus categories; then, we performed zero-shot brain decoding on 1) fMRI, 2) magnetoencephalography (MEG), and 3) electrocorticography (ECoG) data reflecting neural representations of visual stimuli. By using fMRI-based brain-aligned vectors, the zero-shot decoding accuracy all three neuroimaging datasets increased. This finding underscores the potential of leveraging a richer array of brainderived features to increase the performance of brain decoding algorithms., Comment: 40 pages, 5 figures

We observe non-perturbative high harmonic generation in solids driven by a macroscopic quantum state of light, bright squeezed vacuum (BSV), which we generate in a single spatiotemporal mode. The BSV-driven process is considerably more efficient in the generation of high harmonics than classical light of the same mean intensity. Due to its broad photon-number distribution, covering states from $0$ to $2 \times 10^{13}$ photons per pulse, and sub-cycle electric field fluctuations over $\pm1\hbox{V}/\hbox{\r{A}}$, BSV provides access to free carrier dynamics within a much broader range of peak intensities than accessible with classical light. Our findings contribute to recent developments of quantum optics with extreme intensities, moving beyond its traditional focus on low photon numbers, and providing a new method for exploring extreme nonlinearities in solids.

The enhancement of the parity-violating asymmetry in the vicinity of $p$-wave compound nuclear resonances was observed for a variety of medium-heavy nuclei. The enhanced parity-violating asymmetry can be understood using the $s$-$p$ mixing model. The $s$-$p$ mixing model predicts several neutron energy-dependent angular correlations between the neutron momentum $\vec k_n$, neutron spin $\vec\sigma_n$, $\gamma$-ray momentum $\vec k_\gamma$, and $\gamma$-ray polarization $\vec\sigma_\gamma$ in the $(n,\gamma)$ reaction. In this paper, the improved value of the transverse asymmetry of $\gamma$-ray emissions, corresponding to a correlation term $\vec{\sigma}_n\cdot(\vec k_n\times\vec k_\gamma)$ in the $^{139}\mathrm{La}(\vec n,\gamma)^{140}\mathrm{La}$ reaction, and the transverse asymmetries in the transitions to several low excited states of $^{140}\mathrm{La}$ are reported., Comment: 5 pages, 5 figures, 2 tables

We consider the problem of graph searching with prediction recently introduced by Banerjee et al. (2022). In this problem, an agent, starting at some vertex $r$ has to traverse a (potentially unknown) graph $G$ to find a hidden goal node $g$ while minimizing the total distance travelled. We study a setting in which at any node $v$, the agent receives a noisy estimate of the distance from $v$ to $g$. We design algorithms for this search task on unknown graphs. We establish the first formal guarantees on unknown weighted graphs and provide lower bounds showing that the algorithms we propose have optimal or nearly-optimal dependence on the prediction error. Further, we perform numerical experiments demonstrating that in addition to being robust to adversarial error, our algorithms perform well in typical instances in which the error is stochastic. Finally, we provide alternative simpler performance bounds on the algorithms of Banerjee et al. (2022) for the case of searching on a known graph, and establish new lower bounds for this setting., Comment: AISTATS 2024

A quantum channel whose image approximates the set of separable states is called a disentangler, which plays a prominent role in the investigation of variants of the computational model called Quantum Merlin Arthur games, and has potential applications in classical and quantum algorithms for the separability testing and NP-complete problems. So far, two types of a disentangler, constructed based on $\epsilon$-nets and the quantum de Finetti theorem, have been known; however, both of them require an exponentially large input system. Moreover, in 2008, John Watrous conjectured that any disentangler requires an exponentially large input system, called the disentangler conjecture. In this paper, we show that both of the two known disentanglers can be regarded as examples of a strong disentangler, which is a disentangler approximately breaking entanglement between one output system and the composite system of another output system and the arbitrarily large environment. Note that the strong disentangler is essentially an approximately entanglement-breaking channel while the original disentangler is an approximately entanglement-annihilating channel, and the set of strong disentanglers is a subset of disentanglers. As a main result, we show that the disentangler conjecture is true for this subset, the set of strong disentanglers, for a wide range of approximation parameters without any computational hardness assumptions., Comment: 9 pages, 2 figures

Azimuthal anisotropies of high-$p_T$ particles produced in heavy-ion collisions are understood as an effect of a geometrical selection bias. Particles oriented in the direction in which the QCD medium formed in these collisions is shorter, suffer less energy loss, and thus, are over-represented in the final ensemble compared to those oriented in the direction in which the medium is longer. In this work we present the first semi-analytical predictions, including propagation through a realistic, hydrodynamical background, of the azimuthal anisotropies for jets, obtaining a quantitative agreement with available experimental data as function of the jet $p_T$, its cone size $R$ and the collisions centrality. Jets are multi-partonic, extended objects and their energy loss is sensitive to substructure fluctuations. This is determined by the physics of color coherence that relates to the ability of the medium to resolve those partonic fluctuations. Namely, color dipoles whose angle is smaller than a critical angle, $\theta_c$, are not resolved by the medium and they effectively act as a coherent source of energy loss. We find that jet azimuthal anisotropies have a specially strong dependence on coherence physics due to the marked length-dependence of $\theta_c$. By combining our predictions for the collision systems and center of mass energies studied at RHIC and the LHC, covering a wide range of typical values of $\theta_c$, we show that the relative size of jet azimuthal anisotropies for jets with different cone-sizes $R$ follow a universal trend that indicates a transition from a coherent regime of jet quenching to a decoherent regime. These results suggest a way forward to reveal the role played by the physics of jet color decoherence in probing deconfined QCD matter., Comment: 25 pages, 14 figures

We study the time evolution of the density matrix of a high energy quark in the presence of a dense QCD background that is modeled as a stochastic Gaussian color field. At late times, we find that only the color singlet component of the quark's reduced density matrix survives the in-medium evolution and that the density matrix becomes asymptotically diagonal in both transverse position and momentum spaces. In addition, we observe an accelerated entropy growth due to the larger phase space being explored by the quark and that the quantum and classical quark entropies converge at late times. We further observe that the quark state loses all memory of the initial condition. Combined with the fact that the reduced density matrix satisfies Boltzmann-diffusion transport, we conclude that the quark reduced density matrix can be interpreted as a classical phase space distribution., Comment: 4 pages, Proceedings of Quark Matter 2023

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively. This enables the construction of local approximate fast solvers. These local solvers are then combined into an overlapping Schwarz preconditioner, which is utilized in a truncated preconditioned conjugate gradient method. Numerical experiments demonstrate the significant memory storage benefits and a uniformly bounded number of iterations with respect to both mesh size and spline degree., Comment: 24 pages, 11 figures

This is a review of preconditioning techniques based on fast-diagonalization methods for space-time isogeometric discretization of the heat equation. Three formulation are considered: the Galerkin approach, a discrete least-square and a continuous least square. For each formulation the heat differential operator is written as a sum of terms that are kronecker products of uni-variate operators. These are used to speed-up the application of the operator in iterative solvers and to construct a suitable preconditioner. Contrary to the fast-diagonalization technique for the Laplace equation where all uni-variate operators acting on the same direction can be simultaneously diagonalized in the case of the heat equation this is not possible. Luckily this can be done up to an additional term that has low rank allowing for the utilization of arrow-head like factorization or inversion by Sherman-Morrison formula. The proposed preconditioners work extremely well on the parametric domain and, when the domain is parametrized or when the equation coefficients are not constant, they can be adapted and retain good performance characteristics., Comment: arXiv admin note: substantial text overlap with arXiv:1909.07309, arXiv:2311.18461

ChatGPT, a widely-recognized large language model (LLM), has recently gained substantial attention for its performance scaling, attributed to the billions of web-sourced natural language sentences used for training. Its underlying architecture, Transformer, has found applications across diverse fields, including video, audio signals, and robotic movement. %The crucial question this raises concerns the Transformer's generalization-in-learning (GIL) capacity. However, this raises a crucial question about Transformer's generalization in learning (GIL) capacity. Is ChatGPT's success chiefly due to the vast dataset used for training, or is there more to the story? To investigate this, we compared Transformer's GIL capabilities with those of a traditional Recurrent Neural Network (RNN) in tasks involving attractor dynamics learning. For performance evaluation, the Dynamic Time Warping (DTW) method has been employed. Our simulation results suggest that under conditions of limited data availability, Transformer's GIL abilities are markedly inferior to those of RNN.

We investigate the violation of the Leggett-Garg inequalities for a quantum field, focusing on the two-time quasi-probability distribution function of the dichotomic variable with a coarse-grained scalar field. The Leggett-Garg inequalities are violated depending on the quantum state of the field and the size of coarse-graining. We also demonstrate that the violation of the Leggett-Garg inequalities appears even for the vacuum state and the squeezed state by properly constructing the dichotomic variable and the projection operator., Comment: 15 pages, 10 figures