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Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology.

We numerically benchmark 30 optimisers on 372 instances of the variational quantum eigensolver for solving the Fermi-Hubbard system with the Hamiltonian variational ansatz. We rank the optimisers with respect to metrics such as final energy achieved and function calls needed to get within a certain tolerance level, and find that the best performing optimisers are variants of gradient descent such as Momentum and ADAM (using finite difference), SPSA, CMAES, and BayesMGD. We also perform gradient analysis and observe that the step size for finite difference has a very significant impact. We also consider using simultaneous perturbation (inspired by SPSA) as a gradient subroutine: here finite difference can lead to a more precise estimate of the ground state but uses more calls, whereas simultaneous perturbation can converge quicker but may be less precise in the later stages. Finally, we also study the quantum natural gradient algorithm: we implement this method for 1-dimensional Fermi-Hubbard systems, and find that whilst it can reach a lower energy with fewer iterations, this improvement is typically lost when taking total function calls into account. Our method involves performing careful hyperparameter sweeping on 4 instances. We present a variety of analysis and figures, detailed optimiser notes, and discuss future directions., Comment: 42 pages, 30 figures. Associated data can be found at https://doi.org/10.5281/zenodo.13960674

Tenure is a cornerstone of the US academic system, yet its relationship to faculty research trajectories remains poorly understood. Conceptually, tenure systems may act as a selection mechanism, screening in high-output researchers; a dynamic incentive mechanism, encouraging high output prior to tenure but low output after tenure; and a creative search mechanism, encouraging tenured individuals to undertake high-risk work. Here, we integrate data from seven different sources to trace US tenure-line faculty and their research outputs at an unprecedented scale and scope, covering over 12,000 researchers across 15 disciplines. Our analysis reveals that faculty publication rates typically increase sharply during the tenure track and peak just before obtaining tenure. Post-tenure trends, however, vary across disciplines: in lab-based fields, such as biology and chemistry, research output typically remains high post-tenure, whereas in non-lab-based fields, such as mathematics and sociology, research output typically declines substantially post-tenure. Turning to creative search, faculty increasingly produce novel, high-risk research after securing tenure. However, this shift toward novelty and risk-taking comes with a decline in impact, with post-tenure research yielding fewer highly cited papers. Comparing outcomes across common career ages but different tenure years or comparing research trajectories in tenure-based and non-tenure-based research settings underscores that breaks in the research trajectories are sharply tied to the individual's tenure year. Overall, these findings provide a new empirical basis for understanding the tenure system, individual research trajectories, and the shape of scientific output.

Directionality in molecular and biomolecular networks plays a significant role in the accurate represention of the complex, dynamic, and asymmetrical nature of interactions present in protein-ligand binding, signal transduction, and biological pathways. Most traditional techniques of topological data analysis (TDA), such as persistent homology (PH) and persistent Laplacian (PL), overlook this aspect in their standard form. To address this, we present the persistent directed flag Laplacian (PDFL), which incorporates directed flag complexes to account for edges with directionality originated from polarization, gene regulation, heterogeneous interactions, etc. This study marks the first application of the PDFL, providing an in-depth analysis of spectral graph theory combined with machine learning. Besides its superior accuracy and reliability, the PDFL model offers simplicity by requiring only raw inputs without complex data processing. We validated our multi-kernel PDFL model for its scoring power against other state-of-art methods on three popular benchmarks, namely PDBbind v2007, v2013, and v2016. Computational results indicate that the proposed PDFL model outperforms competitors in protein-ligand binding affinity predictions, indicating that PDFL is a promising tool for protein engineering, drug discovery, and general applications in science and engineering.

We introduce a class of left cancellative categories we call ordinal graphs for which there is a functor $d:\Lambda\rightarrow\mathrm{Ord}$ through which elements of $\Lambda$ factor. We use generators and relations to study the Cuntz-Krieger algebra $\mathcal{O}\left(\Lambda\right)$ defined by Spielberg. In particular, we construct a $\mathrm{C}^{*}$-correspondence $X_{\alpha}$ for each $\alpha\in\mathrm{Ord}$ in order to apply Ery\"uzl\"u and Tomforde's condition (S) and prove a Cuntz-Krieger uniqueness theorem for ordinal graphs., Comment: 31 pages, 5 figures

We prove a lower bound on the number of copies needed to test the property of a multipartite quantum state being product across some bipartition (i.e. not genuinely multipartite entangled), given the promise that the input state either has this property or is $\epsilon$-far in trace distance from any state with this property. We show that $\Omega(n / \log n)$ copies are required (for fixed $\epsilon \leq \frac{1}{2}$), complementing a previous result that $O(n / \epsilon^2)$ copies are sufficient. Our proof technique proceeds by considering uniformly random ensembles over such states, and showing that the trace distance between these ensembles becomes arbitrarily small for sufficiently large $n$ unless the number of copies is at least $\Omega (n / \log n)$. We discuss implications for testing graph states and computing the generalised geometric measure of entanglement., Comment: 19 pages, 2 figures

The continuous evolution of the SARS-CoV-2 virus poses a significant challenge to global public health. Of particular concern is the potential resistance to the widely prescribed drug PAXLOVID, of which the main ingredient nirmatrelvir inhibits the viral main protease (Mpro). Here, we developed CAPTURE (direCted flAg laPlacian Transformer for drUg Resistance prEdictions) to analyze the effects of Mpro mutations on nirmatrelvir-Mpro binding affinities and identify potential drug-resistant mutations. CAPTURE combines a comprehensive mutation analysis with a resistance prediction module based on DFFormer-seq, which is a novel ensemble model that leverages a new Directed Flag Transformer and sequence embeddings from the protein and small-molecule-large-language models. Our analysis of the evolution of Mpro mutations revealed a progressive increase in mutation frequencies for residues near the binding site between May and December 2022, suggesting that the widespread use of PAXLOVID created a selective pressure that accelerated the evolution of drug-resistant variants. Applied to mutations at the nirmatrelvir-Mpro binding site, CAPTURE identified several potential resistance mutations, including H172Y and F140L, which have been experimentally confirmed, as well as five other mutations that await experimental verification. CAPTURE evaluation in a limited experimental data set on Mpro mutants gives a recall of 57\% and a precision of 71\% for predicting potential drug-resistant mutations. Our work establishes a powerful new framework for predicting drug-resistant mutations and real-time viral surveillance. The insights also guide the rational design of more resilient next-generation therapeutics.

Storage, organizing, and decorating are an important part of home design. While one can buy commercial items for many of these tasks, this can be costly, and re-use is more sustainable. An alternative is a "home hack", a functional assembly that can be constructed from existing household items. However, coming up with such hacks requires combining objects to make a physically valid design, which might be difficult to test if they are large, require nailing or screwing something to the wall, or the designer has mobility limitations. In this work, we present a design and visualization system for creating workable functional assemblies, FabHacks, which is based on a solver-aided domain-specific language (S-DSL) FabHaL. By analyzing existing home hacks shared online, we create a design abstraction for connecting household items using predefined types of connections. We provide a UI for FabHaL that can be used to design assemblies that fulfill a given specification. Our system leverages a physics-based solver that takes an assembly design and finds its expected physical configuration. Our validation includes a user study showing that users can create assemblies successfully using our UI and explore a range of designs.

We consider models of quantum computation that involve operations performed on some fixed resourceful quantum state. Examples that fit this paradigm include magic state injection and measurement-based approaches. We introduce a framework that incorporates both of these cases and focus on the role of coherence (or superposition) in this context, as exemplified through the Hadamard gate. We prove that given access to incoherent unitaries (those that are unable to generate superposition from computational basis states, e.g. CNOT, diagonal gates), classical control, computational basis measurements, and any resourceful ancillary state (of arbitrary dimension), it is not possible to implement any coherent unitary (e.g. Hadamard) exactly with non-zero probability. We also consider the approximate case by providing lower bounds for the induced trace distance between the above operations and $n$ Hadamard gates. To demonstrate the stability of this result, this is then extended to a similar no-go result for the case of using $k$ Hadamard gates to exactly implement $n>k$ Hadamard gates., Comment: 31 pages, 3 figures. Latest version includes an improved bound for Lemma 19, added references, and data access statement. Accepted in Quantum

Topological data analysis (TDA) has had enormous success in science and engineering in the past decade. Persistent topological Laplacians (PTLs) overcome some limitations of persistent homology, a key technique in TDA, and provide substantial insight to the behavior of various geometric and topological objects. This work extends PTLs to directed flag complexes, which are an exciting generalization to flag complexes, also known as clique complexes, that arise naturally in many situations. We introduce the directed flag Laplacian and show that the proposed persistent directed flag Laplacian (PDFL) is a distinct way of analyzing these flag complexes. Example calculations are provided to demonstrate the potential of the proposed PDFL in real world applications.

We present a conceptual design of a high-performance camera system with applications to neutrino detectors, deep sea exploration, and glaciology. The design combines ultra-sensitive cameras with a number of well-calibrated light sources enclosed in a pressure vessel. The instrument will be capable of withstanding extreme environments such as those encountered in Antarctica or the deep ocean, and be deployable as a standalone system that can be retrieved for deep-sea exploration or glaciology. The camera system is designed to be replicated and deployed in multiple detectors, requiring only modest modifications from one detector to another. The instrument combines a number of capabilities essential for neutrino detector calibrations, including characterization of the scattering and absorption properties of the optical medium, measurement of geometries via photogrammetry, and detector surveillance. The ability to deploy the instrument at different detector sites also offers opportunities for cross-calibration efforts. We present the conceptual design of the instrument and describe plans to produce a prototype., Comment: 8 pages, 3 figures, Presented at the 38th International Cosmic Ray Conference (ICRC2023)

Parametric CAD models encode entire families of shapes that should, in principle, be easy for designers to explore. However, in practice, parametric CAD models can be difficult to manipulate due to implicit semantic constraints among parameter values. Finding and enforcing these semantic constraints solely from geometry or programmatic shape representations is not possible because these constraints ultimately reflect design intent. They are informed by the designer's experience and semantics in the real world. To address this challenge, we introduce a zero-shot pipeline that leverages pre-trained large language and image model to infer meaningful space of variations for a shape. We then re-parameterize a new constrained parametric CAD program that captures these variations, enabling effortless exploration of the design space along meaningful design axes.

Large Computer-Aided Design (CAD) projects usually require collaboration across many different CAD systems as well as applications that interoperate with them for manufacturing, visualization, or simulation. A fundamental barrier to such collaborations is the ability to refer to parts of the geometry (such as a specific face) robustly under geometric and/or topological changes to the model. Persistent referencing schemes are a fundamental aspect of most CAD tools, but models that are shared across systems cannot generally make use of these internal referencing mechanisms, creating a challenge for collaboration. In this work, we address this issue by developing a novel learning-based algorithm that can automatically find correspondences between two CAD models using the standard representation used for sharing models across CAD systems: the Boundary-Representation (B-rep). Because our method works directly on B-reps it can be generalized across different CAD applications enabling collaboration.

This report summarizes the current status of neutrino physics and the broad and exciting future prospects identified for the Neutrino Frontier as part of the 2021 Snowmass Process., Comment: 49 pages, contribution to: 2021 Snowmass Summer Study. Minor updates

The design of man-made objects is dominated by computer aided design (CAD) tools. Assisting design with data-driven machine learning methods is hampered by lack of labeled data in CAD's native format; the parametric boundary representation (B-Rep). Several data sets of mechanical parts in B-Rep format have recently been released for machine learning research. However, large scale databases are largely unlabeled, and labeled datasets are small. Additionally, task specific label sets are rare, and costly to annotate. This work proposes to leverage unlabeled CAD geometry on supervised learning tasks. We learn a novel, hybrid implicit/explicit surface representation for B-Rep geometry, and show that this pre-training significantly improves few-shot learning performance and also achieves state-of-the-art performance on several existing B-Rep benchmarks.

Neutrinos are the most elusive among the known elementary particles, because of their feeble interactions with ordinary matter. They are also the most mysterious, because of their tiny masses that suggest a novel mass generating mechanism, their unknown Dirac or Majorana nature, and their big quantum mixing leading to large-amplitude flavor oscillations. This Topical Group focuses on neutrino properties that are not directly investigated in other Topical Groups of the Neutrino Frontier: in particular, the absolute value of the neutrino masses, the Dirac or Majorana nature of neutrinos, their electromagnetic properties, their lifetime, and hypothetical exotic properties., Comment: Topical Group Report for NF05 (Neutrino Frontier Topical Group on Neutrino Properties) for Snowmass 2021. 51 pages excluding references

We describe our work on inferring the degrees of freedom between mated parts in mechanical assemblies using deep learning on CAD representations. We train our model using a large dataset of real-world mechanical assemblies consisting of CAD parts and mates joining them together. We present methods for re-defining these mates to make them better reflect the motion of the assembly, as well as narrowing down the possible axes of motion. We also conduct a user study to create a motion-annotated test set with more reliable labels., Comment: Contains 5 pages, 2 figures. Presented at the ICML 2022 Workshop on Machine Learning in Computational Design

The effect of quantum steering arises from the judicious combination of an entangled state with a set of incompatible measurements. Recently, it was shown that this form of quantum correlations can be quantified in terms of a dimension, leading to the notion of genuine high-dimensional steering. While this naturally connects to the dimensionality of entanglement (Schmidt number), we show that this effect also directly connects to a notion of dimension for measurement incompatibility. More generally, we present a general connection between the concepts of steering and measurement incompatibility, when quantified in terms of dimension. From this connection, we propose a novel twist on the problem of simulating quantum correlations. Specifically, we show how the correlations of certain high-dimensional entangled states can be exactly recovered using only shared randomness and lower-dimensional entanglement. Finally, we derive criteria for testing the dimension of measurement incompatibility, and discuss the extension of these ideas to quantum channels., Comment: 10 pages, 2 figures

We investigate the compression of quantum information with respect to a given set $\mathcal{M}$ of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from $\mathcal{M}$ and an arbitrary quantum state $\rho$ are recovered exactly by first compressing $\rho$ into a lower dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set $\mathcal{M}$ is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative examples involving mutually unbiased basis, and develop a method based on semi-definite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.

Neutrino telescopes provide strong sensitivity to sterile neutrino oscillations through matter-enhanced oscillation, occurring in the few TeV energy range for eV$^{2}$-scale neutrino mass-squared splittings. Prior searches have focused on $\nu_\mu$ disappearance, which has a particularly strong sensitivity to the mixing angle $\theta_{24}$ via $\nu_\mu\rightarrow\nu_s$ transitions. Nowadays, the $\nu_\mu\rightarrow\nu_e$ and $\nu_\mu\rightarrow\nu_\tau$ appearance channels have been considered less promising at neutrino telescopes, due in part to the much smaller target volume for cascades relative to tracks. This work explores the detectability of these signatures at neutrino telescopes given present constraints on sterile neutrino mixing, and as an example, forecasts the sensitivity of the IceCube Neutrino Observatory to the mixing angles $\theta_{14}$, $\theta_{24}$, and $\theta_{34}$ in the 3+1 sterile neutrino model using the cascade channel with ten years of data. We find that $\nu_\tau$ appearance signatures consistent with the existing IceCube $\nu_\mu$ disappearance best-fit point are discoverable for values of $\theta_{34}$ consistent with world constraints, and that the sterile neutrino parameters favored by the BEST and gallium anomalies are expected to be testable at the 95\% confidence level., Comment: Implemented reviewer comments. Updated format of references, added a new reference, fixed several typos, made several clarifications, updated axis labels on a few plots, added some new discussion, and changed the color of a contour