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Adversarial robustness and generalization are both crucial properties of reliable machine learning models. In this paper, we study these properties in the context of quantum machine learning based on Lipschitz bounds. We derive parameter-dependent Lipschitz bounds for quantum models with trainable encoding, showing that the norm of the data encoding has a crucial impact on the robustness against data perturbations. Further, we derive a bound on the generalization error which explicitly involves the parameters of the data encoding. Our theoretical findings give rise to a practical strategy for training robust and generalizable quantum models by regularizing the Lipschitz bound in the cost. Further, we show that, for fixed and non-trainable encodings, as those frequently employed in quantum machine learning, the Lipschitz bound cannot be influenced by tuning the parameters. Thus, trainable encodings are crucial for systematically adapting robustness and generalization during training. The practical implications of our theoretical findings are illustrated with numerical results.

Quantum computing is a fascinating interdisciplinary research field that promises to revolutionize computing by efficiently solving previously intractable problems. Recent years have seen tremendous progress on both the experimental realization of quantum computing devices as well as the development and implementation of quantum algorithms. Yet, realizing computational advantages of quantum computers in practice remains a widely open problem due to numerous fundamental challenges. Interestingly, many of these challenges are connected to performance, robustness, scalability, optimization, or feedback, all of which are central concepts in control theory. This paper provides a tutorial introduction to quantum computing from the perspective of control theory. We introduce the mathematical framework of quantum algorithms ranging from basic elements including quantum bits and quantum gates to more advanced concepts such as variational quantum algorithms and quantum errors. The tutorial only requires basic knowledge of linear algebra and, in particular, no prior exposure to quantum physics. Our main goal is to equip readers with the mathematical basics required to understand and possibly solve (control-related) problems in quantum computing. In particular, beyond the tutorial introduction, we provide a list of research challenges in the field of quantum computing and discuss their connections to control.

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper, we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits, and they can be used to guarantee fault-tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a novel guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the $3$-qubit Quantum Fourier Transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

We show that the category of quasi-coherent Cartier crystals is equivalent to the category of unit Cartier modules on an F-finite noetherian ring R, and that these equivalent categories have finite global dimension, by showing that every quasi-coherent Cartier crystal has a finite injective resolution. The length of the resolution is uniformly bounded by a bound only depending on R. Our result should be viewed as a generalization of a result of Ma showing that the category of unit R[F]-modules over a F-finite regular ring R has finite global dimension dim R + 1., Comment: 11 pages

1. Citizen and community-science (CS) datasets have great potential for estimating interannual patterns of population change given the large volumes of data collected globally every year. Yet, the flexible protocols that enable many CS projects to collect large volumes of data typically lack the structure necessary to keep consistent sampling across years. This leads to interannual confounding, as changes to the observation process over time are confounded with changes in species population sizes. 2. Here we describe a novel modeling approach designed to estimate species population trends while controlling for the interannual confounding common in citizen science data. The approach is based on Double Machine Learning, a statistical framework that uses machine learning methods to estimate population change and the propensity scores used to adjust for confounding discovered in the data. Additionally, we develop a simulation method to identify and adjust for residual confounding missed by the propensity scores. Using this new method, we can produce spatially detailed trend estimates from citizen science data. 3. To illustrate the approach, we estimated species trends using data from the CS project eBird. We used a simulation study to assess the ability of the method to estimate spatially varying trends in the face of real-world confounding. Results showed that the trend estimates distinguished between spatially constant and spatially varying trends at a 27km resolution. There were low error rates on the estimated direction of population change (increasing/decreasing) and high correlations on the estimated magnitude. 4. The ability to estimate spatially explicit trends while accounting for confounding in citizen science data has the potential to fill important information gaps, helping to estimate population trends for species, regions, or seasons without rigorous monitoring data., Comment: 28 pages, 6 figures

Understanding how environmental characteristics affect bio-diversity patterns, from individual species to communities of species, is critical for mitigating effects of global change. A central goal for conservation planning and monitoring is the ability to accurately predict the occurrence of species communities and how these communities change over space and time. This in turn leads to a challenging and long-standing problem in the field of computer science - how to perform ac-curate multi-label classification with hundreds of labels? The key challenge of this problem is its exponential-sized output space with regards to the number of labels to be predicted.Therefore, it is essential to facilitate the learning process by exploiting correlations (or dependency) among labels. Previous methods mostly focus on modelling the correlation on label pairs; however, complex relations between real-world objects often go beyond second order. In this paper, we pro-pose a novel framework for multi-label classification, High-order Tie-in Variational Autoencoder (HOT-VAE), which per-forms adaptive high-order label correlation learning. We experimentally verify that our model outperforms the existing state-of-the-art approaches on a bird distribution dataset on both conventional F1 scores and a variety of ecological metrics. To show our method is general, we also perform empirical analysis on seven other public real-world datasets in several application domains, and Hot-VAE exhibits superior performance to previous methods., Comment: accepted at AAAI'21 AISI

An important benefit of multi-objective search is that it maintains a diverse population of candidates, which helps in deceptive problems in particular. Not all diversity is useful, however: candidates that optimize only one objective while ignoring others are rarely helpful. This paper proposes a solution: The original objectives are replaced by their linear combinations, thus focusing the search on the most useful tradeoffs between objectives. To compensate for the loss of diversity, this transformation is accompanied by a selection mechanism that favors novelty. In the highly deceptive problem of discovering minimal sorting networks, this approach finds better solutions, and finds them faster and more consistently than standard methods. It is therefore a promising approach to solving deceptive problems through multi-objective optimization., Comment: 7 pages

Bird species' migratory patterns have typically been studied through individual observations and historical records. In recent years however, the eBird citizen science project, which solicits observations from thousands of bird watchers around the world, has opened the door for a data-driven approach to understanding the large-scale geographical movements. Here, we focus on the North American Tree Swallow (\textit{Tachycineta bicolor}) occurrence patterns throughout the eastern United States. Migratory departure dates for this species are widely believed by both ornithologists and casual observers to vary substantially across years, but the reasons for this are largely unknown. In this work, we present evidence that maximum daily temperature is a major factor influencing Tree Swallow occurrence. Because it is generally understood that species occurrence is a function of many complex, high-order interactions between ecological covariates, we utilize the flexible modeling approach offered by random forests. Making use of recent asymptotic results, we provide formal hypothesis tests for predictive significance various covariates and also develop and implement a permutation-based approach for formally assessing interannual variations by treating the prediction surfaces generated by random forests as functional data. Each of these tests suggest that maximum daily temperature has a significant effect on migration patterns., Comment: 23 pages, 7 figures. Work between Cornell Lab of Ornithology and University of Pittsburgh Department of Statistics

Understanding how species are distributed across landscapes over time is a fundamental question in biodiversity research. Unfortunately, most species distribution models only target a single species at a time, despite strong ecological evidence that species are not independently distributed. We propose Deep Multi-Species Embedding (DMSE), which jointly embeds vectors corresponding to multiple species as well as vectors representing environmental covariates into a common high-dimensional feature space via a deep neural network. Applied to bird observational data from the citizen science project \textit{eBird}, we demonstrate how the DMSE model discovers inter-species relationships to outperform single-species distribution models (random forests and SVMs) as well as competing multi-label models. Additionally, we demonstrate the benefit of using a deep neural network to extract features within the embedding and show how they improve the predictive performance of species distribution modelling. An important domain contribution of the DMSE model is the ability to discover and describe species interactions while simultaneously learning the shared habitat preferences among species. As an additional contribution, we provide a graphical embedding of hundreds of bird species in the Northeast US., Comment: 13 pages