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We present adaptive measurement techniques tailored for variational quantum algorithms on near-term small and noisy devices. In particular, we generalise earlier "learning to measure" strategies in two ways. First, by considering a class of adaptive positive operator valued measures (POVMs) that can be simulated with simple projective measurements without ancillary qubits, we decrease the amount of required qubits and two-qubit gates. Second, by introducing a method based on Quantum Detector Tomography to mitigate the effect of noise, we are able to optimise the POVMs as well as to infer expectation values reliably in the currently available noisy quantum devices. Our numerical simulations clearly indicate that the presented strategies can significantly reduce the number of needed shots to achieve chemical accuracy in variational quantum eigensolvers, thus helping to solve one of the bottlenecks of near-term quantum computing., Comment: 10+4 pages, 4 figures

We propose a Model Predictive Control (MPC) for collision avoidance between an autonomous agent and dynamic obstacles with uncertain predictions. The collision avoidance constraints are imposed by enforcing positive distance between convex sets representing the agent and the obstacles, and tractably reformulating them using Lagrange duality. This approach allows for smooth collision avoidance constraints even for polytopes, which otherwise require mixed-integer or non-smooth constraints. We consider three widely used descriptions of the uncertain obstacle position: 1) Arbitrary distribution with polytopic support, 2) Gaussian distributions and 3) Arbitrary distribution with first two moments known. For each case we obtain deterministic reformulations of the collision avoidance constraints. The proposed MPC formulation optimizes over feedback policies to reduce conservatism in satisfying the collision avoidance constraints. The proposed approach is validated using simulations of traffic intersections in CARLA., Comment: Accepted to CDC'22

We propose a Stochastic MPC (SMPC) approach for autonomous driving which incorporates multi-modal, interaction-aware predictions of surrounding vehicles. For each mode, vehicle motion predictions are obtained by a control model described using a basis of fixed features with unknown weights. The proposed SMPC formulation finds optimal controls which serves two purposes: 1) reducing conservatism of the SMPC by optimizing over parameterized control laws and 2) prediction and estimation of feature weights used in interaction-aware modeling using Kalman filtering. The proposed approach is demonstrated on a longitudinal control example, with uncertainties in predictions of the autonomous and surrounding vehicles., Comment: Accepted to AVEC'22

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $\kappa$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr\"odinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{\kappa^2}{\pi^2} \] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\varepsilon$ from the essential spectrum, where $2\varepsilon$ is the width of the waveguide. This is in contrast with the non-relativistic counterpart of this model, for which they are known to be at a finite distance., Comment: 11 pages. To appear on "Indam Quantum Meetings 22" proceedings

We study a human-robot collaborative transportation task in presence of obstacles. The task for each agent is to carry a rigid object to a common target position, while safely avoiding obstacles and satisfying the compliance and actuation constraints of the other agent. Human and robot do not share the local view of the environment. The human policy either assists the robot when they deem the robot actions safe based on their perception of the environment, or actively leads the task. Using estimated human inputs, the robot plans a trajectory for the transported object by solving a constrained finite time optimal control problem. Sensors on the robot measure the inputs applied by the human. The robot then appropriately applies a weighted combination of the human's applied and its own planned inputs, where the weights are chosen based on the robot's trust value on its estimates of the human's inputs. This allows for a dynamic leader-follower role adaptation of the robot throughout the task. Furthermore, under a low value of trust, if the robot approaches any obstacle potentially unknown to the human, it triggers a safe stopping policy, maintaining safety of the system and signaling a required change in the human's intent. With experimental results, we demonstrate the efficacy of the proposed approach.

The rapid progress in quantum computing witnessed in recent years has sparked widespread interest in developing scalable quantum information theoretic methods to work with large quantum systems. For instance, several approaches have been proposed to bypass tomographic state reconstruction, and yet retain to a certain extent the capability to estimate multiple physical properties of a given state previously measured. In this paper, we introduce the Virtual Linear Map Algorithm (VILMA), a new method that enables not only to estimate multiple operator averages using classical post-processing of informationally complete measurement outcomes, but also to do so for the image of the measured reference state under low-depth circuits of arbitrary, not necessarily physical, $k$-local maps. We also show that VILMA allows for the variational optimisation of the virtual circuit through sequences of efficient linear programs. Finally, we explore the purely classical version of the algorithm, in which the input state is a state with a classically efficient representation, and show that the method can prepare ground states of many-body Hamiltonians., Comment: 10 pages, 5 figs. Comments welcome

Scientific and technological advances in medicine and systems biology have unequivocally shown that health and disease must be viewed in the context of the interplay among multiple molecular and environmental factors. Understanding the effects of cellular interconnection on disease progression may lead to the identification of novel disease genes and pathways, and hence influence precision diagnostics and therapeutics. To accomplish this goal, the emerging field of network medicine applies network science approaches to investigate disease pathogenesis, integrating information from relevant Omics databases, including protein-protein interaction, correlation-based, gene regulatory, and Bayesian networks. However, this requires analysing and computing large amounts of data. Moreover, if we are to efficiently search for new drugs and new drug combinations, there is a pressing need for computational methods that could allow us to access the immense chemical compound space until now largely unexplored. Finally, at the microscopic level, drug-target chemistry simulation is ultimately a quantum problem, and hence it requires a quantum solution. As we will discuss, quantum computing may be a key ingredient in enabling the full potential of network medicine. We propose to combine network medicine and quantum algorithms in a novel research field, quantum network medicine, to lay the foundations of a new era of disease prevention and drug design.

We introduce a quantum Monte Carlo method to simulate the reversible dynamics of correlated many-body systems. Our method is based on the Laplace transform of the time-evolution operator which, as opposed to most quantum Monte Carlo methods, makes it possible to access the dynamics at longer times. The Monte Carlo trajectories are realised through a piece-wise stochastic-deterministic reversible evolution where free dynamics is interspersed with two-process quantum jumps. The dynamical sign problem is bypassed via the so-called deadweight approximation, which stabilizes the many-body phases at longer times. We benchmark our method by simulating spin excitation propagation in the XXZ model and dynamical confinement in the quantum Ising chain, and show how to extract dynamical information from the Laplace representation., Comment: 11 pages, 9 figures

In head-to-head racing, an accurate model of interactive behavior of the opposing target vehicle (TV) is required to perform tightly constrained, but highly rewarding maneuvers such as overtaking. However, such information is not typically made available in competitive scenarios, we therefore propose to construct a prediction and uncertainty model given data of the TV from previous races. In particular, a one-step Gaussian process (GP) model is trained on closed-loop interaction data to learn the behavior of a TV driven by an unknown policy. Predictions of the nominal trajectory and associated uncertainty are rolled out via a sampling-based approach and are used in a model predictive control (MPC) policy for the ego vehicle in order to intelligently trade-off between safety and performance when attempting overtaking maneuvers against a TV. We demonstrate the GP-based predictor in closed loop with the MPC policy in simulation races and compare its performance against several predictors from literature. In a Monte Carlo study, we observe that the GP-based predictor achieves similar win rates while maintaining safety in up to 3x more races. We finally demonstrate the prediction and control framework in real-time in a experimental study on a 1/10th scale racecar platform operating at speeds of around 2.8 m/s, and show a significant level of improvement when using the GP-based predictor over a baseline MPC predictor. Videos of the hardware experiments can be found at https://youtu.be/KMSs4ofDfIs., Comment: Submitted to IROS 2023

We leverage game theory and a new vehicle modeling approach to compute overtaking maneuvers for racecars on a nonplanar surface. We solve for equilibria between noncooperative racing agents and demonstrate that by leveraging the novel nonplanar vehicle dynamics, overtaking can be achieved in situations where simpler models can do not provide a winning strategy.

We present a 10 DoF dynamic vehicle model for model-based control on nonplanar road surfaces. A parametric surface is used to describe the road surface, allowing the surface parameterization to describe the pose of the vehicle. We use the proposed approach to compute minimum-time vehicle trajectories on nonplanar surfaces and compare planar and nonplanar models.

The problem of multimodal intent and trajectory prediction for human-driven vehicles in parking lots is addressed in this paper. Using models designed with CNN and Transformer networks, we extract temporal-spatial and contextual information from trajectory history and local bird's eye view (BEV) semantic images, and generate predictions about intent distribution and future trajectory sequences. Our methods outperform existing models in accuracy, while allowing an arbitrary number of modes, encoding complex multi-agent scenarios, and adapting to different parking maps. To train and evaluate our method, we present the first public 4K video dataset of human driving in parking lots with accurate annotation, high frame rate, and rich traffic scenarios., Comment: Published at IEEE ITSC 2022

In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result., Comment: 40 pages.The first part of the proof of Theorem 1.6 has been rewritten. Minor changes. Final version, to appear on Calc. Var. PDE

Dynamic games can be an effective approach to modeling interactive behavior between multiple non-cooperative agents and they provide a theoretical framework for simultaneous prediction and control in such scenarios. In this work, we propose a numerical method for the solution of local generalized Nash equilibria (GNE) for the class of open-loop general-sum dynamic games for agents with nonlinear dynamics and constraints. In particular, we formulate a sequential quadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration. Central to the robustness of our approach is a non-monotonic line search method and a novel merit function for SQP step acceptance. We show that our method achieves linear convergence in the neighborhood of local GNE and we derive an update rule for the merit function which helps to improve convergence from a larger set of initial conditions. We demonstrate the effectiveness of the algorithm in the context of car racing, where we show up to 32\% improvement of success rate when comparing against a state-of-the-art solution approach for dynamic games. \url{https://github.com/zhu-edward/DGSQP}.

We investigate the open dynamics of a chain of interacting spins using the quantized version of the GENERIC equation from classical out-of-equilibrium thermodynamics. We focus on both equilibrium and nonequilibrium scenarios for chains of different sizes. While in the equilibrium case we demonstrate thermal equilibration to the correct many-body Gibbs density matrix, in the nonequilibrium dynamics we show a ballistic-to-diffusive transition in the steady-state energy current and a scaling that is consistent with Fourier's law of heat transfer., Comment: 8 pages, 6 figures

The success of recurrent neural networks (RNNs) has been demonstrated in many applications related to turbulence, including flow control, optimization, turbulent features reproduction as well as turbulence prediction and modeling. With this study we aim to assess the capability of these networks to reproduce the temporal evolution of a minimal turbulent channel flow. We first obtain a data-driven model based on a modal decomposition in the Fourier domain (which we denote as FFT-POD) of the time series sampled from the flow. This particular case of turbulent flow allows us to accurately simulate the most relevant coherent structures close to the wall. Long-short-term-memory (LSTM) networks and a Koopman-based framework (KNF) are trained to predict the temporal dynamics of the minimal-channel-flow modes. Tests with different configurations highlight the limits of the KNF method compared to the LSTM, given the complexity of the flow under study. Long-term prediction for LSTM show excellent agreement from the statistical point of view, with errors below 2% for the best models with respect to the reference. Furthermore, the analysis of the chaotic behaviour through the use of the Lyapunov exponents and of the dynamic behaviour through Poincar\'e maps emphasizes the ability of the LSTM to reproduce the temporal dynamics of turbulence. Alternative reduced-order models (ROMs), based on the identification of different turbulent structures, are explored and they continue to show a good potential in predicting the temporal dynamics of the minimal channel.

We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation (introduced in J. Chem. Phys. 150, 234102, [2019]) to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schr\"odinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-rank matrices. The new approach facilitates accurate simulations of non-equilibrium quantum dynamics in larger and more complex open quantum systems. The performance of the method is demonstrated for a model of a molecular junction exhibiting current-induced mode-selective vibrational excitation., Comment: 12 pages, 4 figures

We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions., Comment: 20 pages. To appear on a special issue of Rendiconti Lincei: Matematica ed Applicazioni dedicated to Antonio Ambrosetti

We present a new approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from Density Matrix Quantum Monte Carlo and incorporates new crucial features. First of all, the dynamics is transferred to the Laplace representation where an exact equation can be derived and solved using a simulation-step that, unlike most Monte Carlo methods, is not a priori physically bounded. Moreover, the spawning events are formulated in terms of two-process stochastic unravellings of quantum master equations, a formalism that is particularly useful when working with density matrices. And last, this is equivalent to an interaction picture, where the free part is integrated exactly and the convergence rate can be greatly increased if the interaction parameter is small. We benchmark our method by applying it to two case-studies in condensed matter physics, show its accuracy and further discuss its efficiency., Comment: 10 pages, 6 figures

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities., Comment: Example 2.1 modified, typos corrected. 20 pages

We propose a Stochastic MPC (SMPC) formulation for autonomous driving at traffic intersections which incorporates multi-modal predictions of surrounding vehicles for collision avoidance constraints. The multi-modal predictions are obtained with Gaussian Mixture Models (GMM) and constraints are formulated as chance-constraints. Our main theoretical contribution is a SMPC formulation that optimizes over a novel feedback policy class designed to exploit additional structure in the GMM predictions, and that is amenable to convex programming. The use of feedback policies for prediction is motivated by the need for reduced conservatism in handling multi-modal predictions of the surrounding vehicles, especially prevalent in traffic intersection scenarios. We evaluate our algorithm along axes of mobility, comfort, conservatism and computational efficiency at a simulated intersection in CARLA. Our simulations use a kinematic bicycle model and multimodal predictions trained on a subset of the Lyft Level 5 prediction dataset. To demonstrate the impact of optimizing over feedback policies, we compare our algorithm with two SMPC baselines that handle multi-modal collision avoidance chance constraints by optimizing over open-loop sequences., Comment: Extended version of ITSC 2022 submission

We address the problem of finding the current position and heading angle of an autonomous vehicle in real-time using a single camera. Compared to methods which require LiDARs and high definition (HD) 3D maps in real-time, the proposed approach is easily scalable and computationally efficient, at the price of lower precision. The new method combines and adapts existing algorithms in three different fields: image retrieval, mapping database, and particle filtering. The result is a simple, real-time localization method using an image retrieval method whose performance is comparable to other monocular camera localization methods which use a map built with LiDARs. We evaluate the proposed method using the KITTI odometry dataset and via closed-loop experiments with an indoor 1:10 autonomous vehicle. The tests demonstrate real-time capability and a 10cm level accuracy. Also, experimental results of the closed-loop indoor tests show the presence of a positive feedback loop between the localization error and the control error. Such phenomena is analysed in details at the end of the article.