Search

The Quantum Approximate Optimization Algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ans\"atze for the maximum-cut (maxcut) problem on connected graphs, while focusing on the generated Lie algebras as well as their invariant subspaces. Specifically, we analyze the standard QAOA ansatz as well as the orbit and the multi-angle ans\"atze. We are able to fully characterize the Lie algebras of the multi-angle ansatz for arbitrary connected graphs, finding that they only fall into one of just six families. Besides the cycle and the path graphs, dimensions of every graph are exponentially large in the system size, meaning that multi-angle ans\"atze are extremely prone to exhibiting barren plateaus. Then, a similar quasi-graph-independent Lie-algebraic characterization beyond the multi-angle ansatz is impeded as the circuit exhibits additional "hidden" symmetries besides those naturally arising from a certain parity-superselection operator and all automorphisms of the considered graph. Disregarding the "hidden" symmetries, we can upper bound the dimensions of the orbit and the standard Lie algebras, and the dimensions of the associated invariant subspaces are determined via explicit character formulas. To finish, we conjecture that (for most graphs) the standard Lie algebras have only components that are either exponential or that grow, at most, polynomially with the system size. This would imply that the QAOA is either prone to barren plateaus, or classically simulable., Comment: 17+24 pages, 7+7 figures, 4+3 tables, comments and feedback are welcome

Quantum sensing is one of the most promising applications for quantum technologies. However, reaching the ultimate sensitivities enabled by the laws of quantum mechanics can be a challenging task in realistic scenarios where noise is present. While several strategies have been proposed to deal with the detrimental effects of noise, these come at the cost of an extra shot budget. Given that shots are a precious resource for sensing -- as infinite measurements could lead to infinite precision -- care must be taken to truly guarantee that any shot not being used for sensing is actually leading to some metrological improvement. In this work, we study whether investing shots in error-mitigation, inference techniques, or combinations thereof, can improve the sensitivity of a noisy quantum sensor on a (shot) budget. We present a detailed bias-variance error analysis for various sensing protocols. Our results show that the costs of zero-noise extrapolation techniques outweigh their benefits. We also find that pre-characterizing a quantum sensor via inference techniques leads to the best performance, under the assumption that the sensor is sufficiently stable., Comment: 17+18 pages, 8+2 figures, 1+1 tables

A unitary state $t$-design is an ensemble of pure quantum states whose moments match up to the $t$-th order those of states uniformly sampled from a $d$-dimensional Hilbert space. Typically, unitary state $t$-designs are obtained by evolving some reference pure state with unitaries from an ensemble that forms a design over the unitary group $\mathbb{U}(d)$, as unitary designs induce state designs. However, in this work we study whether Haar random symplectic states -- i.e., states obtained by evolving some reference state with unitaries sampled according to the Haar measure over $\mathbb{SP}(d/2)$ -- form unitary state $t$-designs. Importantly, we recall that random symplectic unitaries fail to be unitary designs for $t>1$, and that, while it is known that symplectic unitaries are universal, this does not imply that their Haar measure leads to a state design. Notably, our main result states that Haar random symplectic states form unitary $t$-designs for all $t$, meaning that their distribution is unconditionally indistinguishable from that of unitary Haar random states, even with tests that use infinite copies of each state. As such, our work showcases the intriguing possibility of creating state $t$-designs using ensembles of unitaries which do not constitute designs over $\mathbb{U}(d)$ themselves, such as ensembles that form $t$-designs over $\mathbb{SP}(d/2)$., Comment: 6+11 pages, 1+1 figures

We present a classical algorithm for estimating expectation values of arbitrary observables on most quantum circuits across all circuit architectures and depths, including those with all-to-all connectivity. We prove that for any architecture where each circuit layer is equipped with a measure invariant under single-qubit rotations, our algorithm achieves a small error $\varepsilon$ on all circuits except for a small fraction $\delta$. The computational time is polynomial in qubit count and circuit depth for any small constant $\varepsilon, \delta$, and quasi-polynomial for inverse-polynomially small $\varepsilon, \delta$. For non-classically-simulable input states or observables, the expectation values can be estimated by augmenting our algorithm with classical shadows of the relevant state or observable. Our approach leverages a Pauli-path method under Heisenberg evolution. While prior works are limited to noisy quantum circuits, we establish classical simulability in noiseless regimes. Given that most quantum circuits in an architecture exhibit chaotic and locally scrambling behavior, our work demonstrates that estimating observables of such quantum dynamics is classically tractable across all geometries., Comment: Main text: 8 pages, 3 figures. Appendices: 25 pages, 1 figure

Quantum Convolutional Neural Networks (QCNNs) are widely regarded as a promising model for Quantum Machine Learning (QML). In this work we tie their heuristic success to two facts. First, that when randomly initialized, they can only operate on the information encoded in low-bodyness measurements of their input states. And second, that they are commonly benchmarked on "locally-easy'' datasets whose states are precisely classifiable by the information encoded in these low-bodyness observables subspace. We further show that the QCNN's action on this subspace can be efficiently classically simulated by a classical algorithm equipped with Pauli shadows on the dataset. Indeed, we present a shadow-based simulation of QCNNs on up-to $1024$ qubits for phases of matter classification. Our results can then be understood as highlighting a deeper symptom of QML: Models could only be showing heuristic success because they are benchmarked on simple problems, for which their action can be classically simulated. This insight points to the fact that non-trivial datasets are a truly necessary ingredient for moving forward with QML. To finish, we discuss how our results can be extrapolated to classically simulate other architectures., Comment: 11 + 13 pages , 6 + 3 figures, 1 table

The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits., Comment: 9 + 6 pages, 3 figures

We introduce a framework for simulating, on an $(n+1)$-qubit quantum computer, the action of a Gaussian Bosonic (GB) circuit on a state over $2^n$ modes. Specifically, we encode the initial bosonic state's expectation values over quadrature operators (and their covariance matrix) as an input qubit-state. This is then evolved by a quantum circuit that effectively implements the symplectic propagators induced by the GB gates. We find families of GB circuits and initial states leading to efficient quantum simulations. For this purpose, we introduce a dictionary that maps between GB and qubit gates such that particle- (non-particle-) preserving GB gates lead to real (imaginary) time evolutions at the qubit level. For the special case of particle-preserving circuits, we present a BQP-complete GB decision problem, indicating that GB evolutions of Gaussian states on exponentially many modes are as powerful as universal quantum computers. We also perform numerical simulations of an interferometer on $\sim8$ billion modes, illustrating the power of our framework., Comment: 5+14 pages, 3 figures, 1 table

Assuming the availability of a reliable aircraft trajectory planner, this paper presents a probabilistic methodology to detect conflicts between aircraft, in the cruise phase of the flight, in the presence of wind prediction uncertainties quantified by ensemble weather forecasts, which are regarded as realizations of correlated random processes and employed to derive the eastward and northward components of the wind velocity. First, the Karhunen-Lo`eve expansion is used to obtain a series expansion of the wind components in terms of a set of uncorrelated random variables and deterministic coefficients. Then, the uncertainty induced by these uncorrelated random variables in the outputs of the aircraft trajectory planner is quantified by means of the arbitrary polynomial chaos technique. Finally, the probability density function of the great circle distance between each pair of aircraft is derived from the polynomial expansions using a Gaussian kernel density estimator and employed to estimate the probability of conflict. The arbitrary polynomial chaos technique allows the effects of uncertainties in complex nonlinear dynamical system, such as those underlying aircraft trajectory planners, to be quantified with high computational efficiency, only requiring the existence of a finite number of statistical moments of the random variables of the Karhunen-Lo`eve expansion, while avoiding any assumption on their probability distributions. In order to demonstrate the effectiveness of the proposed conflict detection method, numerical experiments are conducted through an optimal control based aircraft trajectory planner for a given wind forecast represented by an ensemble prediction system.

This paper studies the formation mission design problem for commercial aircraft in the presence of uncertainties. Specifically, it considers uncertainties in the departure times of the aircraft and in the fuel burn savings for the trailing aircraft. Given several commercial flights, the problem consists in arranging them in formation or solo flights and finding the trajectories that minimize the expected value of the direct operating cost of the flights. The formation mission design problem is formulated as an optimal control problem of a stochastic switched dynamical system and solved using nonintrusive generalized polynomial chaos based stochastic collocation. The stochastic collocation method converts the stochastic switched optimal control problem into an augmented deterministic switched optimal control problem. With this approach, a small number of sample points of the random parameters are used to jointly solve particular instances of the switched optimal control problem. The obtained solutions are then expressed as orthogonal polynomial expansions in terms of the random parameters using these sample points. This technique allows statistical and global sensitivity analysis of the stochastic solutions to be conducted at a low computational cost. The aim of this study is to establish if, in the presence of uncertainties, a formation mission is beneficial with respect to solo flight in terms of the expected value of the direct operating costs. Several numerical experiments have been conducted in which uncertainties on the departure times and on the fuel saving during formation flight have been considered. The obtained results demonstrate that benefits can be achieved even in the presence of these uncertainties.

In this article, the formation mission design problem for commercial aircraft is studied. Given the departure times and the departure and arrival locations of several commercial flights, the relevant weather forecast, and the expected fuel savings during formation flight, the problem consists in establishing how to organize them in formation or solo flights and in finding the trajectories that minimize the overall direct operating cost of the flights. Each aircraft can fly solo or in different positions inside a formation. Therefore, the mission is modeled as a switched dynamical system, in which the discrete state describes the combination of flight modes of the individual aircraft and logical constraints in disjunctive form establish the switching logic among the discrete states of the system. The formation mission design problem has been formulated as an optimal control problem of a switched dynamical system and solved using an embedding approach, which allows switching decision among discrete states to be modeled without relying on binary variables. The resulting problem is a classical optimal control problem which has been solved using a knotting pseudospectral method. Several numerical experiments have been conducted to demonstrate the effectiveness of this approach. The obtained results show that formation flight has great potential to reduce fuel consumption and emissions.

Parametrized and random unitary (or orthogonal) $n$-qubit circuits play a central role in quantum information. As such, one could naturally assume that circuits implementing symplectic transformation would attract similar attention. However, this is not the case, as $\mathbb{SP}(d/2)$ -- the group of $d\times d$ unitary symplectic matrices -- has thus far been overlooked. In this work, we aim at starting to right this wrong. We begin by presenting a universal set of generators $\mathcal{G}$ for the symplectic algebra $i\mathfrak{sp}(d/2)$, consisting of one- and two-qubit Pauli operators acting on neighboring sites in a one-dimensional lattice. Here, we uncover two critical differences between such set, and equivalent ones for unitary and orthogonal circuits. Namely, we find that the operators in $\mathcal{G}$ cannot generate arbitrary local symplectic unitaries and that they are not translationally invariant. We then review the Schur-Weyl duality between the symplectic group and the Brauer algebra, and use tools from Weingarten calculus to prove that Pauli measurements at the output of Haar random symplectic circuits can converge to Gaussian processes. As a by-product, such analysis provides us with concentration bounds for Pauli measurements in circuits that form $t$-designs over $\mathbb{SP}(d/2)$. To finish, we present tensor-network tools to analyze shallow random symplectic circuits, and we use these to numerically show that computational-basis measurements anti-concentrate at logarithmic depth., Comment: 13+7 pages, 8 figures

Variational quantum computing offers a flexible computational paradigm with applications in diverse areas. However, a key obstacle to realizing their potential is the Barren Plateau (BP) phenomenon. When a model exhibits a BP, its parameter optimization landscape becomes exponentially flat and featureless as the problem size increases. Importantly, all the moving pieces of an algorithm -- choices of ansatz, initial state, observable, loss function and hardware noise -- can lead to BPs when ill-suited. Due to the significant impact of BPs on trainability, researchers have dedicated considerable effort to develop theoretical and heuristic methods to understand and mitigate their effects. As a result, the study of BPs has become a thriving area of research, influencing and cross-fertilizing other fields such as quantum optimal control, tensor networks, and learning theory. This article provides a comprehensive review of the current understanding of the BP phenomenon., Comment: 21 pages, 10 boxes

In this paper, we introduce a novel formulation for camera motion estimation that integrates RGB-D images and inertial data through scene flow. Our goal is to accurately estimate the camera motion in a rigid 3D environment, along with the state of the inertial measurement unit (IMU). Our proposed method offers the flexibility to operate as a multi-frame optimization or to marginalize older data, thus effectively utilizing past measurements. To assess the performance of our method, we conducted evaluations using both synthetic data from the ICL-NUIM dataset and real data sequences from the OpenLORIS-Scene dataset. Our results show that the fusion of these two sensors enhances the accuracy of camera motion estimation when compared to using only visual data., Comment: Accepted to CVPR2024 Workshop on Visual Odometry and Computer Vision Applications

This paper presents two methods for implementing complex-valued sliding mode controllers in three-phase power converters. The paper includes the description of the algorithms and a detailed analysis of the proposed implementations. The methods, that are easy to code and have a low computational burden, retain the sliding mode properties of robustness and fast response and do not require any additional processing often used to decouple the dynamics of the three-phase system. The performance of the methods is compared in numerical simulations, and the algorithms are experimentally tested in a microcontroller using a Hardware-in-the-Loop platform.

Atom probe tomography data is composed of a list of coordinates of the reconstructed atoms in the probed volume. The elemental identity of each atom is derived from time-of-flight mass spectrometry, with no local energetic or chemical information readily available within the mass spectrum. Here, we used a new data processing technique referred to as field evaporation energy loss spectroscopy (FEELS), which analyses the tails of mass peaks. FEELS was used to extract critical energetic parameters that characterize the field evaporation process, which are related to the binding energy of atoms to the surface under intense electrostatic field and dependent of the path followed by the departing atoms during the field evaporation process. We focused our study on different pure face centered cubic metals (Al, Ni, Rh). We demonstrate that the energetic parameters extracted from mass spectra can be mapped in 2D with nanometric resolution. A dependence on the considered crystallographic planes is observed, with sets of planes of low Miller indices showing a lower sensitivity to the intensity of the electric field, which indicates a lower effective attachment energy. The temperature is also an important parameter in particular for Al, which we attribute to an energetic transition between two paths of field evaporation between 25K and 60K close to (002) pole at the specimen's surface. This paper shows that the complex information that can be retrieved from the measured energy loss of surface atoms is important both instrumentally and fundamentally.

Geometric algebra (GA) is a mathematical tool for geometric computing, providing a framework that allows a unified and compact approach to geometric relations which in other mathematical systems are typically described using different more complicated elements. This fact has led to an increasing adoption of GA in applied mathematics and engineering problems. However, the scarcity of symbolic implementations of GA and its inherent complexity, requiring a specific mathematical background, make it challenging and less intuitive for engineers to work with. This prevents wider adoption among more applied professionals. To address this challenge, this paper introduces SUGAR (Symbolic and User-friendly Geometric Algebra Routines), an open-source toolbox designed for Matlab and licensed under the MIT License. SUGAR facilitates the translation of GA concepts into Matlab and provides a collection of user-friendly functions tailored for GA computations, including support for symbolic operations. It supports both numeric and symbolic computations in high-dimensional GAs. Specifically tailored for applied mathematics and engineering applications, SUGAR has been meticulously engineered to represent geometric elements and transformations within two and three-dimensional projective and conformal geometric algebras, aligning with established computational methodologies in the literature. Furthermore, SUGAR efficiently handles functions of multivectors, such as exponential, logarithmic, sinusoidal, and cosine functions, enhancing its applicability across various engineering domains, including robotics, control systems, and power electronics. Finally, this work includes four distinct validation examples, demonstrating SUGAR's capabilities across the above-mentioned fields and its practical utility in addressing real-world applied mathematics and engineering problems., Comment: 33 pages, 6 figures, journal paper submitted to ACM TOMS

We construct, for any $H\in \mathbb{R}$, infinitely many free boundary annuli in geodesic balls of $\mathbb{S}^3$ with constant mean curvature $H$ and a discrete, non-rotational, symmetry group. Some of these free boundary CMC annuli are actually embedded if $H\geq 1/\sqrt{3}$. We also construct embedded, non-rotational, free boundary CMC annuli in geodesic balls of $\mathbb{H}^3$, for all values $H>1$ of the mean curvature $H$., Comment: 42 pages, 7 figures

A basic primitive in quantum information is the computation of the moments $\mathbb{E}_U[{\rm Tr}[U\rho U^\dagger O]^t]$. These describe the distribution of expectation values obtained by sending a state $\rho$ through a random unitary $U$, sampled from some distribution, and measuring the observable $O$. While the exact calculation of these moments is generally hard, if $U$ is composed of local random gates, one can estimate $\mathbb{E}_U[{\rm Tr}[U\rho U^\dagger O]^t]$ by performing Monte Carlo simulations of a Markov chain-like process. However, this approach can require a prohibitively large number of samples, or suffer from the sign problem. In this work, we instead propose to estimate the moments via tensor networks, where the local gates moment operators are mapped to small dimensional tensors acting on their local commutant bases. By leveraging representation theoretical tools, we study the local tensor dimension and we provide bounds for the bond dimension of the matrix product states arising from deep circuits. We compare our techniques against Monte Carlo simulations, showing that we can significantly out-perform them. Then, we showcase how tensor networks can exactly compute the second moment when $U$ is a quantum neural network acting on thousands of qubits and having thousands of gates. To finish, we numerically study the anticoncentration phenomena of circuits with orthogonal random gates, a task which cannot be studied via Monte Carlo due to sign problems., Comment: 15 + 8 pages, 9 figures, updated to published version

This paper describes a holographic mobile-based application designed to help Spanish-speaking children to practice the pronunciation of basic English vocabulary words. The mastery of vocabulary is a fundamental step when learning a language but is often perceived as boring. Producing the correct pronunciation is frequently regarded as the most difficult and complex skill for new learners of English. In order to address these problems this research takes advantage of the power of multi-channel stimuli (sound, image and interaction) in a mobilebased hologram application in order to motivate students and improve their experience of practicing. We adapted the prize-winning HolograFX game and developed a new mobile application to help practice English pronunciation. A 3D holographic robot that acts as a virtual teacher interacts via voice with the children. To test the tool we carried out an experiment with 70 Spanish pre-school children divided into three classes, the control group using traditional methods such as images in books and on the blackboard, and two experimental groups using our drills and practice software. One experimental group used the mobile application without the holographic game and the other experimental group used the application with the holographic game. We performed pre-test and post-test performance assessments, a satisfaction survey and emotion analysis. The results are very promising. They show that the use of the holographic mobile-based application had a significant impact on the children's motivation. It also improved their performance compared to traditional methods used in the classroom.

Content assessment has broadly improved in e-learning scenarios in recent decades. However, the eLearning process can give rise to a spatial and temporal gap that poses interesting challenges for assessment of not only content, but also students' acquisition of core skills such as self-regulated learning. Our objective was to discover students' self-regulated learning processes during an eLearning course by using Process Mining Techniques. We applied a new algorithm in the educational domain called Inductive Miner over the interaction traces from 101 university students in a course given over one semester on the Moodle 2.0 platform. Data was extracted from the platform's event logs with 21629 traces in order to discover students' self-regulation models that contribute to improving the instructional process. The Inductive Miner algorithm discovered optimal models in terms of fitness for both Pass and Fail students in this dataset, as well as models at a certain level of granularity that can be interpreted in educational terms, which are the most important achievement in model discovery. We can conclude that although students who passed did not follow the instructors' suggestions exactly, they did follow the logic of a successful self-regulated learning process as opposed to their failing classmates. The Process Mining models also allow us to examine which specific actions the students performed, and it was particularly interesting to see a high presence of actions related to forum-supported collaborative learning in the Pass group and an absence of those in the Fail group.

The aim of this study was to predict university students' learning performance using different sources of data from an Intelligent Tutoring System. We collected and preprocessed data from 40 students from different multimodal sources: learning strategies from system logs, emotions from face recording videos, interaction zones from eye tracking, and test performance from final knowledge evaluation. Our objective was to test whether the prediction could be improved by using attribute selection and classification ensembles. We carried out three experiments by applying six classification algorithms to numerical and discretized preprocessed multimodal data. The results show that the best predictions were produced using ensembles and selecting the best attributes approach with numerical data.

In this paper we applied data fusion approaches for predicting the final academic performance of university students using multiple-source, multimodal data from blended learning environments. We collected and preprocessed data about first-year university students from different sources: theory classes, practical sessions, on-line Moodle sessions, and a final exam. Our objective was to discover which data fusion approach produced the best results using our data. We carried out experiments by applying four different data fusion approaches and six classification algorithms. The results showed that the best predictions were produced using ensembles and selecting the best attributes approach with discretized data. The best prediction models showed us that the level of attention in theory classes, scores in Moodle quizzes, and the level of activity in Moodle forums were the best set of attributes for predicting students' final performance in our courses.

Citizens are actively interacting with their surroundings, especially through social media. Not only do shared posts give important information about what is happening (from the users' perspective), but also the metadata linked to these posts offer relevant data, such as the GPS-location in Location-based Social Networks (LBSNs). In this paper we introduce a global analysis of the geo-tagged posts in social media which supports (i) the detection of unexpected behavior in the city and (ii) the analysis of the posts to infer what is happening. The former is obtained by applying density-based clustering techniques, whereas the latter is consequence of applying natural language processing. We have applied our methodology to a dataset obtained from Instagram activity in New York City for seven months obtaining promising results. The developed algorithms require very low resources, being able to analyze millions of data-points in commodity hardware in less than one hour without applying complex parallelization techniques. Furthermore, the solution can be easily adapted to other geo-tagged data sources without extra effort., Comment: 11 pages, 12 figures, journal

A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We present strong evidence that commonly used models with provable absence of barren plateaus are also classically simulable, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing quantum computers can be essential for collecting data, our analysis sheds serious doubt on the non-classicality of the information processing capabilities of parametrized quantum circuits for barren plateau-free landscapes. We end by discussing caveats in our arguments, the role of smart initializations and the possibility of provably superpolynomial, or simply practical, advantages from running parametrized quantum circuits., Comment: 14+15 pages, 5 figures, 2 tables, minor corrections added

Major advances in information and communication technologies (ICTs) make citizens to be considered as sensors in motion. Carrying their mobile devices, moving in their connected vehicles or actively participating in social networks, citizens provide a wealth of information that, after properly processing, can support numerous applications for the benefit of the community. In the context of smart communities, the INRISCO proposal intends for (i) the early detection of abnormal situations in cities (i.e., incidents), (ii) the analysis of whether, according to their impact, those incidents are really adverse for the community; and (iii) the automatic actuation by dissemination of appropriate information to citizens and authorities. Thus, INRISCO will identify and report on incidents in traffic (jam, accident) or public infrastructure (e.g., works, street cut), the occurrence of specific events that affect other citizens life (e.g., demonstrations, concerts), or environmental problems (e.g., pollution, bad weather). It is of particular interest to this proposal the identification of incidents with a social and economic impact, which affects the quality of life of citizens.

Symmetries are crucial for tailoring parametrized quantum circuits to applications, due to their capability to capture the essence of physical systems. In this work, we shift the focus away from incorporating symmetries in the circuit design and towards symmetry-aware training of variational quantum algorithms. For this, we introduce the concept of projected derivatives of parametrized quantum circuits, in particular the equivariant and covariant derivatives. We show that the covariant derivative gives rise to the quantum Fisher information and quantum natural gradient. This provides an operational meaning for the covariant derivative, and allows us to extend the quantum natural gradient to all continuous symmetry groups. Connecting to traditional particle physics, we confirm that our covariant derivative is the same as the one introduced in physical gauge theory. This work provides tools for tailoring variational quantum algorithms to symmetries by incorporating them locally in derivatives, rather than into the design of the circuit., Comment: 22+20 pages, 6+1 figures

State-of-the-art techniques for modeling, analysis and control of three-phase electrical systems belong to the real-valued multi-input/multi-output (MIMO) domain, or to the complex-valued nonlinear single-input/single-output (SISO) domain. In order to complement both domains while simplifying complexity and offering new analysis and design perspectives, this paper introduces the application of geometric algebra (GA) principles to the modeling, analysis and control of three-phase electrical systems. The key contribution for the modeling part is the identification of the transformation that allows transferring real-valued linear MIMO systems into GA-valued linear SISO representations (with independence of having a balanced or unbalanced system). Closed-loop stability analysis in the new space is addressed by using intrinsic properties of GA. In addition, a recipe for designing stabilizing and decoupling GA-valued controllers is provided. Numerical examples illustrate key developments and experiments corroborate the main findings.

Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: The dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability., Comment: 5+26 pages, 2+1 figures

Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit's generators., Comment: 11+17 pages, 2+2 figures, updated manuscript and title to published version

Online Mental Health Communities (OMHCs), such as Reddit, have witnessed a surge in popularity as go-to platforms for seeking information and support in managing mental health needs. Platforms like Reddit offer immediate interactions with peers, granting users a vital space for seeking mental health assistance. However, the largely unregulated nature of these platforms introduces intricate challenges for both users and society at large. This study explores the factors that drive peer engagement within counseling threads, aiming to enhance our understanding of this critical phenomenon. We introduce BeCOPE, a novel behavior encoded Peer counseling dataset comprising over 10,118 posts and 58,279 comments sourced from 21 mental health-specific subreddits. The dataset is annotated using three major fine-grained behavior labels: (a) intent, (b) criticism, and (c) readability, along with the emotion labels. Our analysis indicates the prominence of ``self-criticism'' as the most prevalent form of criticism expressed by help-seekers, accounting for a significant 43% of interactions. Intriguingly, we observe that individuals who explicitly express their need for help are 18.01% more likely to receive assistance compared to those who present ``surveys'' or engage in ``rants.'' Furthermore, we highlight the pivotal role of well-articulated problem descriptions, showing that superior readability effectively doubles the likelihood of receiving the sought-after support. Our study emphasizes the essential role of OMHCs in offering personalized guidance and unveils behavior-driven engagement patterns.

In the past few decades, researchers have created a veritable zoo of quantum algorithm by drawing inspiration from classical computing, information theory, and even from physical phenomena. Here we present quantum algorithms for parallel-in-time simulations that are inspired by the Page and Wooters formalism. In this framework, and thus in our algorithms, the classical time-variable of quantum mechanics is promoted to the quantum realm by introducing a Hilbert space of "clock" qubits which are then entangled with the "system" qubits. We show that our algorithms can compute temporal properties over $N$ different times of many-body systems by only using $\log(N)$ clock qubits. As such, we achieve an exponential trade-off between time and spatial complexities. In addition, we rigorously prove that the entanglement created between the system qubits and the clock qubits has operational meaning, as it encodes valuable information about the system's dynamics. We also provide a circuit depth estimation of all the protocols, showing an exponential advantage in computation times over traditional sequential in time algorithms. In particular, for the case when the dynamics are determined by the Aubry-Andre model, we present a hybrid method for which our algorithms have a depth that only scales as $\mathcal{O}(\log(N)n)$. As a by product we can relate the previous schemes to the problem of equilibration of an isolated quantum system, thus indicating that our framework enable a new dimension for studying dynamical properties of many-body systems., Comment: 19+15 pages, 18+1 figures

Classical simulation of quantum dynamics plays an important role in our understanding of quantum complexity, and in the development of quantum technologies. Compared to other techniques for efficient classical simulations, methods relying on the Lie-algebraic structure of quantum dynamics have received relatively little attention. At their core, these simulations leverage the underlying Lie algebra - and the associated Lie group - of a dynamical process. As such, rather than keeping track of the individual entries of large matrices, one instead keeps track of how its algebraic decomposition changes during the evolution. When the dimension of the algebra is small (e.g., growing at most polynomially in the system size), one can leverage efficient simulation techniques. In this work, we review the basis for such methods, presenting a framework that we call "$\mathfrak{g}$-sim", and showcase their efficient implementation in several paradigmatic variational quantum computing tasks. Specifically, we perform Lie-algebraic simulations to train and optimize parametrized quantum circuits, design enhanced parameter initialization strategies, solve tasks of quantum circuit synthesis, and train a quantum-phase classifier., Comment: 17 pages main text, 12 pages appendices

Visual cryptography aims to protect images against their possible illegitimate use. Thus, one can cipher, hash, or add watermarks for protecting copyright, among others. In this paper we provide a new solution to the problem of secret sharing for the case when the secret is an image. Our method combines the Shamir scheme for secret sharing using finite fields of characteristic 2 with the CBC mode of operation of a secure symmetric cryptographic scheme like AES, so that the security relies on that of the mentioned techniques. The resulting shares have the same resolution as that of the original image. The idea of the method could be generalized to other multimedia formats like audio or video, adapting the method to the corresponding encoded information., Comment: 6 pages, 5 figures, preprint

Recent advances in the field of intelligent robotic manipulation pursue providing robotic hands with touch sensitivity. Haptic perception encompasses the sensing modalities encountered in the sense of touch (e.g., tactile and kinesthetic sensations). This letter focuses on multimodal object recognition and proposes analytical and data-driven methodologies to fuse tactile- and kinesthetic-based classification results. The procedure is as follows: a three-finger actuated gripper with an integrated high-resolution tactile sensor performs squeeze-and-release Exploratory Procedures (EPs). The tactile images and kinesthetic information acquired using angular sensors on the finger joints constitute the time-series datasets of interest. Each temporal dataset is fed to a Long Short-term Memory (LSTM) Neural Network, which is trained to classify in-hand objects. The LSTMs provide an estimation of the posterior probability of each object given the corresponding measurements, which after fusion allows to estimate the object through Bayesian and Neural inference approaches. An experiment with 36-classes is carried out to evaluate and compare the performance of the fused, tactile, and kinesthetic perception systems.The results show that the Bayesian-based classifiers improves capabilities for object recognition and outperforms the Neural-based approach.