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Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

In this work a quantum analogue of Bayesian statistical inference is considered. Based on the notion of instrument, we propose a sequential measurement scheme from which observations needed for statistical inference are obtained. We further put forward a quantum analogue of Bayes rule, which states how the prior normal state of a quantum system updates under those observations. We next generalize the fundamental notions and results of Bayesian statistics according to the quantum Bayes rule. It is also note that our theory retains the classical one as its special case. Finally, we investigate the limit of posterior normal state as the number of observations tends to infinity.

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics by proving two results that do not hold in the subclass of monotone automata. First, it is undecidable whether the automaton almost surely fills the space when initialized on a Bernoulli random configuration with density $p$, for some/all $0 < p < 1$. Second, there exists an automaton whose space-filling property depends on $p$ in a non-monotone way.

The security of quantum key distribution (QKD) is severely threatened by discrepancies between realistic devices and theoretical assumptions. Recently, a significant framework called the reference technique was proposed to provide security against arbitrary source flaws, including pulse correlations. Here, we propose an efficient four-phase twin-field QKD using laser pulses adopting the reference technique for security against all possible source imperfections. We present a characterization of source flaws and connect them to experimental data, together with a finite-key analysis. In addition, we demonstrate the feasibility of our protocol through a proof-of-principle experimental implementation and demonstrate a secure key rate of 1.63 kbps with a 20 dB channel loss. Compared with previous QKD protocols with imperfect devices, our work considerably improves both the secure key rate and the transmission distance, and shows application potential in the practical deployment of secure QKD with device imperfections.

We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system+bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long-range interaction systems.

With the advent of open source software, a veritable treasure trove of previously proprietary software development data was made available. This opened the field of empirical software engineering research to anyone in academia. Data that is mined from software projects, however, requires extensive processing and needs to be handled with utmost care to ensure valid conclusions. Since the software development practices and tools have changed over two decades, we aim to understand the state-of-the-art research workflows and to highlight potential challenges. We employ a systematic literature review by sampling over one thousand papers from leading conferences and by analyzing the 286 most relevant papers from the perspective of data workflows, methodologies, reproducibility, and tools. We found that an important part of the research workflow involving dataset selection was particularly problematic, which raises questions about the generality of the results in existing literature. Furthermore, we found a considerable number of papers provide little or no reproducibility instructions -- a substantial deficiency for a data-intensive field. In fact, 33% of papers provide no information on how their data was retrieved. Based on these findings, we propose ways to address these shortcomings via existing tools and also provide recommendations to improve research workflows and the reproducibility of research.

Recent decades, the emergence of numerous novel algorithms makes it a gimmick to propose an intelligent optimization system based on metaphor, and hinders researchers from exploring the essence of search behavior in algorithms. However, it is difficult to directly discuss the search behavior of an intelligent optimization algorithm, since there are so many kinds of intelligent schemes. To address this problem, an intelligent optimization system is regarded as a simulated physical optimization system in this paper. The dynamic search behavior of such a simplified physical optimization system are investigated with quantum theory. To achieve this goal, the Schroedinger equation is employed as the dynamics equation of the optimization algorithm, which is used to describe dynamic search behaviours in the evolution process with quantum theory. Moreover, to explore the basic behaviour of the optimization system, the optimization problem is assumed to be decomposed and approximated. Correspondingly, the basic search behaviour is derived, which constitutes the basic iterative process of a simple optimization system. The basic iterative process is compared with some classical bare-bones schemes to verify the similarity of search behavior under different metaphors. The search strategies of these bare bones algorithms are analyzed through experiments.

After spending 9 years in Quantum Computing and given the impending timeline of developing good quality quantum processing units, it is the moment to rethink the approach to advance quantum computing research. Rather than waiting for quantum hardware technologies to mature, we need to start assessing in tandem the impact of the occurrence of quantum computing in various scientific fields. However, for this purpose, we need to use a complementary but quite different approach than proposed by the NISQ vision, which is heavily focused on and burdened by the engineering challenges. That is why we propose and advocate the PISQ-approach: Perfect Intermediate-Scale Quantum computing based on the already known concept of perfect qubits. This will allow researchers to focus much more on the development of new applications by defining the algorithms in terms of perfect qubits and evaluating them on quantum computing simulators that are executed on supercomputers. It is not a long-term solution but it will allow universities to currently develop research on quantum logic and algorithms and companies can already start developing their internal know-how on quantum solutions.

White noise is a fundamental and fairly well understood stochastic process that conforms the conceptual basis for many other processes, as well as for the modeling of time series. Here we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to give new interesting insights both for modelling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree $n$, thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.