Search

Chemical bonding is a nonlocal phenomenon that binds atoms into molecules. Its ubiquitous presence in chemistry, however, stands in stark contrast to its ambiguous definition and the lack of a universal perspective for its understanding. In this work, we rationalize and characterize chemical bonding through the lens of an equally nonlocal concept from quantum information, the orbital entanglement. We introduce maximally entangled atomic orbitals (MEAOs) whose entanglement pattern is shown to recover both Lewis (two-center) and beyond-Lewis (multicenter) structures, with multipartite entanglement serving as a comprehensive index of bond strength. Our unifying framework for bonding analyses is effective not only for equilibrium geometries but also for transition states in chemical reactions and complex phenomena such as aromaticity. It also has the potential to elevate the Hilbert space atomic partitioning to match the prevalent real-space partitioning in the theory of atoms in molecules. Accordingly, our work opens new pathways for understanding fuzzy chemical concepts using rigorous, quantitative descriptors from quantum information., Comment: 9+7 pages, 5+5 figures

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $\varphi_i$ according to $0 \leq n_i \leq 2$. In this work, we first refine the underlying one-body $N$-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness $\boldsymbol w$ of the $N$-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope $\Sigma_{N,S}(\boldsymbol w) \subset [0,2]^d$. These constraints are independent of $M$ and the number $d$ of orbitals, while their dependence on $N, S$ is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

Deep reinforcement learning has emerged as a powerful tool for obtaining high-performance policies. However, the safety of these policies has been a long-standing issue. One promising paradigm to guarantee safety is a shield, which shields a policy from making unsafe actions. However, computing a shield scales exponentially in the number of state variables. This is a particular concern in multi-agent systems with many agents. In this work, we propose a novel approach for multi-agent shielding. We address scalability by computing individual shields for each agent. The challenge is that typical safety specifications are global properties, but the shields of individual agents only ensure local properties. Our key to overcome this challenge is to apply assume-guarantee reasoning. Specifically, we present a sound proof rule that decomposes a (global, complex) safety specification into (local, simple) obligations for the shields of the individual agents. Moreover, we show that applying the shields during reinforcement learning significantly improves the quality of the policies obtained for a given training budget. We demonstrate the effectiveness and scalability of our multi-agent shielding framework in two case studies, reducing the computation time from hours to seconds and achieving fast learning convergence.

Decision trees, owing to their interpretability, are attractive as control policies for (dynamical) systems. Unfortunately, constructing, or synthesising, such policies is a challenging task. Previous approaches do so by imitating a neural-network policy, approximating a tabular policy obtained via formal synthesis, employing reinforcement learning, or modelling the problem as a mixed-integer linear program. However, these works may require access to a hard-to-obtain accurate policy or a formal model of the environment (within reach of formal synthesis), and may not provide guarantees on the quality or size of the final tree policy. In contrast, we present an approach to synthesise optimal decision-tree policies given a deterministic black-box environment and specification, a discretisation of the tree predicates, and an initial set of states, where optimality is defined with respect to the number of steps to achieve the goal. Our approach is a specialised search algorithm which systematically explores the (exponentially large) space of decision trees under the given discretisation. The key component is a novel trace-based pruning mechanism that significantly reduces the search space. Our approach represents a conceptually novel way of synthesising small decision-tree policies with optimality guarantees even for black-box environments with black-box specifications., Comment: 8 pages main text incl. references, 2 pages appendix

We consider the problem of synthesizing safety strategies for control systems, also known as shields. Since the state space is infinite, shields are typically computed over a finite-state abstraction, with the most common abstraction being a rectangular grid. However, for many systems, such a grid does not align well with the safety property or the system dynamics. That is why a coarse grid is rarely sufficient, but a fine grid is typically computationally infeasible to obtain. In this paper, we show that appropriate state-space transformations can still allow to use a coarse grid at almost no computational overhead. We demonstrate in three case studies that our transformation-based synthesis outperforms a standard synthesis by several orders of magnitude. In the first two case studies, we use domain knowledge to select a suitable transformation. In the third case study, we instead report on results in engineering a transformation without domain knowledge.

A control system consists of a plant component and a controller which periodically computes a control input for the plant. We consider systems where the controller is implemented by a feedforward neural network with ReLU activations. The reachability problem asks, given a set of initial states, whether a set of target states can be reached. We show that this problem is undecidable even for trivial plants and fixed-depth neural networks with three inputs and outputs. We also show that the problem becomes semi-decidable when the plant as well as the input and target sets are given by automata over infinite words.

It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts such as entanglement and correlation into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on `electron correlation' leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.

The simultaneous treatment of static and dynamical correlations in strongly-correlated electron systems is a critical challenge. In particular, finding a universal scheme for identifying a single-particle orbital basis that minimizes the representational complexity of the many-body wavefunction is a formidable and longstanding problem. As a substantial contribution towards its solution, we show that the total orbital correlation actually reveals and quantifies the intrinsic complexity of the wavefunction,once it is minimized via orbital rotations. To demonstrate the power of this concept in practice, an iterative scheme is proposed to optimize the orbitals by minimizing the total orbital correlation calculated by the tailored coupled cluster singles and doubles (TCCSD) ansatz. The optimized orbitals enable the limited TCCSD ansatz to capture more non-trivial information of the many-body wavefunction, indicated by the improved wavefunction and energy. An initial application of this scheme shows great improvement of TCCSD in predicting the singlet ground state potential energy curves of the strongly correlated C$_{\rm 2}$ and Cr$_{\rm 2}$ molecule., Comment: Close to published version

We propose a novel approach to the problem of controller design for environments modeled as Markov decision processes (MDPs). Specifically, we consider a hierarchical MDP a graph with each vertex populated by an MDP called a "room". We first apply deep reinforcement learning (DRL) to obtain low-level policies for each room, scaling to large rooms of unknown structure. We then apply reactive synthesis to obtain a high-level planner that chooses which low-level policy to execute in each room. The central challenge in synthesizing the planner is the need for modeling rooms. We address this challenge by developing a DRL procedure to train concise "latent" policies together with PAC guarantees on their performance. Unlike previous approaches, ours circumvents a model distillation step. Our approach combats sparse rewards in DRL and enables reusability of low-level policies. We demonstrate feasibility in a case study involving agent navigation amid moving obstacles., Comment: 19 pages main text, 17 pages Appendix (excluding references)

The extension of the Rayleigh-Ritz variational principle to ensemble states $\rho_{\mathbf{w}}\equiv\sum_k w_k |\Psi_k\rangle \langle\Psi_k|$ with fixed weights $w_k$ lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state $\rho_{\mathbf{w}}$ as well as the individual eigenstates $|\Psi_k\rangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-\Delta{E}_{\mathbf{w}} \leq \Delta Q \leq d_+ \Delta{E}_{\mathbf{w}}$ on the errors $\Delta Q $ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of $d_{\pm}\equiv d_{\pm}^{(Q)}(\mathbf{w},\mathbf{E})$ provide valuable insights into the optimal choice of the auxiliary weights $w_k$ in practical applications., Comment: 23+7 pages, 9 figures, to appear on Quantum

Automated active space selection is arguably one of the most challenging and essential aspects of multiconfigurational methods. In this work we propose an effective quantum information-assisted complete active space optimization (QICAS) scheme. What sets QICAS apart from other correlation-based selection schemes is (i) the use of unique measures from quantum information that assess the correlation in electronic structures in an unambiguous and predictive manner, and (ii) an orbital optimization step that minimizes the correlation discarded by the active space approximation. Equipped with these features QICAS yields for smaller correlated molecules sets of optimized orbitals with respect to which the CASCI energy reaches the corresponding CASSCF energy within chemical accuracy. For more challenging systems such as the Chromium dimer, QICAS offers an excellent starting point for CASSCF by greatly reducing the number of iterations required for numerical convergence. Accordingly, our study validates a profound empirical conjecture: the energetically optimal non-active spaces are predominantly those that contain the least entanglement., Comment: Close to published version

Safe and optimal controller synthesis for switched-controlled hybrid systems, which combine differential equations and discrete changes of the system's state, is known to be intricately hard. Reinforcement learning has been leveraged to construct near-optimal controllers, but their behavior is not guaranteed to be safe, even when it is encouraged by reward engineering. One way of imposing safety to a learned controller is to use a shield, which is correct by design. However, obtaining a shield for non-linear and hybrid environments is itself intractable. In this paper, we propose the construction of a shield using the so-called barbaric method, where an approximate finite representation of an underlying partition-based two-player safety game is extracted via systematically picked samples of the true transition function. While hard safety guarantees are out of reach, we experimentally demonstrate strong statistical safety guarantees with a prototype implementation and UPPAAL STRATEGO. Furthermore, we study the impact of the synthesized shield when applied as either a pre-shield (applied before learning a controller) or a post-shield (only applied after learning a controller). We experimentally demonstrate superiority of the pre-shielding approach. We apply our technique on a range of case studies, including two industrial examples, and further study post-optimization of the post-shielding approach.

We study the problem of computing the preimage of a set under a neural network with piecewise-affine activation functions. We recall an old result that the preimage of a polyhedral set is again a union of polyhedral sets and can be effectively computed. We show several applications of computing the preimage for analysis and interpretability of neural networks.

The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ans\"atze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble $\rho_{\boldsymbol{w}}$ state with spectral weights $\boldsymbol{w}$, their measurement would just collapse $\rho_{\boldsymbol{w}}$ with probabilities $w_j$ to one of its eigenstates $|\Psi_j \rangle$. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights $\boldsymbol{w}$ on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE., Comment: 21 pages, 8 figures. The title has been changed

In [arXiv:2207.03377] the first closed formula of a faithful entanglement measure applicable to realistic electron systems has been derived. In the present work, we build on this key achievement with the ultimate goal of guiding the development of quantum technologies. For this, we first elucidate the process of entanglement swapping in electron systems such as atoms, molecules or solid bodies. This clearly demonstrates the necessity of both the reference to localized few-orbital subsystems and the implementation of the number-parity superselection rule. Accordingly, in virtue of Wick's theorem, we then provide a fully analytical study of the true physical entanglement between sites in free electron chains. In that sense, we break the common paradigm of restricting such analytical analyses to unitarily invariant settings, i.e. bipartitions of the chain into rather impractical, macroscopically large subsystems. We then upgrade this model to a hydrogen ring of interacting electrons and construct the sought-after localized orbitals. For both systems, we confirm the presence of long-distance entanglement, provided the filling fractions are sufficiently low/high., Comment: Close to published version

The leading terms in the large-$R$ asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H$_2$ molecule with the internuclear separation $R$ is derived thanks to the solution of the phase dilemma at the $R \to \infty$ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes of the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-$R$ asymptotics for the occupation numbers of the weakly occupied NOs and the $C_6$ dispersion coefficient. The highly accurate approximates for the radial components of the $p$-type "half-space" orbitals and the corresponding occupation numbers (that decay like $R^{-6}$), which are available for the first time thanks to the development of the present formalism, have some unexpected properties., Comment: close to published version; selected as "Editor's Pick"

To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT) we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its possible natural variables and explain how symmetries could be exploited. In particular, for systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of $v$-representability is relative to the scope and choice of variable. All these fundamental concepts are then comprehensively discussed and illustrated for the Hubbard dimer and its generalization to arbitrary pair interactions $W$. For this, we derive by analytical means the pure and ensemble functionals with respect to both the real- and complex-valued Hilbert space. The comparison of various functionals allows us to solve the underlying $v$-representability problems analytically and the dependence of its solution on the pair interaction is demonstrated. Intriguingly, the gradient of each universal functional is found to always diverge repulsively on the boundary of the domain. In that sense, this key finding emphasizes the universal character of the fermionic exchange force, recently discovered and proven in the context of translationally-invariant one-band lattice models.

A linear pseudo-Boolean constraint (LPB) is an expression of the form $a_1 \cdot \ell_1 + \dots + a_m \cdot \ell_m \geq d$, where each $\ell_i$ is a literal (it assumes the value 1 or 0 depending on whether a propositional variable $x_i$ is true or false) and $a_1, \dots, a_m, d$ are natural numbers. An LPB represents a Boolean function, and those Boolean functions that can be represented by exactly one LPB are called threshold functions. The problem of finding an LPB representation of a Boolean function if possible is called threshold recognition problem or threshold synthesis problem. The problem has an $O(m^7 t^5)$ algorithm using linear programming, where $m$ is the dimension and $t$ the number of terms in the DNF input. It has been an open question whether one can recognise threshold functions through an entirely combinatorial procedure. Smaus has developed such a procedure for doing this, which works by decomposing the DNF and "counting" the variable occurrences in it. We have implemented both algorithms as a thesis project. We report here on this experience. The most important insight was that the algorithm by Smaus is, unfortunately, incomplete.

We present symQV, a symbolic execution framework for writing and verifying quantum computations in the quantum circuit model. symQV can automatically verify that a quantum program complies with a first-order specification. We formally introduce a symbolic quantum program model. This allows to encode the verification problem in an SMT formula, which can then be checked with a delta-complete decision procedure. We also propose an abstraction technique to speed up the verification process. Experimental results show that the abstraction improves symQV's scalability by an order of magnitude to quantum programs with 24 qubits (a 2^24-dimensional state space)., Comment: This is the extended version of a paper with the same title that appeared at FM 2023. Tool available at doi.org/10.5281/zenodo.7400321

We initiate the recently proposed $\boldsymbol{w}$-ensemble one-particle reduced density matrix functional theory ($\boldsymbol{w}$-RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of $\boldsymbol{w}$-RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional's domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories., Comment: Application of w-RDMFT which has been proposed in arXiv:2204.12715; close to published version

We propose an algorithmic approach for synthesizing linear hybrid automata from time-series data. Unlike existing approaches, our approach provides a whole family of models. Each model in the family is guaranteed to capture the input data up to a precision error {\epsilon}, in the following sense: For each time series, the model contains an execution that is {\epsilon}-close to the data points. Our construction allows to effectively choose a model from this family with minimal precision error {\epsilon}. We demonstrate the algorithm's efficiency and its ability to find precise models in two case studies.

Entanglement is one of the most fascinating concepts of modern physics. In striking contrast to its abstract, mathematical foundation, its practical side is, however, remarkably underdeveloped. Even for systems of just two orbitals or sites no faithful entanglement measure is known yet. By exploiting the spin symmetries of realistic many-electron systems, we succeed in deriving a closed formula for the relative entropy of entanglement between electron orbitals. Its broad applicability in the quantum sciences is demonstrated: (i) in light of the second quantum revolution, it quantifies the true physical entanglement by incorporating the crucial fermionic superselection rule (ii) an analytic description of the long-distance entanglement in free electron chains is found, refining Kohn's locality principle (iii) the bond-order wave phase in the extended Hubbard model can be confirmed, and (iv) the quantum complexity of common molecular bonding structures could be marginalized through orbital transformations, thus rationalizing zero-seniority wave function ansatzes.

We present a novel approach to efficiently compute tight non-convex enclosures of the image through neural networks with ReLU, sigmoid, or hyperbolic tangent activation functions. In particular, we abstract the input-output relation of each neuron by a polynomial approximation, which is evaluated in a set-based manner using polynomial zonotopes. While our approach can also can be beneficial for open-loop neural network verification, our main application is reachability analysis of neural network controlled systems, where polynomial zonotopes are able to capture the non-convexity caused by the neural network as well as the system dynamics. This results in a superior performance compared to other methods, as we demonstrate on various benchmarks.

The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To support this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a toolbox for systematically exploring, quantifying and dissecting correlation effects in quantum chemical systems. By utilizing the geometric picture of quantum states we compare -- on a unified basis and in an operationally meaningful way -- total, quantum and classical correlation and entanglement in molecular ground states. To maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is provided, leading to a paradigm-shifting insight: A single covalent bond equates to the entanglement $2\ln(2)$. This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories., Comment: Close to published version

Motivated by the Penrose-Onsager criterion for Bose-Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh-Ritz variational principle to ensemble states with spectrum $\boldsymbol{w}$ and prove a corresponding generalization of the Hohenberg-Kohn theorem: The underlying one-particle reduced density matrix determines all properties of systems of $N$ identical particles in their $\boldsymbol{w}$-ensemble states. Then, to circumvent the $v$-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy-Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body $\boldsymbol{w}$-ensemble $N$-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli's exclusion principle for fermions and recently discovered generalizations thereof., Comment: In particular, bosonic counterpart of fermionic w-RDMFT (arXiv:2106.02560, arXiv:2106.03918) is presented; v2: title has been changed, new section proving a generalization of the Hohenberg-Kohn theorem for w-RDMFT has been added

In [Phys. Rev. Lett. 128, 013001 (2022)] a novel ground state method was proposed. It has been suggested that this $i$-DMFT would be a method within one-particle reduced density matrix functional theory (DMFT), capable of describing accurately molecules at various geometries with an information-theoretical nature. We reassess this work and its suggestions from a conceptual and practical point of view, leading to the following conclusions: i) A method which assigns to each molecule $\mathcal{M}$ its own functional $\mathcal{F}_{\!\mathcal{M}}$ is not a functional theory (striking violation of "universality") ii) even for the simplest systems $i$-DMFT yields incorrect one-particle reduced density matrices and iii) the use of an information-theoretical concept to describe molecular dissociation limits was not essential. The latter insight may help to fix the deficiency of $i$-DMFT to not reproduce correctly the smaller occupation numbers and thus to not recover the important dynamic correlations., Comment: Comments are very welcome!

State-average calculations based on mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states is also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out., Comment: 9 pages, additional calculation

We study the verification problem for closed-loop dynamical systems with neural-network controllers (NNCS). This problem is commonly reduced to computing the set of reachable states. When considering dynamical systems and neural networks in isolation, there exist precise approaches for that task based on set representations respectively called Taylor models and zonotopes. However, the combination of these approaches to NNCS is non-trivial because, when converting between the set representations, dependency information gets lost in each control cycle and the accumulated approximation error quickly renders the result useless. We present an algorithm to chain approaches based on Taylor models and zonotopes, yielding a precise reachability algorithm for NNCS. Because the algorithm only acts at the interface of the isolated approaches, it is applicable to general dynamical systems and neural networks and can benefit from future advances in these areas. Our implementation delivers state-of-the-art performance and is the first to successfully analyze all benchmark problems of an annual reachability competition for NNCS.

We present the first review of methods to overapproximate the set of reachable states of linear time-invariant systems subject to uncertain initial states and input signals for short time horizons. These methods are fundamental to state-of-the-art reachability algorithms for long time horizons, which proceed in two steps: First they use such a method to discretize the system for a short time horizon, and then they efficiently obtain a solution of the new discrete system for the long time horizon. Traditionally, both qualitative and quantitative comparison between different reachability algorithms has only considered the combination of both steps. In this paper we study the first step in isolation. We perform a variety of numerical experiments for six fundamental discretization methods from the literature. As we show, these methods have different trade-offs regarding accuracy and computational cost and, depending on the characteristics of the system, some methods may be preferred over others. We also discuss preprocessing steps to improve the results and efficient implementation strategies.

LazySets.jl is a Julia library that provides ways to symbolically represent sets of points as geometric shapes, with a special focus on convex sets and polyhedral approximations. LazySets provides methods to apply common set operations, convert between different set representations, and efficiently compute with sets in high dimensions using specialized algorithms based on the set types. LazySets is the core library of JuliaReach, a cutting-edge software addressing the fundamental problem of reachability analysis: computing the set of states that are reachable by a dynamical system from all initial states and for all admissible inputs and parameters. While the library was originally designed for reachability and formal verification, its scope goes beyond such topics. LazySets is an easy-to-use, general-purpose and scalable library for computations that mix symbolics and numerics. In this article we showcase the basic functionality, highlighting some of the key design choices., Comment: published in the Proceedings of the JuliaCon Conferences 2021

In this article we introduce a solution method for a special class of nonlinear initial-value problems using set-based propagation techniques. The novelty of the approach is that we employ a particular embedding (Carleman linearization) to leverage recent advances of high-dimensional reachability solvers for linear ordinary differential equations based on the support function. Using a global error bound for the Carleman linearization abstraction, we are able to describe the full set of behaviors of the system for sets of initial conditions and in dense time.

In [Phys. Rev. Lett. 127, 023001 (2021)] a reduced density matrix functional theory (RDMFT) has been proposed for calculating energies of selected eigenstates of interacting many-fermion systems. Here, we develop a solid foundation for this so-called $\boldsymbol{w}$-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights $\boldsymbol{w}$ in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case $\boldsymbol{w}=(1,0,\ldots)$. Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed., Comment: published version; in particular various derivations of the letter arXiv:2106.02560 are provided

We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size., Comment: to appear in Phys. Rev. Lett

Deep neural networks (DNNs) are increasingly applied in safety-critical domains, such as self-driving cars, unmanned aircraft, and medical diagnosis. It is of fundamental importance to certify the safety of these DNNs, i.e. that they comply with a formal safety specification. While safety certification tools exactly answer this question, they are of no help in debugging unsafe DNNs, requiring the developer to iteratively verify and modify the DNN until safety is eventually achieved. Hence, a repair technique needs to be developed that can produce a safe DNN automatically. To address this need, we present SpecRepair, a tool that efficiently eliminates counter-examples from a DNN and produces a provably safe DNN without harming its classification accuracy. SpecRepair combines specification-based counter-example search and resumes training of the DNN, penalizing counter-examples and certifying the resulting DNN. We evaluate SpecRepair's effectiveness on the ACAS Xu benchmark, a DNN-based controller for unmanned aircraft, and two image classification benchmarks. The results show that SpecRepair is more successful in producing safe DNNs than comparable methods, has a shorter runtime, and produces safe DNNs while preserving their classification accuracy., Comment: This is the extended version of a paper with the same title that appeared at SPIN 2022

From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of $1$-fermion and $N$-fermion density matrices, therefore it coincides with the convex hull of the pure $N$-representable $1$-fermion density matrices. Consequently, the description of ground state physics through $1$-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the $1$-body $N$-representability problem to ensemble states with fixed spectrum $\mathbf{w}$, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the $1$-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of $\mathbf{w}$. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes., Comment: 59 pages, 17 figures, 3 appendices Version 2: several minor changes and fixes, now the source files include two rst files containing the obtained generalized exclusion principles as linear inequalities

Formal design of embedded and cyber-physical systems relies on mathematical modeling. In this paper, we consider the model class of hybrid automata whose dynamics are defined by affine differential equations. Given a set of time-series data, we present an algorithmic approach to synthesize a hybrid automaton exhibiting behavior that is close to the data, up to a specified precision, and changes in synchrony with the data. A fundamental problem in our synthesis algorithm is to check membership of a time series in a hybrid automaton. Our solution integrates reachability and optimization techniques for affine dynamical systems to obtain both a sufficient and a necessary condition for membership, combined in a refinement framework. The algorithm processes one time series at a time and hence can be interrupted, provide an intermediate result, and be resumed. We report experimental results demonstrating the applicability of our synthesis approach.

One-particle reduced density matrix functional theory would potentially be the ideal approach for describing Bose-Einstein condensates. It namely replaces the macroscopically complex wavefunction by the simple one-particle reduced density matrix, therefore provides direct access to the degree of condensation and still recovers quantum correlations in an exact manner. We eventually initiate and establish this novel theory by deriving the respective universal functional $\mathcal{F}$ for general homogeneous Bose-Einstein condensates with arbitrary pair interaction. Most importantly, the successful derivation necessitates a particle-number conserving modification of Bogoliubov theory and a solution of the common phase dilemma of functional theories. We then illustrate this novel approach in several bosonic systems such as homogeneous Bose gases and the Bose-Hubbard model. Remarkably, the general form of $\mathcal{F}$ reveals the existence of a universal Bose-Einstein condensation force which provides an alternative and more fundamental explanation for quantum depletion., Comment: close to published version

Neural-network classifiers achieve high accuracy when predicting the class of an input that they were trained to identify. Maintaining this accuracy in dynamic environments, where inputs frequently fall outside the fixed set of initially known classes, remains a challenge. The typical approach is to detect inputs from novel classes and retrain the classifier on an augmented dataset. However, not only the classifier but also the detection mechanism needs to adapt in order to distinguish between newly learned and yet unknown input classes. To address this challenge, we introduce an algorithmic framework for active monitoring of a neural network. A monitor wrapped in our framework operates in parallel with the neural network and interacts with a human user via a series of interpretable labeling queries for incremental adaptation. In addition, we propose an adaptive quantitative monitor to improve precision. An experimental evaluation on a diverse set of benchmarks with varying numbers of classes confirms the benefits of our active monitoring framework in dynamic scenarios., Comment: published at RV 2021

Efficiently handling time-triggered and possibly nondeterministic switches for hybrid systems reachability is a challenging task. In this paper we present an approach based on conservative set-based enclosure of the dynamics that can handle systems with uncertain parameters and inputs, where the uncertainties are bound to given intervals. The method is evaluated on the plant model of an experimental electro-mechanical braking system with periodic controller. In this model, the fast-switching controller dynamics requires simulation time scales of the order of nanoseconds. Accurate set-based computations for relatively large time horizons are known to be expensive. However, by appropriately decoupling the time variable with respect to the spatial variables, and enclosing the uncertain parameters using interval matrix maps acting on zonotopes, we show that the computation time can be lowered to 5000 times faster with respect to previous works. This is a step forward in formal verification of hybrid systems because reduced run-times allow engineers to introduce more expressiveness in their models with a relatively inexpensive computational cost.

The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan--Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems., Comment: some typos corrected, page makeup improved, 62 pages

A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements of numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement previously seen. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated., Comment: Close to published version

The realization of Bose-Einstein condensation in ultracold trapped gases has led to a revival of interest in that fascinating quantum phenomenon. This experimental achievement necessitated both extremely low temperatures and sufficiently weak interactions. Particularly in reduced spatial dimensionality even an infinitesimal interaction immediately leads to a departure to quasi-condensation. We propose a system of strongly interacting bosons which overcomes those obstacles by exhibiting a number of intriguing related features: (i) The tuning of just a single control parameter drives a transition from quasi-condensation to complete condensation, (ii) the destructive influence of strong interactions is compensated by the respective increased mobility, (iii) topology plays a crucial role since a crossover from one- to `infinite'-dimensionality is simulated, (iv) a ground state gap opens which makes the condensation robust to thermal noise. Remarkably, all these features can be derived by analytical and exact numerical means despite the non-perturbative character of the system., Comment: close to published version; title has been changed

Based on a generalization of Hohenberg-Kohn's theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix $\gamma$ as a natural variable but still recovers quantum correlations in an exact way it is particularly well-suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying $v$-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals for this $N$-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. The respective gradient forces are found to diverge in the regime of Bose-Einstein condensation, $\nabla_{\gamma} \mathcal{F} \propto 1/\sqrt{1-N_{\mathrm{BEC}}/N}$, providing a natural explanation for the absence of complete BEC in nature.

The interplay between electron interaction and geometry in a molecular system can lead to rather paradoxical situations. The prime example is the dissociation limit of the hydrogen molecule: While a significant increase of the distance $r$ between the two nuclei marginalizes the electron-electron interaction, the exact ground state does, however, not take the form of a single Slater determinant. By first reviewing and then employing concepts from quantum information theory, we resolve this paradox and its generalizations to more complex systems in a quantitative way. To be more specific, we illustrate and prove that thermal noise due to finite, possibly even just infinitesimally low, temperature $T$ will destroy the entanglement beyond a critical separation distance $r_{\mathrm{crit}}$($T$) entirely. Our analysis is comprehensive in the sense that we simultaneously discuss both total correlation and entanglement in the particle picture as well as in the orbital/mode picture. Our results reveal a conceptually new characterization of static and dynamical correlation in ground states by relating them to the (non)robustness of correlation with respect to thermal noise., Comment: title has been modified, close to published version

Neural networks have demonstrated unmatched performance in a range of classification tasks. Despite numerous efforts of the research community, novelty detection remains one of the significant limitations of neural networks. The ability to identify previously unseen inputs as novel is crucial for our understanding of the decisions made by neural networks. At runtime, inputs not falling into any of the categories learned during training cannot be classified correctly by the neural network. Existing approaches treat the neural network as a black box and try to detect novel inputs based on the confidence of the output predictions. However, neural networks are not trained to reduce their confidence for novel inputs, which limits the effectiveness of these approaches. We propose a framework to monitor a neural network by observing the hidden layers. We employ a common abstraction from program analysis - boxes - to identify novel behaviors in the monitored layers, i.e., inputs that cause behaviors outside the box. For each neuron, the boxes range over the values seen in training. The framework is efficient and flexible to achieve a desired trade-off between raising false warnings and detecting novel inputs. We illustrate the performance and the robustness to variability in the unknown classes on popular image-classification benchmarks., Comment: accepted at ECAI 2020