arXiv subjects

Christian Schilling

Publications and source records attributed to Christian Schilling.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Probabilistic Shielding by Learning MDPs for Safe Reinforcement Learning

Probabilistic shielding is a technique for safe reinforcement learning (RL). Typically, a static observer -- called the shield -- constrains the learning agent's actions to those for which acting safely remains feasible. Traditionally, the shield is computed from the transition probabilities of the underlying Markov decision process (MDP). Thus, this technique is not applicable when the MDP model is not given a priori, which, unfortunately, is the case in typical RL applications. In this paper, we study the problem of computing a shield in the setting where the transition graph of the MDP is known, but the transition probabilities are unknown. Our approach integrates probabilistic shielding with online model learning: as the RL agent explores the environment, we estimate the transition probabilities. From this estimate, we compute a shield. While the shield may be conservative initially, it adapts as the model estimate becomes more precise. Thus, the shield improves in tandem with the RL agent. This paradigm of adaptive probabilistic shielding raises a number of challenges, such as when to recompute the shield and how to balance between exploration and safety during learning. We empirically evaluate multiple variants of this paradigm across several environments.

cs.LG↗

arXiv2026

Scalable Reachability Analysis of Linear Continuous Systems with Property-Driven Time-Step Adaptation

We study safety verification for linear time-invariant systems with bounded inputs in continuous time. The standard approach reduces to a reachability analysis in two steps: first discretize time and then apply a forward analysis in the discretized system. Existing algorithms use either a fixed time step or an adaptive time step that changes based on the approximation error compared to the underlying continuous system. In this paper, we present an efficient reachability algorithm that adapts the time step based on a given safety property. Essentially, our algorithm makes the largest possible time step such that it can still prove safety. For this approach to be scalable in practice, we discuss several optimizations such as avoiding the repeated expensive calculation of the matrix exponential during discretization and a careful balance how we tame the approximation error stemming from the states and the inputs. This allows our algorithm to yield a moderate approximation error even when using a large time step, thus requiring much fewer steps than prior algorithms. We demonstrate the effectiveness and scalability on the large-scale SLICOT benchmark suite, where our algorithm consistently outperforms other state-of-the-art approaches.

eess.SY↗

arXiv2026

Capturing electron correlation at mean-field cost: Assessment of i-DMFT and the underlying correlation conjecture

Accurately treating strong electron correlation in quantum chemistry typically requires multireference wave-function methods with steep computational scaling. The recently proposed i-DMFT method promises near configuration-interaction accuracy at mean-field cost by invoking an empirical linear relation between correlation energy and entropy (Collins' conjecture), whose validity remains unclear. We systematically assess this relation across a range of di- and polyatomic molecules, including diverse bond types, third-row elements, different types of geometric distortions, and excited states. We find that the conjectured linearity holds for bond-breaking processes dominated by electron redistribution within orbital pairs, but breaks down for heterolytic dissociation and excited states. In simple molecules, i-DMFT provides a reasonable description of total energies, but does not reliably reproduce reduced density matrices or individual energy components. It further degrades in more complex cases such as ethylene. Based on these results, we formulate criteria for the validity of the conjecture and outline implications for entropy-based reduced density matrix functionals.

physics.chem-ph↗

arXiv2026

Chemical bonding concepts emerge naturally from maximally entangled atomic orbitals

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 provides a new framework for understanding fuzzy chemical concepts using rigorous, quantitative descriptors from quantum information.

quant-ph↗

arXiv2026

Quantum correlations in molecules: from quantum resourcing to chemical bonding

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.

quant-ph↗

arXiv2026

Robust Zonotopic Control

We propose a zonotopic framework for synthesizing a single robust state feedback controller that is certified to stabilize every plant inside a matrix zonotope, describing linearly varying parameters or parametric uncertainty. Common robust design strategies rely on checking many vertex models or on complex gain-scheduling, leading to high offline computation and implementation complexity. Our approach finds a single gain that is provably valid across the entire parameter domain, which is simpler to implement and can reduce conservatism by exploiting the structure of the zonotope. We formulate the robust synthesis as a single convex program tailored to the zonotope representation and incorporate practical performance requirements (actuator constraints, decay rate, disturbance attenuation) into the same synthesis stage. In numerical experiments on a representative 4-state example, our controller provides larger stability coverage across the parameter domain, attains comparable transient performance and control effort to more complex designs, and significantly reduces the number and scale of offline synthesis problems required by other robust approaches, compared to common-vertex gain, $H_{\infty}$, and $μ$-synthesis baselines.

eess.SY↗

arXiv2026

Unified Framework for Functional Theories of Quantum Systems

We introduce and study a unified framework for density-functional theory and its variants for quantum systems on finite-dimensional Hilbert spaces. These theories seek to reduce the complexity inherent in the many-body quantum problem by describing ground states through reduced variables. The central ingredients of our unified framework are a generalized choice of basic observables, whose expectation values define precisely those reduced variables, and a fixed part of the Hamiltonian characterizing the class of quantum systems under consideration. It is this minimal structure, which we call the scope of a functional theory, that is necessary and sufficient for the formulation of a functional theory. In particular, it allows one to define the universal functionals, establish their convexity and differentiability properties, address representability questions, and prove a Hohenberg-Kohn-type uniqueness result. A purification construction also relates ensemble and weighted-ensemble functionals to the pure-state variant. Particular emphasis is placed on functional theories with Lie-algebra observable structures, connecting the variational framework to symplectic geometry. The result of this work is a systematic mathematical formulation in which structural results can be proved once and applied across a broad class of finite-dimensional functional theories.

math-ph↗

arXiv2026

Safe and Near-Optimal Gate Control: A Case Study from the Danish West Coast

Ringkoebing Fjord is an inland water basin on the Danish west coast separated from the North Sea by a set of gates used to control the amount of water entering and leaving the fjord. Currently, human operators decide when and how many gates to open or close for controlling the fjord's water level, with the goal to satisfy a range of conflicting safety and performance requirements such as keeping the water level in a target range, allowing maritime traffic, and enabling fish migration. Uppaal Stratego. We then use this digital twin along with forecasts of the sea level and the wind speed to learn a gate controller in an online fashion. We evaluate the learned controllers under different sea-level scenarios, representing normal tidal behavior, high waters, and low waters. Our evaluation demonstrates that, unlike a baseline controller, the learned controllers satisfy the safety requirements, while performing similarly regarding the other requirements.

eess.SY↗

arXiv2026

A Geometric Approach to Strongly Correlated Bosons: From $N$-Representability to the Generalized BEC Force

Building on recent advances in reduced density matrix theory, we develop a geometric framework for describing strongly correlated lattice bosons. We first establish that translational symmetry, together with a fixed pair interaction, enables an exact functional formulation expressed solely in terms of momentum occupation numbers. Employing the constrained-search formalism and exploiting a geometric correspondence between $N$-boson configuration states and their one-particle reduced density matrices, we derive the general form of the ground-state functional. Its structure highlights the omnipresent significance of one-body $N$-representability: (i) the domain is exactly determined by the $N$-representability conditions; (ii) at its boundary, the gradient of the functional diverges repulsively, thereby generalizing the recently discovered Bose-Einstein condensate (BEC) force; and (iii) an explicit expression for this boundary force follows directly from geometric arguments. These key results are demonstrated analytically for few-site lattice systems, and we illustrate the broader significance of our functional form in defining a systematic hierarchy of functional approximations.

quant-ph↗

arXiv2025

Solving one-body ensemble N-representability problems with spin

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $φ_i$ 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 $Σ_{N,S}(\boldsymbol w) \subset [0,2]^d$. 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.

quant-ph↗

arXiv2025

Uppaal Coshy: Automatic Synthesis of Compact Shields for Hybrid Systems

We present Uppaal Coshy, a tool for automatic synthesis of a safety strategy -- or shield -- for Markov decision processes over continuous state spaces and complex hybrid dynamics. The general methodology is to partition the state space and then solve a two-player safety game, which entails a number of algorithmically hard problems such as reachability for hybrid systems. The general philosophy of Uppaal Coshy is to approximate hard-to-obtain solutions using simulations. Our implementation is fully automatic and supports the expressive formalism of Uppaal models, which encompass stochastic hybrid automata. The precision of our partition-based approach benefits from using finer grids, which however are not efficient to store. We include an algorithm called Caap to efficiently compute a compact representation of a shield in the form of a decision tree, which yields significant reductions.

cs.LO↗

arXiv2025

Compositional Shielding and Reinforcement Learning for Multi-Agent Systems

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.

cs.LO↗

arXiv2025

Refining ensemble $N$-representability of one-body density matrices from partial information

The $N$-representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body $N$-representability problems that incorporate partial knowledge of the one-body reduced density matrices (1RDMs) within an ensemble of $N$-fermion states with fixed weights $w_i$. Specifically, we propose a systematic relaxation that reduces the refined problem -- where full 1RDMs are fixed for certain ensemble elements -- to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn's problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble $N$-representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.

quant-ph↗

arXiv2025

Spin adaptation of the cumulant expansions of reduced density matrices

We develop a systematic framework for the spin adaptation of the cumulants of p-particle reduced density matrices (RDMs), with explicit constructions for p = 1 to 3. These spin-adapted cumulants enable rigorous treatment of both S_z and S^2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation -- referred to as complete S-representability -- can be enforced by constraining the variances of S_z and S^2, which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size -- size-extensive -- making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods (DFT), one-particle RDM functional theories (RDMFT), and two-particle RDM methods. We further extend the approach to spin-orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.

physics.chem-ph↗

arXiv2025

Composing Reinforcement Learning Policies, with Formal Guarantees

We propose a novel framework to controller design in environments with a two-level structure: a known high-level graph ("map") in which each vertex is populated by a Markov decision process, called a "room". The framework "separates concerns" by using different design techniques for low- and high-level tasks. We apply reactive synthesis for high-level tasks: given a specification as a logical formula over the high-level graph and a collection of low-level policies obtained together with "concise" latent structures, we construct a "planner" that selects which low-level policy to apply in each room. We develop a reinforcement learning procedure to train low-level policies on latent structures, which unlike previous approaches, circumvents a model distillation step. We pair the policy with probably approximately correct guarantees on its performance and on the abstraction quality, and lift these guarantees to the high-level task. These formal guarantees are the main advantage of the framework. Other advantages include scalability (rooms are large and their dynamics are unknown) and reusability of low-level policies. We demonstrate feasibility in challenging case studies where an agent navigates environments with moving obstacles and visual inputs.

cs.AI↗

arXiv2025

A toolbox of spin-adapted generalized Pauli constraints

We establish a toolbox for studying and applying spin-adapted generalized Pauli constraints (GPCs) in few-electron quantum systems. By exploiting the spin symmetry of realistic $N$-electron wave functions, the underlying one-body pure $N$-representability problem simplifies, allowing us to calculate the GPCs for larger system sizes than previously accessible. We then uncover and rigorously prove a superselection rule that highlights the significance of GPCs: whenever a spin-adapted GPC is (approximately) saturated - referred to as (quasi)pinning - the corresponding $N$-electron wave function assumes a simplified structure. Specifically, in a configuration interaction expansion based on natural orbitals only very specific spin configuration state functions may contribute. To assess the nontriviality of (quasi)pinning, we introduce a geometric measure that contrasts it with the (quasi)pinning induced by simple (spin-adapted) Pauli constraints. Applications to few-electron systems suggest that previously observed quasipinning largely stems from spin symmetries.

quant-ph↗

arXiv2025

In Search of Trees: Decision-Tree Policy Synthesis for Black-Box Systems via Search

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.

cs.AI↗

arXiv2024

Ground and Excited States from Ensemble Variational Principles

The extension of the Rayleigh-Ritz variational principle to ensemble states $ρ_{\mathbf{w}}\equiv\sum_k w_k |Ψ_k\rangle \langleΨ_k|$ 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 $ρ_{\mathbf{w}}$ as well as the individual eigenstates $|Ψ_k\rangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-Δ{E}_{\mathbf{w}} \leq ΔQ \leq d_+ Δ{E}_{\mathbf{w}}$ on the errors $ΔQ $ 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.

quant-ph↗