The Bose-Einstein condensate dark matter model, where dark matter can be thought of as a non-relativistic, Newtonian gravitational condensate, has recently attracted a great deal of interest. In the present study, we explore the possibility that the dark matter could exist in the form of a self-bound quantum droplet formed by ultradilute quantum Bose mixtures under the action of Lee-Huang-Yang corrections at zero temperature. To this end, we derive an extended equation of state by using the nonrelativistic self-consistent Hartree-Fock-Bogoliubov theory and the hydrodynamic approach. The solutions of the obtained equations of state show that the key parameters of the dark matter halos such as the density, mass, and radius are sensitive to the interspecies interaction and to the quantum fluctuation strength. The stability and the dynamical evolution of the droplet Galactic halos are analyzed by considering small perturbations of the quantum hydrodynamical equations. In order to increase the reliability of our predictions we compare them with some observed data for the Galactic rotation curves.

In this article, we connect statistical inference for spatial point processes with the analysis of waiting pedestrian crowds through two interconnected contributions. First, on the methodological side we develop an inference procedure for semiparametric spatial point process models leveraging replicated spatial patterns, i.e., multiple approximately independent realizations from the same process. Second, we show that spatial point processes provide a suitable modeling framework for waiting pedestrians, capturing two key aspects: spatial inhomogeneity driven by location attractiveness and repulsive interactions between pedestrians. These two components are central to the inference problem itself, since spatial point process modeling hinges on disentangling background intensity from interaction. Although replicated spatial patterns are rare in point process literature, they are available here through a unique real-life pedestrian dataset, thereby directly linking the methodological development to the physical application. We use the proposed methods to fit and evaluate determinantal and Gibbs point processes in a simulation study and a real-world case study. Despite persistent challenges in decoupling the influences of inhomogeneity from interaction, these models are able to reproduce key empirical features of waiting pedestrians.

In this article, we introduce several classes of permutation polynomials over $\mathbb{F}_{q^2}$. More precisely, we characterize permutation polynomials of the forms $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$ over $\mathbb{F}_{q^2}$. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums. We also present the compositional inverses of the permutation polynomials obtained in this paper.

We report multi-angle reflectivity measurements in the extreme-ultraviolet (XUV) range for mono- and bilayer MoS$_2$ on a Si$_3$N$_4$ substrate. Using a single-sheet 2D conductivity model, we extract the complex optical response of the MoS$_2$ bilayer between 25 and 90 eV and derive an effective refractive index by introducing a thickness equal to the interlayer spacing. The MoS$_2$ monolayer response is consistently reproduced either by halving the 2D conductivity or the effective thickness, indicating a robust scaling with layer number. The resulting optical constants display a broad resonance at the Mo N$_{2,3}$ edge with no signatures of sharp core-exciton features despite the reduced dimensionality. First-principles calculations reproduce the experimental results and show that local-field (Hartree) effects dominate the XUV response, while screened-exchange (SEX) contributions remain weak and mainly induce spectral shifts. Our analysis demonstrates that excitonic effects play a minor role in the XUV optical response of atomically thin MoS$_2$, highlighting key differences with respect to the visible and infrared regimes, and calling for a reassessment of the use of Mo-based transition metal dichalcogenides in attosecond spectroscopy and XUV excitonics.

Light-matter interactions are often modeled via the Dicke model, namely, by two-level systems coupled to a cavity mode. Alas, the threshold for superradiance is often experimentally inaccessible or hindered by light's diamagnetic term. Here, within the Dicke setting, we consider self-interacting light in a cavity, modeled by a photonic Kerr nonlinearity. We show that negative Kerr nonlinearity gives rise to a low-threshold superradiant phase with spin inversion. While unstable in a closed system, cavity dissipation stabilizes this lit phase, opening avenues for lasing and bath-engineered phases.

Browser automation frameworks are essential tools for security and privacy research on the web, yet bot detection scripts increasingly probe their artifacts, threatening measurement validity as automated browsers may be blocked or served different content. Prior work measures detection deployment, while we measure blocking-induced sample loss. Through a literature survey of top-tier security, privacy, and web measurement venues, we find that 83% of papers omit any discussion of bot detection blocking. To address this gap, we conduct a measurement study of 10,000 websites across four browser configurations (40K page visits in total) to quantify detection prevalence and employed techniques. Using custom instrumentation to detect when sites probe for automation, we develop a taxonomy of bot detection techniques and measure how often they appear in practice. Chromium headless encounters a 15% soft block rate compared to 7% for other configurations. Across all conditions, 82% of blocks are attributable to bot detection (59% vendor-confirmed, 23% inferred from condition-dependent blocking), predominantly by providers with integrated bot detection such as Cloudflare (37% block rate) and Akamai (26%). A header spoofing experiment establishes that 75% of Chromium-headless-only blocks are caused by header-level signals alone, yet JavaScript-based environment probing is more extensive than current blocking rates suggest. These findings demonstrate that bot detection creates systematic, provider-correlated sample loss that the web measurement community neither measures nor reports. The downstream effect on specific measurement outcomes remains future work.

In this paper, we study finite dimensional Lie algebras with respect to the length of their chief series. We examine separately the semisimple, solvable, and mixed cases over fields of characteristic zero, and we establish several structural results describing Lie algebras of prescribed chief length. In the solvable and mixed settings, the problem is reduced to the study of irreducible modules arising from suitable quotient actions. In positive characteristic, we investigate semisimple Lie algebras by means of Block s structural theorem and obtain several results concerning Lie algebras of chief length two.

In this note, we provide several equivalent descriptions of the tropicalization of definable sets in a polynomially bounded o-minimal expansion of a real closed field. We show that the tropicalization can be described in multiple ways: as the image under the valuation map, as the image under the tropicalization map of the archimedean points in the o-minimal spectrum, in terms of initial degenerations of the definable set and initial nonnegativity cones, or via the real tropicalization of polynomials belonging to the corresponding nonnegativity cone. Moreover, we prove analogous results for the fine tropicalization of definable sets, where the valuation map is replaced by the RV sort map, which records not only the valuation of a point but also its angular component.

Superconducting quantum processors could significantly profit from enhanced connectivity together with precise control of interactions and gates between qubits. Here we investigate plaquettes of four qubits that are coupled via a central tunable coupling circuit, so that not only gates between qubits connected by an edge of the plaquette can be executed but also between qubits across the diagonal. By numerically and analytically analyzing parametrically driven processes, we explore $\sqrt{\text{iSWAP}}$-gates between any pair of qubits, also across the diagonal, as well as three-qubit interactions and gates. For experimentally available circuit parameters, we for example find $\sqrt{\text{iSWAP}}$-gates with a gate time of 50 ns and 99.9\% fidelity, which is decreased to 99.4\% if two such gates are executed in parallel on disjoint qubit pairs in the plaquette. For three-qubit gates we find fidelities of 95\% fidelity at a gate time of 200 ns.

The Lyapunov exponent quantifies the sensitivity of a dynamical system to perturbations, and the full Lyapunov spectrum extends this to every orthogonal direction in tangent space. For cellular automata the spectrum is almost always approximated numerically, and the approximation is delicate. We show that the affine rules, those whose update is a XOR of a subset of the inputs together with a constant, admit an exact Lyapunov spectrum. An affine rule has a configuration-independent Boolean Jacobian, so the spectrum reduces to the logarithms of the singular values of a single constant matrix, with no simulation and no limit involved. Two cases carry a closed form. For an affine cellular automaton on a periodic lattice the Jacobian is a multilevel circulant matrix, and the spectrum is the discrete Fourier transform of the rule's gradient stencil, valid in any spatial dimension. For the parity rule on an arbitrary graph the Jacobian is the adjacency matrix itself, so the Lyapunov spectrum is the logarithm of the absolute adjacency spectrum, and the maximal exponent is the logarithm of the spectral radius. The long-time amplitude of a single-site perturbation then scales with the eigenvector centrality of the seeded node. Reading the periodic lattice as the Cayley graph of an abelian group unifies the two cases. Because they are exact, the affine spectra also serve as benchmarks: they reveal numerical artefacts in previously reported spectra and turn the informal correspondence between spectral radius and dynamical sensitivity into an exact identity.

The open ocean lacks systematic in situ wind observations, and satellite scatterometer calibration depends on collocated surface measurements largely absent away from coastlines. Compact wave-sensing drifters already retrieve the open-ocean wind vector, but at modest accuracy -- about 1-2 m/s in speed and with unreliable direction at low wind. We show that a compact freely drifting GNSS/IMU drifter (MELODI) improves both components at a small fraction of the cost of moored platforms. Wind speed is read from the full shape of the measured wave acceleration spectrum rather than a single equilibrium-range level. A supervised model (Wind Inversion using Tikhonov Regularization, WITR -- a regularized regression with a gated residual correction), trained against scatterometer winds and evaluated by leave-one-buoy-out cross-validation (ASCAT MetOp-B/C, HY-2B/C; ERA5 used during feature development), reaches an RMSE of 0.90 m/s and defines the empirical skill ceiling of the feature set; it serves both directly as a wind retrieval and as a teacher model for distillation. Symbolic regression then distills this teacher into compact, interpretable closed-form laws suited to onboard implementation -- a spectrum-only law (RMSE about 1.0 m/s) and a motion-enhanced reduced drag law (0.93-0.94 m/s) -- that reproduce the teacher's skill within the present validation uncertainty. Wind direction is recovered independently from IMU-derived directional wave moments in the wind-sea band, with a mean absolute error of 9.4 degrees against scatterometer winds. These accuracies are scatterometer-consistent (comparable to published scatterometer-buoy differences) and close to a factor-of-two improvement over the traditional single-band Toba inversion; winds are retrieved plausibly up to 18 m/s, with higher winds flagged as lower confidence.

The covariant formulation of stochastic thermodynamics requires treating the stochastic work as a 4-vector, posing significant challenges for quantum systems due to the non-commutativity. We introduce a new quasiprobability distribution for the work 4-vector, which combines the Wigner and Margenau-Hill quasiprobabilities. This extends the covariant fluctuation theorems from classical to quantum regime. We illustrate our findings with a scalar field driven by classical particles with a generalized version of trace formula. Our work establishes a quasiprobability approach to studying relativistic quantum thermodynamics in a covariant way.

We propose a kernel-based estimator of the velocity field governing the transport and diffusion of $d$-dimensional interacting particles. Assuming the initial positions are i.i.d. with law $μ_0$, we establish consistency of the estimator with an explicit mean-squared error rate of order $h^2 + N^(-2/(d+2))$, where $h$ denotes the time-discretization step and N the number of particles. The analysis covers two distinct settings: the Fokker-Planck equation, where we recover the underlying potential function, and the McKean-Vlasov equation, where we deconvolve the interaction kernel driving the mean-field dynamics.

We prove rigidity and classification results for the quasilinear Liouville equation associated with the $n$-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles. We also prove a finite-mass rigidity theorem under the corresponding sharp asymptotic lower bound. We show that any logarithmic lower bound forces positive asymptotic volume ratio and one-endedness of the manifold. Finally, we construct solutions on nonflat manifolds with nonnegative Ricci curvature showing the sharpness of our hypotheses.

Similarity search lies at the heart of many modern applications, ranging from databases to deep learning to data series analysis. As such, a vast effort has been invested in developing algorithms, data structures and implementations to speed up this crucial subroutine. To empirically validate these approaches, several benchmarking efforts have been initiated covering a wide array of datasets. In this paper, we observe that usually little control is exercised on the hardness of the workloads with which methods are tested and compared. To address this issue, we first evaluate several query hardness measures with respect to their ability to capture the empirical hardness of a query, i.e. the effort invested by an index data structure to provide an answer. Then, we propose two methods, deemed \HephAnn and \HephGrad, for synthesizing query workloads so that they meet a user-specified hardness target. Both methods allow to produce workloads with the desired hardness: we find that \HephGrad is faster, while \HephAnn makes fewer assumptions on the target hardness measure. The resulting workloads can be used to gain insights into the behavior of similarity search algorithms.

We investigate how tidal locking affects the magnetic field generation and magnetospheric protection of Earth-like planets orbiting the habitable zone of M dwarfs. While the role of stellar activity has been widely studied, the impact of tidal locking on the planetary dynamo remains largely unexplored. We develop a framework that couples the stellar wind with two distinct dynamo paradigms: direct rotational scaling of a dipolar field, and energy-flux scaling where the local Rossby number governs the magnetic geometry, capturing a rapid transition from a dipolar to a multipolar configuration. We employ a Constant Time Lag model to capture the continuous tidal spin-down of the planets, evaluating two distinct planetary interiors: a highly dissipative Modern Earth and a less dissipative, rapidly rotating Early Earth. We find that tidal locking acts as a severe negative feedback on planetary magnetism across both paradigms, where the dipolar field frequently collapses before full tidal synchronization is reached. At the inner boundary, nearly all the planets are rapidly locked and the intrinsic magnetic fields are significantly attenuated. The outer boundary provides a favorable environment only for early M dwarfs ($M_\star \gtrsim 0.32~M_\odot$) hosting Early Earth. Here, the combination of non-synchronous rotation and residence outside the sub-Alfvénic regime provides atmospheric protection likelihoods of $97-100\%$. For mid-to-late M dwarfs, the combination of inevitable tidal locking and a sub-Alfvénic environment results in a total collapse of the magnetic shield, reducing the likelihood of atmospheric protection to essentially zero across the entire habitable zone.

We establish boundary Hölder regularity for weak solutions to a class of nonlinear elliptic Dirichlet problems with nonstandard growth posed on Reifenberg flat domains. More precisely, for any prescribed exponent \(α\in(0,1)\), we show that weak solutions are \(α\)-Hölder continuous up to the boundary provided that the Reifenberg flatness parameter is sufficiently small. The proof combines an iterative boundary decay argument with an ABP-type maximum principle in the Orlicz--Sobolev setting.

We introduce a Grid-Based Multinode Shepard Collocation Method (GBMSC) for two-dimensional elliptic boundary value problems on rectangular domains. The method combines Shepard-type partition functions with tensor-product Lagrange interpolation on overlapping local Cartesian subgrids. The Cartesian structure avoids the local unisolvency search required by multinode Shepard constructions on scattered data, while the local character of the approximation yields sparse collocation matrices and moderate conditioning. We establish the main structural properties of the method, including the cardinal property, tensor-product polynomial reproduction, and local interpolation estimates. Numerical experiments for Poisson problems with Dirichlet and mixed boundary conditions confirm the polynomial reproduction property up to round-off accuracy and show regular convergence for smooth non-polynomial solutions. Comparisons with Multinode Shepard Collocation Method (MSC) and Kansa's collocation methods indicate that the proposed discretization preserves high accuracy with significantly better conditioning than the RBF-based schemes considered.

Recent DAMPE measurements of individual cosmic-ray components, including carbon, oxygen, and iron, reveal distinct spectral softenings below the knee. The energy spectrum, mass composition, and anisotropy together provide key probes of cosmic-ray origin and propagation. By incorporating the individual elemental spectra reported by DAMPE, we derive a more complete $\langle \ln A\rangle$ distribution, which smoothly connects to the higher-energy $\langle \ln A\rangle$ measurements from LHAASO and exhibits a pronounced bump-like feature. This feature indicates a transition from the conventional Galactic cosmic ray source population to a nearby-source-dominated regime. We show that a spatially dependent propagation model with a nearby-source contribution can consistently reproduce the observed spectra, mass composition, and anisotropy. This suggests a unified picture in which Galactic cosmic rays below the knee arise from multiple source populations jointly constrained by these observables. Leveraging the precise component-resolved spectra from DAMPE, we further predict the transition energies in the anisotropy phase and amplitude for different mass components. Future component-resolved anisotropy measurements by LHAASO will provide a crucial test of this scenario.

In 1986, Sussmann proposed an expansion of the Chen series as an infinite product of exponentials of Hall basis elements multiplied by coefficients defined through a simple induction. He proved that this product converges locally in time for bilinear control systems associated with matrices or bounded operators. In this note, we prove that Sussmann's product also converges locally in time for control-affine systems driven by analytic vector fields. The difficulty is that one must keep track of the structure of the involved Lie brackets, as the natural decay of the explicit coefficients is not enough to counterbalance the natural growth of the iterated Lie brackets.

A microwave linear analog computer (MiLAC) is a tunable microwave network that performs computation through wave propagation in the analog domain. In beamforming, data streams pass through a reconfigurable admittance network and emerge as antenna signals. For communications, MiLACs are preferably lossless and reciprocal to avoid power dissipation and non-reciprocal components, but these constraints limit the analog beamformers they can realize. Fully-connected MiLACs offer broad flexibility at the cost of a quadratic number of tunable admittances in the antenna count. Stem-connected MiLACs reduce this scaling to linear and preserve point-to-point capacity, but their role in multiuser downlink beamforming and under bounded, discrete hardware constraints has remained open. This paper addresses both questions for the multiuser multiple-input single-output downlink. We show that a stem-connected MiLAC can realize every beamformer on the complex Stiefel manifold and prove that, when $N\ge 2K-1$, this Stiefel-restricted design achieves the same sum-rate as the fully-connected MiLAC, where $N$ and $K$ are the numbers of transmit antennas and users. We then develop a weighted minimum mean-square error solver with a Riemannian Stiefel update, together with a closed-form projection baseline and an alternating refinement for bounded, discrete susceptances. Simulations show that the stem-connected MiLAC matches fully-connected MiLAC performance, approaches the fully digital sum-rate upper bound without symbol-rate digital processing, and recovers most of the loss caused by direct hardware-grid quantization.

Ultra-broadband anti-jamming communication represents a promising approach to secure and robust information transfer through spread-spectrum techniques, effectively combatting malicious interference and eavesdropping. Rydberg atoms, enhanced by waveguide coupling, facilitate ultra-broadband spectrum sensing without traditional RF components. This framework provides an experimental platform for ultra-wide anti-jamming communication. Here, we demonstrate real-time signal demodulation based on frequency-hopping spread spectrum (FHSS) in a waveguide-coupled Rydberg receiver, achieving ultra-broad frequency-hopping covering 100 kHz to 20 GHz and a hopping rate of 100 khop/s. When confined to a standard operational band (e.g., the 2.4 GHz ISM band), our system achieves a high channel density of 8 channels per MHz. Beyond this, by leveraging its ultra-broad and continuous bandwidth, the system supports over 150,000 channels. Experimental results reveal a 51 dB enhancement in narrowband interference tolerance compared with single-frequency systems, confirming its outstanding anti-jamming capability. The reported system demonstrates significant potential for secure communications based on quantum technology, especially communication in complex electromagnetic environments.

We introduce a notion of labelled cycle for normal labelled spaces and prove a reduction theorem for Leavitt labelled path algebras. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labelled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz--Krieger uniqueness theorem for Leavitt labelled path algebras, and show that these algebras are semiprime and semiprimitive over fields.

The growth of primordial density fluctuations in the early Universe leads to an inhomogeneous dark matter distribution at high redshift, where semi-annihilation processes of the form $χχ\rightarrow χ^c ν$ can occur with a sizable rate. Using a state-of-the-art model for the cosmological boost factor, we compute the resulting redshift-dependent flux of boosted dark matter particles generated by semi-annihilation, and we study the implications of the boosted component for structure formation and direct detection experiments. We find that the cosmological contribution can enhance the sensitivity of current dark matter searches by up to three orders of magnitude, and may allow future experiments to probe scattering cross sections in the femtobarn range for dark matter masses between $\sim 10$ MeV and $\sim1$ GeV.

We study logarithmic topological Hochschild homology of the truncated Brown--Peterson spectra $\BP{n}$. We first describe prelog structures on $\BP{n}$ and analyze the spectral sequences used to compute the $V(n)$-homotopy groups of the associated logarithmic topological Hochschild homology spectra. For general $n$, we establish some structural properties of these spectral sequences that are needed for the computations. We then specialize to the case $n=2$ and carry out the calculation explicitly, determining the $V(2)$-homotopy groups of the corresponding logarithmic topological Hochschild homology spectra.

Pulse profile modelling is a relativistic ray-tracing technique used to infer neutron star mass, radius, and surface hotspot properties from X-ray pulsations. Pulse profile modelling has been widely applied to rotation-powered millisecond pulsars, where the local environment is relatively empty. Application to accreting millisecond pulsars is complicated by the geometry of the local accretion flow, including disc occultation of surface emission. In this work, we extend an established pulse profile modelling code, X-PSI, to incorporate accretion disc occultation in accreting millisecond pulsar pulse profile modelling. We quantify how disc occultation depends on system geometry and evaluate its impact on parameter inference. We find that disc occultation is primarily governed by the viewing inclination and can significantly reshape pulse profiles at moderate to high inclinations. Using synthetic Neutron Star Interior Composition Explorer datasets, we investigate parameter recovery for two representative hotspot configurations. For hotspots close to the rotational poles, statistically acceptable fits can yield posteriors that deviate noticeably from the true parameters. In contrast, in a case with hotspots located closer to the equator we find more reliable parameter recovery. We further find that neglecting disc occultation can introduce spurious posterior modes with comparable statistical support, potentially affecting the interpretation of inferred neutron star parameters, suggesting that this effect should be included in accreting millisecond pulsar pulse profile modelling.

Electric charge fluctuations $χ_n^Q$ allow comparisons between theory and experiment, but are elusive on the lattice due to severe cutoff effects. We use a 4HEX action to obtain $χ_2^Q$ and, for the first time ever, $χ_4^Q$ in the continuum limit. We find disagreement with the hadron resonance gas (HRG) model, which we cannot explain with finite volume effects. We include light meson interactions in the HRG model via the S-matrix, reducing the tension for $χ_4^Q$, but worsening the agreement for $χ_2^Q$. We propose measuring the ratio $χ_4^Q/χ_2^Q$ at the LHC to investigate this tension.

Hot and ultra-hot Jupiters offer a unique laboratory to study atmospheric chemistry at the population level using ground-based high-resolution spectroscopy. Fe and Ti are key tracers of thermal and chemical structure, yet they exhibit different observational trends across the population. We present a homogeneous reanalysis of high-resolution transmission spectra of ten hot and ultra-hot Jupiters observed with VLT/ESPRESSO and Gemini-N/MAROON-X. We search for neutral Fe and Ti absorption and perform injection-recovery tests using models spanning a range of Ti-depletion levels and T-p profiles. For direct comparison across observations, we introduce the relative cross-correlation metric, $Δ$Ti-Fe. We detect Fe in 7 and Ti in 4 planets above 5$σ$. Across the population, $Δ$Ti-Fe decreases sharply towards lower equilibrium temperatures. Under the assumption of equal Ti depletion across planets, isothermal models fail to reproduce this trend, instead requiring a temperature-dependent depletion of Ti that increases toward cooler planets, consistent with cold-trapping processes in cooler atmospheres. Models with inverted T-p profiles naturally reproduce the decline without invoking temperature-dependent depletion. There, Ti is converted into TiO in deeper, cooler layers and then removed from the gas phase through condensation, leading to strong suppression of the observable atomic Ti signal. Nevertheless, even in the gradient models, overall depletion of Ti relative to Fe is required to match the hottest planets. Our results demonstrate that observable refractory chemistry is governed by the interplay of molecular partitioning, ionisation, condensation, and cold-trapping processes, as well as the vertical structure of ultra-hot Jupiter atmospheres. Additional observations will be necessary to distinguish between temperature-dependent cold-trapping and overall depletion.(abbrev.)

The characterisation of quantum phases in strongly correlated systems is a crucial milestone for the deployment of quantum sensors. In this work, we present a Physics-Informed Variational Quantum Classifier (VQC) designed to detect the topological phase transition between the Fermi polaron quasiparticle and the molecular bound state. Unlike conventional Machine Learning approaches, our quantum architecture is constructed via the Trotterised time-evolution of an effective Hamiltonian, ensuring that the learnable parameters correspond to interpretable physical quantities. We show that the VQC efficiently discovers the optimal interferometric protocol, specifically the evolution time and effective bath interactions required to maximise the visibility of Ramsey fringes, thereby clearly distinguishing the Bose-Einstein Condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes. Furthermore, we report the validation of this classifier on the QRed superconducting quantum processor (BSC-CNS). Despite the intrinsic hardware noise and decoherence, the VQC preserves the relative ordering of the topological phases. We demonstrate that the physics-informed architecture achieves a linear gate complexity $\mathcal{O}(N)$, bypassing the exponential memory wall of classical simulation and ensuring scalability to many-body regimes.

Tuning spin-orbit coupling is essential in controlling both spin and charge in confined semiconductor nanostructures, yet it is rarely a truly controllable parameter. Here, we show control over the spin-orbit Hamiltonian in quantum dots and the resulting quantum states by tailoring the confinement potential with atomic-scale precision. Using scanning tunnelling microscopy and spectroscopy, we pattern individual Cs ions into designer quantum dot structures on the surface of indium antimonide, in which electrons from a two-dimensional electron gas are confined with chosen in-plane electric-field gradients. We then quantify the atomic level structure, both spatially resolving the orbital character of the electronic states and their magnetic-field evolution. We demonstrate that the level structure, including the induced zero-field splitting, can be tailored by the designed geometry of the local electric fields. These effects can be described using a Hamiltonian that allows consistent treatment of the confinement-induced spin-orbit coupling beyond the conventional Bychkov-Rashba description. This Hamiltonian is derived from a multiband k.p model and takes the energy dependence of the relevant physical parameters into account. Such precise control of spin-orbit coupling in semiconductor quantum dots is relevant to quantum and spintronic technologies.