The setup is the classic one of textbook optics. In a laboratory frame, denoted (L), a laser beam of wavelength λ passes through a slit of width d, where the slit dimensions are on the order of the wavelength so that diffraction is strong. The diffracted intensity as a function of observation angle θ follows the standard Fraunhofer result, proportional to the square of a sinc function, with the argument set by the transverse wave-vector mismatch. Minima occur when the sine of the observation angle equals integer multiples of λ divided by d. The angular width of the central peak can be estimated from Heisenberg’s uncertainty principle, and Bourdier shows this estimate agrees with the standard Fourier optics result. In the laboratory frame, expressed in the appropriate angular variable, the two first minima on either side of the central maximum sit symmetrically about the peak.

The novelty of the work lies in introducing a second inertial frame, (L′), moving at constant velocity V along the z-axis relative to the laboratory. Bourdier applies the Lorentz transformation to the four-velocity of diffracted photons and to the wave four-vector, deriving the relativistic aberration relations that connect propagation angles in the two frames. These relations are nonlinear: equal angular intervals in the laboratory frame do not map to equal angular intervals in the moving frame. It is precisely this nonlinearity that breaks the symmetry of the diffraction pattern. The transformation is expressed in compact tensorial form using the Lorentz matrix, and the author emphasizes that the same spacetime transformation applies whether the light propagates in vacuum or in a material medium, although the refractive index must then be treated carefully as a property of the medium’s rest frame rather than as a simple Lorentz scalar.

For the case of normal incidence, the analysis uncovers a striking special situation. There exists a particular boost velocity, determined by the condition that the sine of the first diffraction minimum angle equals V divided by c, in which one of the low-intensity directions adjacent to the central peak becomes exactly perpendicular to the boost axis in the moving frame. Combining this with the diffraction condition yields the elegant result that V/c equals λ/d. Since significant diffraction requires the slit width to be no more than about ten wavelengths, the relevant velocities are at least a tenth of the speed of light, placing the effect firmly in the relativistic regime. Under this condition, the transformed wave vector of photons diffracted toward that particular minimum becomes parallel to the transverse axis in the moving frame, a result confirmed independently by transforming the energy-momentum four-vector of the photon.

The direction of peak intensity transforms as well. In the laboratory frame, the central maximum corresponds to forward propagation along the original beam direction. After the Lorentz boost, this direction is tilted, with the sine of the new peak angle equal to minus V/c. Bourdier verifies this using the transformation of the electromagnetic fields themselves: the Poynting vector, which describes the flow of energy in the wave, acquires a transverse component in the moving frame. The energy flow is therefore no longer aligned with the original propagation direction but propagates obliquely. This is not a violation of relativity but a direct manifestation of relativistic aberration, the same effect responsible for the well-known Penrose-Terrell appearance of relativistically moving objects.

A crucial point established in the paper’s appendices is that the diffraction mechanism itself is untouched. The phase factor governing interference between contributions from different points of the aperture is the Lorentz-invariant scalar product of the wave four-vector and the spacetime position. Because this phase is invariant, the diffraction integral in the boosted frame has exactly the same sinc-type functional form as in the laboratory frame. What changes is only the mapping between the observation angle and the transverse wave-vector component, which is distorted by aberration. The maxima and minima of the pattern remain well defined under the transformation, but their angular spacing is redistributed. Most of the diffracted power in the moving frame lies between two directions that are no longer equidistant from the peak, and a relativistic correction term breaks the symmetry that existed in the laboratory frame.

The paper also treats oblique incidence, where the incoming wave strikes the slit at an angle. In the laboratory frame, the central maximum then lies along the incidence direction, and the two adjacent minima are again symmetric in sine space. Choosing the boost velocity so that the central maximum becomes normal to the slit in the moving frame, Bourdier derives the transformed positions of the two minima and shows they are no longer symmetric about the peak. Their sine values differ by a relativistic correction, and the asymmetry again traces back to the nonlinearity of the aberration formula. In both configurations, normal and oblique incidence, the conclusion is the same: the symmetry is not destroyed but reparametrized, and the apparent asymmetry is a kinematic effect of observation from a moving frame.

The magnitude of the effect is quantified explicitly. Applying the aberration transformation to the two symmetric minima at plus and minus the first minimum angle, and expanding to first order in V/c for small diffraction angles, the author finds that the angular separation between the transformed minima is approximately minus two times V/c. The asymmetry therefore scales linearly with the ratio of the boost velocity to the speed of light. The transformation of the intensity distribution itself is handled through photon number conservation, which requires that the intensity times the angular interval be preserved. This yields a Jacobian factor relating the intensity in the moving frame to that in the laboratory frame, and the invariance properties of the radiation distribution function ensure that maxima and minima are preserved under the transformation.

Although the study is analytical, Bourdier outlines how the predictions could be tested. A numerical reconstruction of the diffraction pattern in both frames, applying the derived angular transformation and Jacobian to the Fraunhofer profile, would directly visualize the distortion and confirm the predicted asymmetry of the minima. Experimentally, observing the effect with a genuinely relativistically moving slit would be extraordinarily difficult, but the author suggests analogue approaches. Optical systems involving moving or effectively moving interfaces, such as plasma environments or time-dependent photonic media with a drift velocity, could produce angular redistributions formally analogous to those derived here. Alternatively, a dynamically controlled optical setup could deliberately reparametrize the outgoing angular distribution according to the aberration law, emulating the observable consequence of a Lorentz boost without physically accelerating an aperture.

The broader significance of the work is conceptual as much as practical. Wherever diffraction occurs in the presence of substantial relative motion between a radiating structure and the observer, whether an optical aperture, a radiating interface, or a drifting plasma structure, the observed profile may lose its symmetry even though the underlying diffraction law is unchanged in the structure’s own rest frame. Relativistic distortions of angular radiation patterns could in principle serve as diagnostic signatures of motion in optical and plasma systems. The study offers a clean, simple example of how a familiar wave-optics phenomenon is reshaped by the kinematics of special relativity, reminding physicists that even the most textbook patterns carry the fingerprints of spacetime structure when viewed from a moving frame.

Subject of Research: Relativistic distortion of slit diffraction patterns under Lorentz transformation

Article Title: Relativistic asymmetry of diffraction patterns induced by Lorentz transformation

Article References: Bourdier, A. (2026). Relativistic asymmetry of diffraction patterns induced by Lorentz transformation. Results in Physics, 88, Article 108750. https://doi.org/10.1016/j.rinp.2026.108750

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108750

Keywords: diffraction, special relativity, Lorentz transformation, relativistic aberration, Fraunhofer pattern, wave optics, sinc function, Poynting vector, photon four-vector, laser propagation, plasma optics, asymmetry

The core challenge the researchers set out to solve is deceptively simple to state: given a start point and a goal in a cluttered 3D environment, generate a trajectory that is short, smooth, dynamically feasible, and energy-efficient, all at the same time. These objectives frequently conflict with one another. The shortest path may hug obstacles so tightly that a real aircraft could not safely follow it. The smoothest path may waste energy in wide, sweeping arcs. Traditional optimization methods can balance these goals but often struggle in unknown or changing environments, where they must be re-run from scratch whenever conditions shift.

Deep reinforcement learning has emerged in recent years as an attractive end-to-end alternative. In this paradigm, an artificial intelligence agent learns to fly by trial and error, receiving rewards for progress toward the goal and penalties for collisions, erratic motion, or excessive energy use. Over many training episodes, the agent internalizes a policy that maps what it observes directly to the actions it should take. The appeal is obvious: once trained, such a policy can react to new situations in milliseconds without recomputing a full trajectory. The drawback, as the new paper emphasizes, is that learning from scratch is painfully slow. Random exploration in a vast three-dimensional action space means the agent spends most of its early training bumping into obstacles or wandering aimlessly, a problem the authors describe as blind exploration combined with low sample efficiency.

To inject common sense into this process, the team turned to the artificial potential field, a concept that has guided robot navigation since the 1980s. In an artificial potential field, the goal exerts an attractive force that pulls the vehicle toward it, while obstacles exert repulsive forces that push it away. The drone is imagined as a ball rolling downhill on a landscape sculpted by these forces, and at every instant the field suggests a sensible direction of travel. The elegance of the approach is that the guidance is state-dependent: as the drone’s position and surroundings change, the forces change with them, always pointing toward safer, more productive regions of space.

Rather than replacing the learning algorithm with this classical method, the researchers fused the two. The state-dependent forces generated by the artificial potential field are integrated directly with the policy of a Soft Actor–Critic agent, a state-of-the-art deep reinforcement learning algorithm prized for its stability and its ability to balance exploration against exploitation. Soft Actor–Critic maximizes both the expected reward and the entropy, or randomness, of the agent’s behavior, which prevents it from collapsing prematurely into a mediocre strategy. By blending the potential field’s heuristic push into the action-selection process, the hybrid system no longer explores blindly. From the very first training episode, the agent is nudged in directions that the physics suggests are promising, while retaining the freedom to deviate when the heuristic is wrong, as it can be in local minima where attractive and repulsive forces cancel out.

The second innovation addresses a different weakness: memory. Standard actor–critic networks treat each moment in isolation, deciding what to do based only on the current observation. A flying vehicle, however, is a dynamical system whose future depends on its recent past. A sudden change in heading that was perfectly safe at low speed may be catastrophic at high speed, and the network cannot know the difference if it has no access to the sequence of states that led to the present moment. To capture these temporal dependencies, the authors embedded a long short-term memory, or LSTM, structure into both the actor and the critic networks. LSTMs are recurrent neural networks equipped with gating mechanisms that allow them to retain information over many time steps and to forget what is no longer relevant, giving the agent an effective working memory of its own flight history.

The practical consequence of this architectural choice is smoother, more dynamically feasible trajectories. Because the policy can perceive trends, accelerations, and oscillations in the state sequence rather than single snapshots, it learns to produce control commands that flow naturally from one to the next, avoiding the jerky, high-frequency corrections that plague memoryless policies and that translate directly into wasted energy and mechanical stress on real airframes. The combination of potential field guidance and recurrent memory gives the method its name: APF–LSTM–SAC.

The team validated the approach in comprehensive tests within complex simulated three-dimensional environments, comparing it against pure deep reinforcement learning baselines. The results were striking. The hybrid method achieved improvements in convergence speed of at least 15.14 percent, meaning the agent reached competent flight policies substantially faster than its unguided counterparts, and it reduced energy consumption by at least 10.36 percent, a figure that matters enormously for battery-powered aircraft whose mission endurance is measured in minutes. Faster training also carries a practical dividend: fewer simulated flight hours are needed before a policy is deployable, which lowers the computational cost of developing autonomous capabilities for new vehicle types or new environments.

The significance of the work extends beyond the specific percentages. It exemplifies a growing trend in robotics toward physics-informed machine learning, in which decades of classical control theory and heuristic reasoning are used to scaffold, rather than be replaced by, modern data-driven methods. Pure learning systems must rediscover from scratch lessons that engineers already know, such as the fact that obstacles should be avoided and goals approached. By encoding those lessons as inductive biases inside the learning pipeline, researchers can preserve the adaptability of reinforcement learning while dramatically shrinking the search space the algorithm must explore. The potential field component supplies a sensible prior; the LSTM supplies temporal awareness; and the Soft Actor–Critic framework supplies robust, entropy-regularized optimization that can gracefully reconcile the two.

The authors note that the datasets generated and analyzed during the study are available from the corresponding author on reasonable request, and the work was supported by the National Natural Science Foundation of China, the CAAC Key Laboratory of General Aviation Operation, and the Fundamental Research Funds for the Central Universities. As drones take on ever more ambitious roles, from delivering medical supplies to surveying disaster zones, the ability to plan safe, efficient three-dimensional trajectories autonomously will only grow in importance. Hybrid approaches like APF–LSTM–SAC suggest that the fastest route to capable autonomous flight may not be to make learning algorithms bigger, but to make them wiser, by letting the accumulated physics of navigation light the way through the darkness of blind exploration.

Subject of Research: UAV three-dimensional path planning using artificial potential field guidance and an LSTM-enhanced Soft Actor–Critic deep reinforcement learning algorithm

Article Title: UAV 3D Path Planning Based on Artificial Potential Field Guidance and LSTM-Enhanced Soft Actor–Critic

Article References: Liu, J., Zhou, H., Lei, X., Tu, X., Hua, H., & Wang, X. (2026). UAV 3D Path Planning Based on Artificial Potential Field Guidance and LSTM-Enhanced Soft Actor–Critic. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01292-7

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01292-7

Keywords: UAV path planning, deep reinforcement learning, artificial potential field, Soft Actor–Critic, long short-term memory, drone navigation, trajectory optimization, energy efficiency, 3D path planning, autonomous flight, machine learning, robotics

The work, carried out by Chong Wu, Junfei Chen, Zhiping Wang and Zhixiang Huang of Anhui University in Hefei, China, relies on a nonlinear optical process known as three-wave mixing, staged inside a cloud of cold atoms. In three-wave mixing, two input fields interact within a medium that possesses a second-order nonlinear response, and the sum of their energies and frequencies emerges as a third field. When one of the inputs is a microwave field and the other is a carefully chosen optical control beam, the output is a new optical field whose frequency sits far above the microwave domain but whose spatial character is inherited from the microwave field that seeded it. In effect, the atoms act as a transducer that reads the microwave beam and rewrites it in optical script.

The ingenuity of the scheme lies in the energy level structure the authors chose. They consider a multilevel atomic system in which two of the transitions are driven by optical laser fields while a third, much lower frequency transition couples to the microwave field. The microwave field in question is not an ordinary beam: it carries orbital angular momentum, the property more familiarly associated with twisted laser beams whose wavefronts wind around the propagation axis like a helix. A field with orbital angular momentum of order l has a phase that winds 2l times around the beam axis and an intensity profile that vanishes on the axis, producing a ring-shaped or vortex structure. Because this winding number can, in principle, take any integer value, orbital angular momentum offers a practically unbounded alphabet of spatial modes for encoding information.

When the twisted microwave field drives the appropriate transition inside the cold atomic ensemble, it imprints its angular phase structure onto the atomic coherence, the collective quantum state shared by the atoms. The nonlinear coupling then transfers that imprint to the generated optical field. Crucially, the authors show that this transfer is coherent, meaning the phase relationship between the input and output fields is preserved throughout the process. Coherence is what separates a genuine quantum transducer from a lossy photocopy: it is the property that would allow the structural information of the microwave field to be recovered, manipulated, or used in later quantum operations at the optical frequency.

To quantify how faithfully the structure survives the frequency conversion, the team turned to a familiar tool from image processing: the structural similarity index, a metric originally developed to assess how closely two images resemble each other as perceived by human vision. By computing the intensity and phase distributions of the input microwave field and the generated optical field and comparing them pixel by pixel, the researchers demonstrate high-similarity transfer of both pieces of information under their chosen energy level scheme. The intensity rings of the vortex microwave beam reappear as intensity rings in the optical output, and the helical phase winding is reproduced with high fidelity, a result that holds across a range of orbital angular momentum values.

The physics behind this fidelity traces back to the way three-wave mixing preserves angular momentum. In any nonlinear frequency conversion process, conservation laws constrain the interaction: energy must balance among the three waves, and so must angular momentum. When the microwave input carries orbital angular momentum l and the optical control fields carry their own defined angular momenta, the generated optical field must absorb the difference, emerging with a well-defined topological charge determined by the input modes. Because the atomic medium is cold and nearly stationary, Doppler broadening and motional decoherence, the usual enemies of coherent conversion in warm vapors, are strongly suppressed. That cleanliness is what allows the structural information, encoded in delicate spatial phase variations, to survive a jump in frequency of many orders of magnitude.

The significance of the result becomes clear when one considers why researchers want microwave-to-optical conversion in the first place. Superconducting qubits, among the most advanced quantum computing platforms, operate at microwave frequencies and at temperatures near absolute zero. Quantum memories based on atomic ensembles, meanwhile, often interact most naturally with optical light. Connecting these platforms demands a converter that can translate between the two regimes while preserving quantum states. Earlier experiments, including demonstrations in cold rubidium ensembles using Rydberg states and coherent population trapping, established that efficient microwave-to-optical conversion is achievable in atomic systems. What distinguishes the new proposal is its explicit focus on structured fields: rather than converting a simple plane-wave signal, it converts a beam whose information content lives in its spatial shape.

That focus opens a distinct set of possibilities. Twisted light has become a workhorse of modern optics, enabling terabit-scale free-space data links, mode-division multiplexing in fibers, high-dimensional quantum cryptography, and entanglement of photons carrying large angular momenta. If microwave fields carrying orbital angular momentum can be coherently lifted into the optical domain, the spatial-mode alphabet of twisted light becomes available to microwave quantum technologies. The authors note that their scheme provides a way to realize orbital angular momentum transmission and spiral phase regulation directly in cold atoms, capabilities they suggest could find applications in quantum information processing, where spatial modes can multiply the information capacity of a single photon or serve as robust carriers for quantum keys.

The proposal also connects to a growing body of work on manipulating vortices in quantum systems, from optical vortices imprinted on Bose-Einstein condensates to quantum memories that store spatial structure in atomic ensembles. Prior studies have shown coherent transfer of optical vortices within atomic media and quantum storage of orbital angular momentum entanglement, but extending these capabilities to microwave frequencies has remained largely unexplored. By demonstrating that the structural similarity between a twisted microwave input and its optical output can be kept high, the Anhui University team effectively extends the toolbox of structured light down into the microwave regime and back up again, tracing a complete coherent pathway between the two worlds.

As with any theoretical scheme, the path from calculation to laboratory demonstration will demand careful experimental work: preparing cold atomic ensembles with the right level structure, delivering shaped microwave fields with well-defined orbital angular momentum, and characterizing the generated optical field with the phase-sensitive techniques developed for structured light. But the reward would be substantial. A converter that faithfully translates the intensity and phase structure of microwave fields into light would give quantum engineers a new degree of freedom in designing hybrid networks, linking microwave processors to optical channels while letting information ride on the twisting of the wave itself. In a field where every preserved qubit and every untarnished phase front counts, high-similarity conversion is not an incremental improvement; it is an invitation to encode quantum information in dimensions that neither microwaves nor light alone could exploit.

Subject of Research: Coherent microwave-to-optical frequency conversion of orbital angular momentum fields via three-wave mixing in cold atoms

Article Title: High-similarity microwave-to-optical frequency conversion via three-wave mixing

Article References: High-similarity microwave-to-optical frequency conversion via three-wave mixing. (n.d.). https://doi.org/10.1007/s11128-026-05330-x

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05330-x

Keywords: microwave-to-optical conversion, three-wave mixing, orbital angular momentum, cold atoms, quantum information processing, structured light, frequency conversion, spiral phase, atomic ensemble, quantum transducer, structural similarity, vortex beam

The core idea builds on variational quantum classifiers, the workhorse architecture of the noisy intermediate-scale quantum era. These models encode classical data into quantum states using parameterized circuits, then extract predictions from measurement outcomes, with the circuit parameters tuned by classical optimizers through techniques such as the parameter-shift rule for quantum gradients. Because each user in a federated setting typically holds only a small, idiosyncratic slice of data, a single personalized model trained in isolation tends to generalize poorly. The new framework tackles this by training user-specific variational quantum models on disjoint local datasets and then combining their predictions through an ensemble, borrowing a strategy as old as classical machine learning itself: many weak learners, aggregated wisely, can outperform any one of them alone.

What distinguishes this work is the rigor of its privacy treatment. Each user privatizes their labels locally, before anything leaves their device, using the randomized response mechanism, a classical technique in which the true label is reported with some probability and a random alternative otherwise. Crucially, each user may choose an individual privacy budget, denoted epsilon-i, which quantifies how much information about any single record the privatized output can leak. This local differential privacy model is stricter than the centralized variant used by large technology companies, because no trusted curator ever sees unprivatized data. The privacy guarantee is established mathematically at the user’s side, before any communication occurs, which means the server aggregating the models need not be trusted at all.

The formal analysis rests on two pillars of differential privacy theory. The first is the post-processing property, which the authors prove in an appendix: any computation performed on already-privatized data cannot weaken the privacy guarantee, no matter how elaborate the downstream machinery. The second is parallel composition, which states that when independent privacy mechanisms are applied to disjoint datasets, the overall privacy loss is governed by the largest individual budget rather than the sum of all budgets. Because each user’s data lives in its own disjoint partition and every supervision signal in the strict regime is itself privatized before use, the protected-label stream inherits record-level epsilon-i local differential privacy, and the entire ensemble-level guarantee is bounded by the maximum epsilon across all participating users. In other words, the privacy cost of collaboration is set by the least private participant, not by the crowd.

Within this privacy-consistent regime, the researchers compare two ways of merging the personalized quantum models. The first is voting-based aggregation, in which the ensemble simply takes a majority or weighted vote over the predictions of the individual quantum classifiers. The second is a learned aggregation module, a small trainable component that decides how much to trust each member model’s output when producing the final prediction. Learned aggregation can be more expressive, but it introduces a subtlety: calibrating such a module typically requires supervision, and if that supervision comes from clean, unprivatized labels, the strict formal privacy scope no longer covers the whole pipeline. The authors are careful to frame this as an optional extension, calibrated on a small clean validation set, that sits outside the end-to-end privacy guarantee.

Empirically, the results reveal a clear division of labor between the two aggregation strategies. When the pipeline remains privacy-consistent from start to finish, with every label privatized before use, voting emerges as the most stable and reliable choice, since it never requires additional clean supervision that could compromise the guarantee. When reliable clean calibration labels are available and the strict formal privacy scope is relaxed accordingly, the learned aggregation module becomes the most effective, exploiting its extra flexibility to weight the ensemble members intelligently. This practical guidance, that the right aggregation rule depends on the supervision regime, gives practitioners a concrete decision rule rather than a one-size-fits-all prescription.

The study does not stop at binary classification benchmarks. The authors extend their experiments to multiclass tasks, where the randomized response mechanism must handle more than two possible labels and the noise floor rises accordingly. They also examine partial participation, the realistic scenario in which only a subset of users contributes to the ensemble in any given round, and they probe the scalability of the framework as the number of participants grows. Simulated noise experiments characterize how the privatization probability interacts with model accuracy, mapping out the trade-off curve between privacy budgets and predictive performance. Together, these experiments delineate the operating envelope of the method, showing where it thrives and where the privacy noise begins to erode the ensemble’s advantage.

Perhaps most striking for a field still dominated by simulation, the team replicated key aspects of their study on real IBM Quantum hardware in a pilot study. Running variational quantum circuits on today’s noisy devices is a stern test, since decoherence, gate errors, and readout noise compound with the deliberate noise injected by privacy randomization. The fact that the framework remained competitive under these compounded imperfections suggests a certain robustness that pure-theory studies often lack. It also aligns with a broader lesson from the quantum machine learning literature, including work on generalization from few training data, that ensembling and careful aggregation can compensate for the limitations of individual models trained on scarce, noisy data, which is precisely the regime that near-term quantum hardware imposes.

The broader significance of this work lies in its timing. Quantum machine learning is maturing from proof-of-concept demos toward applications in finance, healthcare, and high-energy physics, domains where the data is exactly the kind that regulators and users insist on protecting. Prior studies have explored differential privacy for quantum machine learning in centralized settings, and quantum local differential privacy has been analyzed from an information-theoretic perspective, but the question of how to combine personalized quantum models across many mutually distrusting parties had remained open. By proving that the ensemble inherits a clean max-epsilon guarantee under parallel composition, and by validating the approach both in simulation and on hardware, the Fordham-led team has supplied a template for privacy-preserving collaborative quantum learning that other groups can build on immediately.

There are, of course, limits that the authors themselves acknowledge. The strict privacy-consistent regime demands that every downstream supervision signal be privatized, which constrains how sophisticated the aggregation layer can be; the moment clean labels enter the picture, the formal guarantee must be renegotiated. The randomized response mechanism also imposes an accuracy tax that grows as privacy budgets shrink, and the framework’s performance ultimately depends on the quality and diversity of the local datasets each user contributes. Still, the study, supported in part by the National Science Foundation under Grants 2335788, 2343535, and 2555384, marks a meaningful step toward quantum machine learning systems that respect the privacy of the people whose data makes them possible. As quantum hardware improves and federated deployments become practical, frameworks like this one may define the standard by which trustworthy quantum AI is judged: powerful, personalized, and provably private.

Subject of Research: Collaborative training of personalized quantum classifiers under local differential privacy with ensemble aggregation

Article Title: Ensembling personalized quantum models with local differential privacy

Article References: Ensembling personalized quantum models with local differential privacy. (n.d.). https://doi.org/10.1007/s42484-026-00440-2

Image Credits: AI Generated

DOI: 10.1007/s42484-026-00440-2

Keywords: quantum machine learning, local differential privacy, ensemble learning, variational quantum circuits, randomized response, federated learning, privacy-preserving machine learning, quantum classifiers, parallel composition, IBM Quantum, label privacy, NISQ era

At the heart of the study is the Doyle–Fuller–Newman (DFN) model, sometimes called the pseudo-two-dimensional model, which has been the workhorse of physics-based battery simulation since the early 1990s. The DFN model describes a cell as a one-dimensional sandwich of a porous graphite negative electrode, a separator, and a porous positive electrode made of layered lithium nickel cobalt oxide, all between metallic current collectors. Lithium ions migrate through the electrolyte and intercalate into or deintercalate from spherical active particles, while electrons travel through the external circuit. The model resolves coupled electrolyte transport, solid-phase lithium diffusion, charge conservation, and interfacial Butler–Volmer reaction kinetics, striking a pragmatic balance between physical realism and computational tractability. Unlike fully three-dimensional microstructure-resolved simulations, which demand high-performance computing and massive parallelization, the DFN model can be evaluated rapidly enough to be sampled thousands of times, a prerequisite for the statistical inference at the core of the new framework.

The central problem the team attacked is parameterization. Accurate values for transport and thermal coefficients are essential for predicting capacity, polarization, and temperature rise, especially at high discharge rates where coupled transport and thermal effects dominate. Yet these effective properties depend on microstructural features such as porosity, tortuosity, particle connectivity, and contact resistances that evolve during manufacturing and aging. Direct experimental measurement is difficult, reported values vary widely across the literature, and empirical tuning remains common. Worse, the parameters are strongly coupled: changing one can compensate for changes in another, making independent identification from terminal voltage data notoriously ill-posed. This is precisely the kind of inverse problem where Bayesian methods shine, because treating parameters as random variables conditioned on data yields full posterior distributions, credible intervals, and identifiability assessments rather than fragile point estimates.

The researchers identified two fundamental challenges that previous approaches had not solved together. The first is multiscale parameter identification. Some DFN parameters, such as electrode porosities, electronic conductivities, and thermal conductivities, are continuum-scale quantities that can legitimately be inferred from cell-level voltage measurements. Others, notably the open-circuit potential and solid-state lithium diffusivity, are intrinsic material properties governed by atomistic thermodynamics and lithium migration mechanisms, and are only weakly constrained by voltage data. Attempting to estimate everything simultaneously inflates parameter correlation and destroys practical identifiability. The team’s solution was elegant: they prescribed the atomistic quantities using first-principles density functional theory calculations performed with the Vienna Ab initio Simulation Package (VASP), and reserved Bayesian inference for the six effective electrode parameters that the discharge data can genuinely inform.

The first-principles component is a substantial contribution in its own right. Using VASP with the projector augmented-wave framework and the Perdew–Burke–Ernzerhof functional, the researchers computed stoichiometry-dependent open-circuit potentials from total-energy differences between lithiated configurations, referencing metallic body-centred-cubic lithium. The resulting curves, calculated at ten lithium stoichiometries, reproduce the sloping behavior and magnitude of established experimental reference profiles for both electrodes. For diffusivity, they resolved the lithium energy landscape along a basal-plane hopping pathway in graphite using a constrained scan, extracting a migration barrier of approximately 0.43 electron volts. Plugging this barrier into an Arrhenius expression with a physically motivated hopping prefactor yields a room-temperature solid-phase diffusivity of about 9.75 times ten to the minus fifteen square meters per second, consistent with values used in established DFN parameterizations. The authors are candid about the limitations of this atomistic-to-continuum transfer, noting that zero-temperature approximations and unresolved microstructural effects mean the mapping is physically informed rather than exact.

The second challenge is structural model discrepancy. Even with perfect parameters, the DFN model embodies idealizations, including homogeneous electrode microstructures, simplified thermal coupling, ideal interfacial kinetics, and neglected degradation mechanisms such as solid-electrolyte interphase growth, lithium plating, and particle cracking. These errors cannot be eliminated by parameter tuning alone. Rather than replacing the physics with a machine-learning surrogate, which would sacrifice interpretability, the team preserved the full DFN formulation and trained a small feed-forward neural network to learn only the residual discrepancy between the model prediction and the experimental voltage. The network takes the six uncertain parameters and time as inputs and outputs a parameter- and time-dependent correction that is added to the DFN prediction, forming a corrected forward model. Crucially, the network’s weights remain fixed during sampling, so Bayesian inference operates exclusively on the physical parameters.

A further methodological safeguard addresses a subtle but important statistical issue: neural-network surrogates introduce their own approximation error, which, if ignored, produces overconfident posteriors. The researchers quantified this error as the validation mean squared error on held-out data and added it to the observation noise variance, forming an augmented effective variance in the Gaussian likelihood. Independent discrepancy surrogates were trained for each discharge rate, and the temporal domain was partitioned into disjoint training and testing subsets to limit information leakage between the discrepancy-learning stage and the posterior evaluation. The sampling itself employed the delayed rejection adaptive Metropolis algorithm, run for one thousand iterations with two hundred discarded as burn-in, using uniform priors on physically admissible bounds drawn from the well-known Ecker 2015 experimental parameterization of a graphite–nickel-cobalt-oxide cell.

The validation results are striking. Working with experimental discharge data digitized from published Ecker benchmark curves and simulated in the open-source PyBaMM framework with particle mechanics enabled, the team tested the framework at both a transport-limited high-rate regime and a transport-relaxed moderate-rate regime. The baseline DFN model using literature reference parameters mispredicted voltage with a root-mean-square error of roughly 0.10 volts. Bayesian calibration alone improved matters substantially, but the full hybrid framework, combining calibrated parameters with the learned discrepancy correction, drove the error down to approximately 0.01 volts, a tenfold improvement. The experimental data fell almost entirely within the 95 percent posterior credible interval, whose narrow width signaled well-identified parameters, with mild widening near end-of-discharge reflecting genuine transport limitations. The inferred values, including porosities clustering near 0.30 to 0.32 and conductivities in physically plausible ranges, agreed well with reference measurements, indicating the framework finds real physics rather than overfitting.

Rigorous diagnostics underpinned these claims. Gelman–Rubin statistics converged rapidly toward unity, confirming well-mixed Markov chains, while pairwise posterior distributions showed compact, approximately unimodal regions with only weak-to-moderate parameter correlations. A Spearman rank correlation analysis found all parameter-observable correlations below 0.15 in magnitude, supporting robust identifiability. An ablation study comparing inference with and without the neural-network correction showed that the discrepancy model tightens the posteriors and stabilizes the chains, compensating systematic model-form error without absorbing the influence of the physical parameters. The authors transparently acknowledge that complete statistical independence between discrepancy training and posterior evaluation is not achievable when both draw on the same experimental trajectory, and they frame their results as evidence of improved practical identifiability rather than formal structural identifiability. Extending the framework to temperature and impedance data, unified multi-rate surrogates, and hierarchical priors that propagate first-principles uncertainty remains future work. Even so, the study demonstrates a compelling template for digital battery engineering: keep the physics, let quantum calculations anchor the material constants, let a small neural network absorb what the physics misses, and let Bayesian statistics keep everyone honest about what is actually known.

Subject of Research: A hybrid machine learning and Bayesian inversion framework for calibrating physics-based lithium-ion battery models using first-principles parameters and neural-network discrepancy correction.

Article Title: A hybrid machine learning–Bayesian inversion framework for physics-based lithium-ion battery models

Article References: Khodadadian, E., Mirsian, S., Khodadadian, A., & Noii, N. (2026). A hybrid machine learning–Bayesian inversion framework for physics-based lithium-ion battery models. Machine Learning with Applications, 25, Article 100995. https://doi.org/10.1016/j.mlwa.2026.100995

Image Credits: AI Generated

DOI: 10.1016/j.mlwa.2026.100995

Keywords: lithium-ion batteries, machine learning, Bayesian inversion, Doyle–Fuller–Newman model, density functional theory, neural networks, uncertainty quantification, parameter identification, PyBaMM, VASP, model discrepancy, electrochemical modeling

The core difficulty with position verification is that ordinary distance measurements are fundamentally based on trust. Radar, GPS, and time-of-flight ranging all rely on the assumption that the device being located responds honestly and that the signals it returns are genuine. A spoofing attacker can exploit this by relaying messages, predicting challenge responses, or replaying recorded signals faster than physics should allow. Classical cryptography alone cannot close these loopholes, because any classical challenge-response scheme can, in principle, be simulated or forwarded by a sufficiently capable adversary. What the new work shows is that by encoding challenges in quantum states, specifically coherent states, the kind of light produced by ordinary lasers, and by enforcing the relativistic speed limit on information, a verifier can confirm a prover’s location with security guarantees that no classical protocol can match.

The protocol builds on a concept known as relativistic position verification, which has been discussed theoretically for more than a decade. The idea is elegant in its use of special relativity. A verifier sends a challenge signal to the claimed position of a prover, and the prover must respond within a strict time window determined by the speed of light. If the prover is genuinely at the claimed position, the round-trip timing works out precisely. If the prover is anywhere else, the finite speed of light makes it impossible to receive the challenge and return a correct response in time. The catch, historically, has been that a single relativistic check can be defeated by multiple colluding adversaries who surround the claimed position and share information, provided the challenge itself carries no quantum advantage. The new protocol confronts this collusion problem directly by making the challenge a quantum state that cannot be perfectly copied or measured without disturbance.

Coherent states occupy a special place in quantum optics. They are the closest quantum states to classical light, describing the output of an ideal laser, and they are famously robust: a beam splitter tapping a fraction of a coherent state leaves the remaining light in another coherent state, undisturbed. This resilience is precisely why coherent states are the workhorses of optical communication. But it also means they cannot be protected by the no-cloning theorem in the same dramatic way that single photons can. The researchers’ insight was to design a verification scheme in which the security does not depend on detecting eavesdropping through disturbance, but rather on the statistical structure of the coherent-state challenge combined with relativistic timing constraints. An attacker who tries to intercept, measure, and forward the challenge gains only limited information within the light-speed-bounded window, and that limitation translates into a quantifiable, provable bound on the probability of successful spoofing.

In the experimental demonstration, the team implemented the protocol using quantum optical equipment in a laboratory setting that emulated the geometry of a multi-verifier network. Verifiers at separated reference points prepared coherent-state challenges and sent them toward the claimed position of a prover. The prover, located at the intersection point defined by the overlapping signals, was required to perform a joint measurement on the arriving light and return a response that depended on the full quantum content of the challenges. The verifiers then checked both the correctness of the response and its arrival time against the relativistically mandated schedule. Only a prover genuinely at the claimed spacetime point, with access to the complete quantum information carried by the coherent pulses, could satisfy both conditions simultaneously with high probability.

The measurement statistics from the experiment confirmed the central theoretical prediction. Honest provers at the correct location passed the verification test with high probability, while simulated attacks, in which adversaries positioned away from the claimed point attempted to collaborate and cheat the timing and content checks, succeeded only with probabilities bounded well below the levels required to break the protocol. The experiment thereby elevated relativistic position verification from a theoretical proposal with idealized assumptions to a demonstrated capability built from realistic optical components. Because coherent states are exactly what standard telecommunications lasers produce, the result carries an unusually direct path toward practical deployment in fiber networks and free-space optical links.

The security implications extend across several domains of modern technology. In satellite navigation, position verification could harden global positioning systems against spoofing attacks, a threat that has been demonstrated against civilian GPS receivers and poses risks to aviation, maritime shipping, and autonomous vehicles. In distributed computing and blockchain systems, verifiable position could anchor the physical identity of nodes, preventing adversaries from masquerading as geographically distributed participants. In quantum networks, where future quantum internet nodes will exchange entanglement and secret keys over continental distances, confirming that a node is physically where it claims to be adds a layer of authentication that no certificate or cryptographic key alone can provide. The protocol essentially gives the physical layer of communication a cryptographic guarantee derived from the laws of nature rather than from computational assumptions.

What makes the achievement scientifically notable is the way it reconciles two apparently competing demands. Quantum protocols for position verification have often relied on fragile single-photon states or entanglement, which are difficult to distribute over long distances and easily degraded by loss. Relativistic schemes, conversely, have been robust in their signals but vulnerable to collusion attacks. By choosing coherent states, the experimenters selected the most loss-tolerant, telecom-compatible quantum states available, and then recovered security against collusion through a careful protocol design that exploits the interplay between the quantum statistics of the light and the strict causal structure imposed by relativity. The result is a scheme whose ingredients are mundane, laser light and precise clocks, but whose guarantees are anything but.

The work also contributes to a broader conceptual shift in quantum information science: the recognition that spacetime structure itself is a computational and cryptographic resource. Just as entanglement enables tasks impossible classically, the light cone structure of relativistic spacetime constrains what any attacker can know and when they can know it, and protocols that weave quantum states through this causal fabric inherit guarantees rooted in physics. Position verification is perhaps the most natural application of this principle, because position is defined by spacetime, but researchers have begun exploring related ideas in secure timing, delegated quantum computation, and verifiable quantum communication. The new demonstration provides an experimental anchor for this emerging field, showing that the theory can be reduced to working hardware.

Challenges remain before such systems secure real-world infrastructure. Laboratory demonstrations operate over short distances with controlled timing, whereas field deployment will demand picosecond-level clock synchronization across verifier stations, management of atmospheric and fiber-induced noise, and careful analysis of loss, which affects coherent states in ways that must be folded into the security proofs. Scaling the number of verifiers and the distance to the prover will require engineering advances in optical timing distribution and high-speed quantum-light detection. Yet the direction is clear. The experiment demonstrates that verifying where someone is, without trusting anything they say, can be achieved by combining the most ordinary light in the universe with the most fundamental speed limit in nature. In doing so, it turns a century of physics, from Einstein’s postulates to the quantum theory of light, into a practical answer to a deceptively simple question: can you prove where you are? The answer, it turns out, is yes, if your answer travels at the speed of light and carries the quiet statistical fingerprint of a coherent state.

Subject of Research: Relativistic position verification using coherent quantum states of light

Article Title: Relativistic position verification with coherent states

Article References: Fan-Yuan, G.-J., Shan, Y.-G., Zhang, C., Wang, Y.-L., Fan, Y.-X., Xie, W.-X., He, D.-Y., Wang, S., Yin, Z.-Q., Chen, W., Fu, S.-N., Guo, G.-C., & Han, Z.-F. (2026). Relativistic position verification with coherent states. Nature Physics. https://doi.org/10.1038/s41567-026-03439-5

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03439-5

Keywords: position verification, coherent states, quantum optics, relativistic cryptography, spoofing attacks, speed of light, quantum communication, laser light, quantum networks, spacetime, time-of-flight, security protocols

Utrecht occupies a special place in medieval Dutch ceramic history. The earliest evidence for pottery production there dates to the late 12th century CE, and by the 14th century the town had joined Haarlem, Leiden and Breda as a major manufacturing centre supplying both local markets and wider regional trade. Its products have been recovered as far afield as Alkmaar, Haarlem, Dordrecht and Amsterdam. The workshops themselves clustered in the suburbs of Bemuurde Weerd and Tolsteeg, north and south of the town walls, and along the banks of the Vecht river, where excavations have uncovered nine vertical updraft kilns with brick or clay floors, round or oval in plan and measuring up to five metres long.

The excavated waste heaps tell a story of technological transition. The first production phase at Zeedijk, dated 1150 to 1175 CE, yielded almost exclusively grey, round-based jars. After an apparent hiatus of roughly a century, the second phase, from 1275 to 1350 CE, saw a wider repertoire including lead-glazed reddish tripod forms, jugs, bowls and pans, though grey jars remained dominant. By the third phase, represented by the Oosterkade workshop (1350 to 1400 CE) and riverside workshops such as Anthoniedijk, Hogelanden and Lauwerecht (1375 to 1425 CE), lead-glazed redware had become far more prominent, and unglazed and lead-glazed floor tiles were also being produced.

To characterise the raw materials and firing technology, the team subjected the 59 samples, plus one clay sample from a waste pit at Bemuurde Weerd, to a battery of techniques: polarised light optical microscopy, wavelength-dispersive X-ray fluorescence spectrometry, X-ray diffraction and scanning electron microscopy with energy dispersive X-ray spectrometry. Thin-section petrography revealed two main fabric groups, a Coarse Group with large, moderately to poorly sorted quartz inclusions and a Fine Group with smaller, better-sorted inclusions. The size, rounded shape and bimodal distribution of the coarse quartz grains suggest they were deliberately added as temper, most likely derived from fluvial deposits, consistent with the Holocene river clays of the region.

The chemical data, measured on 26 major, minor and trace elements at the Fitch Laboratory of the British School at Athens, showed a strongly homogeneous, silico-aluminous dataset pointing to local clay sources. Principal component analysis identified three compositional groups, with a large, homogeneous group A accounting for more than 70 percent of the samples. All the Utrecht products were made from calcium-poor, iron-rich clay, with calcium contents below 2.5 percent, and the chemical similarity between the fired clay sample and the pottery, despite differences in calcium, may itself be evidence of quartz tempering. X-ray diffraction confirmed the mineral assemblage of quartz, illite-muscovite, K-feldspar and plagioclase, with redware bodies generally containing more hematite than greyware.

Firing temperatures emerged as consistently low. Most of the ceramics, 44 of them, were fired below 800 degrees Celsius, while a handful containing the high-temperature minerals gehlenite, diopside and mullite may have reached roughly 850 to 900 degrees or slightly above. The co-occurrence of surviving calcite and dolomite with these high-temperature phases implies short soaking times in the kiln. The glazes told their own story: single-layered, transparent coatings up to about 250 micrometres thick, of very high to high-lead type, with lead oxide contents between roughly 51 and 69 weight percent. Comparisons of corrected glaze and body compositions indicate that potters mixed lead oxide with silica before application, and that the yellowish-brown to greenish colour came from iron in the glaze over the reddish ceramic body.

The most novel element of the study was the application of small-angle neutron scattering, or SANS, to the question of how the vessels were formed. Measured non-destructively at the YS-SANS instrument of the Budapest Neutron Centre, 38 jar samples yielded data on the orientation and alignment of nanoscale domains in the ceramic fabric, which record the forces applied during forming. Thirty-two samples showed high isotropy values, indicating disorganised internal structures characteristic of percussion-building techniques such as pinching, moulding or tamper-and-concave-anvil forming. Combined with the wheel-made traces on rims and necks, this points to a two-stage strategy the authors call percussion-wheeling: the body formed by percussion, then the neck and rim refined on a rotational device.

Only six samples showed the low isotropy and significant tilting angles diagnostic of other techniques. One redware jar from the second phase at Zeedijk proved to be coil-built and wheel-shaped, while five jars, from both Zeedijk and Oudenoord, were genuinely wheel-thrown, three with clockwise and two with anticlockwise wheel rotation. Notably, all the wheel-thrown examples date to the second production phase after about 1275 CE, confirming a previously observed typo-technological shift from grey hand-formed jars to reddish tripod forms. Contrary to common expectations, the potters did not favour fine fabrics for wheel-throwing; nearly all the wheel-thrown jars were made from the coarse fabric. The persistence of percussion-wheeling across the 100-year hiatus between the first and second phases suggests a conservative, culturally embedded technological tradition, one perhaps also suited to producing the round-based jar shapes that were difficult to throw on a wheel.

The study also cautions against reading forming techniques from surface features alone. Interior depressions often interpreted as fingertip impressions from moulding appeared on wheel-thrown vessels too, and may instead reflect hands supporting the vessel wall during brushing, while interior ridges below the neck, sometimes taken as evidence of added coils, also occurred on wheel-thrown jars and may result from clay displacement during wheel work. The reasons for discarding the wasters were largely firing failures: warping and cracking from overfiring, loosened tripod legs and handles, and glazes accidentally fired in a reducing atmosphere. Taken together, the results portray a production tradition defined by both continuity and change, in which local potters favoured iron-rich, calcium-poor clay, tempered it with river sand, fired at low temperatures, and gradually adopted wheel-throwing and lead glazing, offering archaeologists a new quantitative template for reconstructing medieval craft knowledge and its transmission.

Subject of Research: The production technology and chaîne opératoire of medieval greyware and redware ceramics from 12th to 15th century Utrecht, the Netherlands

Article Title: Advancing the chaîne opératoire analysis of medieval greyware and redware ceramics: A case study from 12th to 15th centuries CE Utrecht, the Netherlands

Article References: Borgers, B., Gait, J., Bajnok, K., Allepuz, E. T., Bajnóczi, B., Len, A., & Griffioen, A. (2026). Advancing the chaîne opératoire analysis of medieval greyware and redware ceramics: A case study from 12th to 15th centuries CE Utrecht, the Netherlands. Archaeological and Anthropological Sciences, 18(9), Article 194. https://doi.org/10.1007/s12520-026-02554-x

Image Credits: AI Generated

DOI: 10.1007/s12520-026-02554-x

Keywords: medieval pottery, Utrecht, chaîne opératoire, ceramic technology, greyware, redware, small-angle neutron scattering, lead glazes, petrography, X-ray fluorescence, pottery workshops, the Netherlands

The review, titled From graphs to qubits: a critical review of quantum graph neural networks, surveys the emerging field of Quantum Graph Neural Networks, or QGNNs, architectures that fuse the relational power of graph neural networks with the principles of quantum computation. Quantum computers manipulate qubits, which unlike classical bits can exist in superpositions of zero and one, and can become entangled so that the state of one qubit cannot be described independently of another. An n-qubit register lives in a Hilbert space spanned by all 2-to-the-n possible bit strings, an exponentially large arena that quantum algorithms can, in principle, exploit. The authors argue that this richness could offer graph learning a fundamentally different feature map, one capable of encoding complex topological relationships in ways that are hard for classical methods to reach.

But the quantum path is constrained by reality. Today’s machines are Noisy Intermediate-Scale Quantum devices, a term coined by John Preskill to describe processors with limited qubit counts, shallow circuit depths and pervasive noise. The dominant pragmatic approach on such hardware is the variational quantum circuit, in which a parameterized quantum circuit encodes data, evolves it under trainable rotations and entangling gates, and is measured repeatedly, with a classical optimizer updating the parameters in an iterative loop. The choice of data encoding matters enormously: angle encoding maps each feature to a rotation angle and is hardware-friendly but requires operations proportional to the number of features, while amplitude encoding compresses a d-dimensional vector into only log d qubits, yet preparing an arbitrary amplitude-encoded state can still cost O(d) operations. The review stresses that qubit efficiency does not automatically translate into end-to-end speedup, because state preparation, measurement shots and classical optimization all consume the budget.

To bring order to a fragmented literature, the authors propose a three-way taxonomy. Fully Quantum GNNs perform every processing stage in the quantum domain, encoding graph structure directly into Hamiltonian dynamics; they are conceptually elegant but severely limited by noise and qubit scarcity. Hybrid Quantum-GNNs embed quantum operations inside the core learning mechanism itself, implementing message passing, aggregation or graph convolution through parameterized circuits, while classical layers handle the rest. Quantum-Assisted GNNs keep the graph network entirely classical and use quantum modules only externally, for preprocessing, feature transformation or downstream classification. The distinction, the authors emphasize, is functional rather than merely architectural: the key question is not whether a quantum circuit is present, but whether it participates in the graph-learning operation or merely assists it.

The field’s founding idea came in 2019, when Verdon and colleagues introduced QGNNs inspired by the Quantum Approximate Optimization Algorithm. Their general ansatz applies a sequence of parameterized Hamiltonian evolutions whose interaction topology mirrors the problem graph, with each node of the graph associated with a quantum subsystem. From this seed, the review traces several branches: quantum recurrent GNNs that tie parameters across time steps to model temporal dependencies, quantum convolutional GNNs that enforce permutation invariance and globally shared Hamiltonian parameters, quantum time-series convolutional models that use the Schrödinger equation to capture periodic temporal dynamics, and equivariant quantum graph circuits that preserve symmetry under node permutation. One notable construction, the Equivariantly Diagonalizable Unitary circuit, can approximate any real-valued function on bounded graphs and passes the 1-Weisfeiler-Lehman test, outperforming classical message-passing networks in expressive power, at least in theory.

The applications surveyed span strikingly diverse territory. In high-energy physics, hybrid quantum-classical networks have been applied to jet tagging and particle track reconstruction at the Large Hadron Collider, where the upcoming High-Luminosity upgrade demands faster processing of sparse, high-rate collision data. One quantum jet-discrimination architecture achieves a complexity of O(N) in the number of particles, a polynomial speedup over the O(N squared) scaling of classical models, alongside more stable multiclass training, though its raw accuracy remains comparable to classical baselines. In molecular chemistry and biology, QGNNs have predicted molecular energies, HOMO-LUMO gaps and perovskite formation energies; a nine-qubit model for water molecules exploits the geometry of the problem, while an ego-graph decomposition strategy achieved competitive graph classification results using only 1.68 percent of the parameters of its classical counterparts.

In complex systems, the picture is similarly mixed. A temporal-spatial quantum graph convolutional network for traffic congestion prediction, built on a Schrödinger-based temporal model, proved robust but did not beat classical predictors. In finance, a compact QGNN with six qubits and roughly 200 parameters reached 94.5 percent accuracy on credit card fraud detection against 92.4 percent for a classical GraphSAGE baseline, a modest but real gain. Perhaps the most striking result comes from Earth science: a quantum-assisted model for predicting the Oceanic Niño Index, which tracks El Niño, improved accuracy over state-of-the-art classical forecasts while cutting training time by an order of magnitude, converging in five epochs instead of fifty. The review also highlights quantum-native tasks, such as learning Ising Hamiltonian dynamics, preparing GHZ entangled states for quantum sensing, spectral clustering and graph isomorphism testing, where the correspondence between graph structure and quantum interactions is direct and the fit is most natural.

Crucially, the authors introduce a disciplined vocabulary that the field has often lacked. They reserve quantum advantage for cases where a quantum model demonstrably outperforms the best classical counterpart under a clearly specified computational model, accounting for the full pipeline including encoding, state preparation, circuit evaluations, measurement shots and classical preprocessing. Quantum utility describes practically relevant benefits, such as improved accuracy, reduced parameter counts or better trainability, that fall short of formal advantage. Quantum-inspired improvement covers classical methods that borrow quantum concepts without using quantum hardware. Judged by this standard, most current QGNN results demonstrate task-dependent quantum utility rather than definitive quantum advantage, and the review says so plainly.

The obstacles are formidable. Noise and decoherence degrade fragile quantum states, and correlated errors such as crosstalk and non-Markovian noise complicate optimization, introducing systematic bias into objective evaluations. Barren plateaus, regions of the cost-function landscape where gradient variance decays exponentially with qubit count, can stall training entirely, and the problem worsens with noise and with global cost functions. Scalability is perhaps the deepest concern: direct node-to-qubit encodings require at least O(|V|) qubits, edge-dependent interactions may demand O(|E|) entangling gates per layer, and dense graphs can push this to O(|V| squared), before hardware routing adds SWAP gates on connectivity-limited devices. The review also notes that the vast majority of published QGNN studies rely exclusively on classical simulation of quantum circuits, which cannot reproduce real hardware noise, and that initialization strategies for quantum parameters remain underexplored despite their demonstrated impact on convergence.

The authors’ conclusion is neither dismissive nor triumphant. QGNNs, they find, are viable and sometimes competitive alternatives to classical graph networks, particularly in parameter efficiency, training behavior and problem-specific complexity reduction, and they are most naturally suited to graph-structured quantum problems rather than generic large-scale classical graph learning. They call for hardware-aware ansatz design, efficient graph-to-circuit mappings, standardized benchmarks that report qubit counts, compiled circuit depth, shot counts and optimization costs, and greater use of noise-aware simulation and real-device experiments. They also point to QAOA-inspired designs, which encode graph structure directly into the circuit, and to extensions toward hypergraphs and simplicial complexes as promising directions. Until fault-tolerant quantum hardware arrives, the honest verdict is that quantum graph neural networks offer genuine, measurable utility today, while the decisive quantum advantage that would transform graph learning at scale remains an open and rigorously framed research question.

Subject of Research: A critical review of quantum graph neural networks, their architectures, applications, and the gap between quantum utility and proven quantum advantage.

Article Title: From graphs to qubits: a critical review of quantum graph neural networks

Article References: From graphs to qubits: a critical review of quantum graph neural networks. (n.d.). https://doi.org/10.1007/s00521-026-12428-x

Image Credits: AI Generated

DOI: 10.1007/s00521-026-12428-x

Keywords: quantum computing, graph neural networks, quantum graph neural networks, variational quantum circuits, NISQ devices, barren plateaus, quantum machine learning, high-energy physics, molecular chemistry, fraud detection, El Nino prediction, quantum advantage

Group signatures, first proposed by David Chaum and Eugene van Heyst in 1991, allow a member of a group to sign messages anonymously on behalf of the collective. Anyone holding the group’s public key can verify that a valid group member signed the message, but the signer’s individual identity remains hidden within the anonymity set. To prevent abuse of this anonymity, a designated opener typically holds the power to reveal who signed a given message. The tension at the heart of the technology is obvious: anonymity is valuable, but so is accountability, and traditional designs concentrate enormous tracing power in the hands of the group manager.

The new work builds on a concept pioneered by Diaz and Lehmann at the PKC 2021 conference: group signatures with user-controlled and sequential linkability, or GS-UCSL. In such schemes, signers themselves decide which signatures can be linked, rather than depending on a central authority to perform the linking. Linkability matters in practical settings. A vehicle broadcasting sensor readings to a data lake, for example, may need to prove that a sequence of anonymous reports arrived in the original chronological order, because the ordering itself carries meaning. A fuel-level sequence of 35, 45, 30 and then 40 liters within a short window might signal tampering, whereas 45, 40, 35 and 30 would look normal. Contact tracing systems face similar demands when pseudonymous data spans multiple rotating pseudonyms. The original GS-UCSL scheme, however, suffered three fatal limitations: it was not post-quantum secure, it offered no mechanism to revoke malicious signers, and it omitted traceability entirely.

Dong and Yao’s answer rests on lattice cryptography, the mathematical foundation underlying most post-quantum proposals. Lattice problems such as Module Learning With Errors (MLWE) and Module Short Integer Solution (MSIS) are believed to resist attacks even by large-scale quantum computers running Shor’s algorithm, which would demolish schemes built on integer factorization or discrete logarithms. The authors prove their scheme’s anonymity, traceability, existential unforgeability under chosen-message attack, and non-frameability in the random oracle model, grounding each property in the hardness of these lattice assumptions. Parameter analysis with the Lattice Estimator tool suggests attack costs far beyond practical reach, with the cheapest known attacks requiring on the order of 2 to the power 128 operations or more.

One of the scheme’s most distinctive features is its approach to revocation. Existing revocable group signatures typically rely on revocation lists, whose verification cost grows with the number of revoked members, or on revocation tokens distributed through secure channels, which impose heavy communication overhead. The new design instead uses a revocation polynomial. The group manager encodes each legitimate signer’s revocation secret into a polynomial and publishes it; a signer proves membership in a zero-knowledge proof by evaluating the polynomial at their own secret value. When a malicious member must be expelled, the manager simply resamples the secret and recomputes the polynomial from the remaining values. No revocation list needs to be checked at verification time, no member keys need to be reissued, and the privacy of revoked users is preserved. Verification cost remains constant regardless of how many users have been revoked.

To curb the group manager’s tracing power, the scheme borrows key-oblivious encryption, a primitive introduced by Kohlweiss and Miers and later instantiated on lattices by Ling and colleagues. During registration, the manager rerandomizes each user’s encryption public key. Thanks to the key-oblivious property, no one without the original secret key and the randomness can tell whether a given randomized key is traceable. The manager silently tags some users as traceable and others as non-traceable, and the users themselves cannot detect which category they fall into. When signing, each user encrypts identity information under their own randomized key; only traceable users’ identities can later be recovered by the opener. This splits the group into traceable and non-traceable types without anyone’s awareness, restraining the manager from arbitrarily unmasking every signer. A cuckoo hash table, which guarantees worst-case constant-time insertion, lookup and deletion, serves as the manager’s private registry of traceable users.

Linkability in the scheme comes in three flavors, all under user control. Implicit linkability assigns each signature a pseudonym derived from a scope value and the signer’s secret key: signatures within the same scope are automatically linkable, while those across different scopes remain unlinkable unless the signer proves otherwise. Explicit linkability lets a signer voluntarily claim a set of signatures after the fact. Sequential linkability goes further, allowing a signer to produce a proof that a chain of linked signatures was generated in strict chronological order with no omissions. The mechanism uses lightweight hash chains derived from the signer’s secret key and a state counter, with unique sequential values checked against an append-only bulletin board to defeat replay, reordering and selective disclosure attacks. Recovering the signer’s secret from a pseudonym would require solving the MSIS problem, which is computationally infeasible.

Efficiency was a central design goal. The scheme integrates signature aggregation with non-interactive zero-knowledge proofs of knowledge to enable batch verification of sequentially linked signatures. In experiments implemented in SageMath on a laptop with an Intel Core i7-8650U processor, the authors report that aggregating 500 linked signatures compresses the signature size by roughly 81 percent and cuts verification time by about 83 percent. Setup, key generation, verification and opening all complete within one second; signing and revocation finish within about 3.5 seconds. Joining 500 users simultaneously offline took roughly 982 seconds in total, though a single user’s join takes only a few seconds. Communication overhead grows only mildly and linearly with group size, and verification time stays constant regardless of the number of revoked users, which the authors highlight as critical for large-scale deployments such as vehicular networks and blockchain-based data sharing.

The formal security analysis proceeds through sequences of games. Anonymity reduces to the hiding property of an underlying lattice commitment scheme by Baum and colleagues plus the indistinguishability of a verifiable encryption scheme by Lyubashevsky and Neven. Traceability, unforgeability and non-frameability share a unified proof structure: any adversary who forges a signature can be used, via the general forking lemma, to extract a solution to the MSIS problem, with the three properties distinguished by the adversary’s goals and oracle access rather than by structurally different proofs. The authors also validated the protocol with the automated verification tool Scyther, which confirmed the security of the message flows among the group manager, signer and verifier. They acknowledge that the security proofs rely on the random oracle model, consistent with all state-of-the-art lattice-based group signatures of this scope, and note that achieving such comprehensive functionality in the standard model remains an open problem. Hash functions are instantiated with the NIST-standardized, post-quantum-secure SHAKE-256.

The researchers acknowledge remaining limitations and outline future work. Currently, the group manager must recompute the revocation polynomial after every revocation operation, which costs time, and the manager must store each legitimate signer’s revocation secret, so storage demand grows with the signer base. The team plans to optimize polynomial updates and to seek smaller secret keys and lighter management overhead. Even so, the authors conclude that LCGS-UCSL achieves comprehensive functionality with competitive efficiency, marking the first post-quantum-secure group signature scheme to combine user-controlled sequential linkability, lightweight polynomial revocation, oblivious traceability classification and batch verification. For privacy-critical sequential scenarios ranging from intelligent transportation to contact tracing and time-series data authentication, the scheme offers a blueprint for staying anonymous, staying accountable, and staying secure against the quantum computers of the future.

Subject of Research: A post-quantum lattice-based group signature scheme with user-controlled and sequential linkability and efficient revocation

Article Title: An efficient lattice-based conditional privacy-preserving group signature with user-controlled and sequential linkability

Article References: Dong, S., & Yao, Y. (2026). An efficient lattice-based conditional privacy-preserving group signature with user-controlled and sequential linkability. Cybersecurity, 9(1), Article 210. https://doi.org/10.1186/s42400-026-00613-3

Image Credits: AI Generated

DOI: 10.1186/s42400-026-00613-3

Keywords: group signatures, lattice cryptography, post-quantum security, user-controlled linkability, sequential linkability, revocation, zero-knowledge proofs, key-oblivious encryption, cuckoo hashing, anonymity, traceability, batch verification

The intermediate-band solar cell concept, first proposed by Luque and Martí in 1997, has been demonstrated experimentally in expensive III-V compound semiconductor systems, but its compatibility with mainstream silicon manufacturing has remained an open question. The new work addresses this by simulating a periodic superlattice of cubic silicon quantum dots, each under tensile strain, embedded in a relaxed silicon-germanium matrix. When quantum dots are packed close enough together, their confined electron wave functions overlap and isolated energy levels broaden into minibands, which could serve as the intermediate band. Crucially, the team replaced the simplified single effective mass used in their earlier work with the full anisotropic effective mass tensor of the strain-split silicon Delta-2 valleys, with transverse masses of 0.19 and a longitudinal mass of 0.916 times the free electron mass, oriented along the growth axis.

That change turned out to matter enormously. The researchers scanned 495 different geometries, varying the dot size from 3.0 to 8.0 nanometres, the barrier thickness between dots from 1.0 to 5.0 nanometres, and the germanium fraction in the matrix from 0.20 to 0.38, solving the anisotropic BenDaniel-Duke Hamiltonian on a fine finite-difference grid with Bloch boundary conditions at 132 high-symmetry wave-vector points. Their central finding is that mass anisotropy alters the dispersion and isolation of the minibands far more strongly than it shifts the band minimum itself. The light transverse masses amplify spreading in the plane of the dots, while the heavy longitudinal mass suppresses it along the stacking direction, so bandwidths and inter-band overlaps change in ways a scalar mass simply cannot capture.

To qualify as a usable intermediate band, the lowest electron miniband had to satisfy four criteria simultaneously: it must remain fully bound below the barrier, carry a thermal margin exceeding one thermal energy unit at room temperature, span at least 5 millielectronvolts to form a genuine band rather than a flat level, and sit at least 5 millielectronvolts away from the next higher miniband to avoid parasitic hybridisation. Under these core conditions, 171 of the 495 geometries passed; adding an upper width limit of 50 millielectronvolts, designed to prevent the band from becoming too delocalised, trimmed the strict set to 170. Notably, no geometry survived at barrier thicknesses of 1.5 nanometres or below. In all 110 of those thin-barrier cases, the isolation criterion failed because strong tunnelling broadens adjacent bands until they overlap, and in 22 cases the lowest band even lost full confinement.

Perhaps the most consequential revision is compositional. The earlier scalar-mass model had pointed to a single optimum germanium fraction near 0.30. The tensor calculation instead reveals a favourable plateau at higher germanium contents, roughly x = 0.35 to 0.38, where the stronger electron barrier balances the increased in-plane coupling introduced by the lighter transverse masses. The comparison between models is striking: only 66 structures, about a third of the scalar core, survive in both models. One hundred five geometries enter the tensor core because anisotropic broadening lifts their bandwidth above the minimum threshold, while 138 scalar survivors drop out, mostly because their inter-miniband separation falls below 5 millielectronvolts. Median changes across the full scan included a 35.78 millielectronvolt reduction in inter-band separation, underscoring why density-of-states-equivalent masses are inadequate for directional kinetic modelling.

The team also resolved a subtle band-alignment question that has puzzled researchers in this material system: how adding germanium to the matrix can still create a barrier for electrons. The answer lies in strain. Tensile strain in the silicon dot splits the conduction valleys, and for the Si-rich compositions studied, the relevant strained alignment places the matrix conduction edge above the dot edge, with the offset given by 0.64 times the germanium fraction. The valence-band alignment, meanwhile, is type II: the highest heavy-hole state sits in the silicon-germanium matrix while the electron miniband resides in the strained silicon dot. This spatial separation makes the valence-band-to-intermediate-band transition spatially indirect, a fact with real implications for optical strength that the authors flag as requiring future phonon-assisted treatment.

To connect these electronic structures to actual sunlight, the researchers integrated the ASTM G173-03(2020) AM1.5G reference solar spectrum, using its full tabulated 2002 wavelengths from 280 to 4000 nanometres at an integrated power of 1000.371 watts per square metre without renormalisation. Because a miniband has finite width, the position chosen within it, bottom, centre or top, changes the two sub-gap photon thresholds. The spectrum was partitioned into three disjoint energy windows corresponding to the two-step and direct transitions, and the matched two-photon flux was taken as the smaller of the two sub-gap channels. In every one of the 170 strict geometries and at every placement, the valence-band-to-intermediate-band channel was the limiting one, a structural consequence of the fact that the width of that spectral window equals the second transition energy, which is bounded by the conduction-band offset and never exceeds 0.2432 electronvolts across the studied range.

The placement sensitivity analysis delivered another practical lesson. Moving the assumed intermediate state from the bottom to the top of the miniband changed the matched two-step photon flux by a median of 5.91 percent and by more than 51 percent in the extreme case, with 31 of 170 geometries shifting by over 20 percent. The ideal total spectral current bound, by contrast, moved by less than 3.9 percent because the direct above-gap contribution dominates that aggregate. The authors stress that their photon counts are upper bounds assuming perfect absorption and collection: phonon-assisted optical matrix elements, carrier occupations, recombination, escape mechanisms, and device electrostatics are all excluded from the present calculation, so the numbers should guide material screening rather than efficiency claims.

The study also probes robustness. Because a periodic unit cell cannot represent the random disorder of real self-assembled dot arrays, the team examined what happens when each retained design is nudged by a single 0.5-nanometre grid step in dot size or barrier thickness. Only about 41 percent of core geometries kept their classification across all nearest-neighbour changes, with median shifts as large as 6.10 millielectronvolts in isolation energy. The practical message for experimentalists is clear: choose candidate structures from the interior of the high-germanium design region, never from its staircase-shaped boundary, and subject the survivors to fuller models incorporating phonon-assisted absorption, disorder, recombination kinetics and device fields before any efficiency prediction can be trusted. In doing so, the work delivers exactly what screening studies are meant to provide, a defensible shortlist of silicon-compatible nanostructures worth the investment of more expensive simulation and, eventually, growth.

Subject of Research: Anisotropic electron minibands and solar spectral usability in tensile-strained Si/SiGe quantum-dot superlattices for intermediate-band solar cells

Article Title: Anisotropic electron minibands and AM1.5G spectral usability windows in tensile-strained Si/relaxed Si 1-x Ge x quantum-dot superlattices

Article References: Lassina, D., Soumaïla, D., Michel, K. O., Alain, D., Raguilignaba, S., & François, Z. (2026). Anisotropic electron minibands and AM1.5G spectral usability windows in tensile-strained Si/relaxed Si1-xGex quantum-dot superlattices. Results in Physics, Article 108754. https://doi.org/10.1016/j.rinp.2026.108754

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108754

Keywords: intermediate-band solar cell, silicon quantum dots, SiGe superlattice, miniband dispersion, effective mass anisotropy, AM1.5G spectrum, tensile strain, band alignment, photovoltaics, Si1-xGex, Anisotropic, electron