Papers

Scaling fault-tolerant superconducting quantum computers will likely require distributed architectures built from multiple manufacturable quantum processing units. A central question is whether noisy and slow inter-chip operations impose substantial spacetime overhead or orchestration burdens compared with monolithic architectures. To answer this question, we present a hardware-grounded architectural co-design together with a comprehensive resource-estimation protocol for surface-code-based modular processors. The design confines inter-chip latency and noise to module boundaries, preventing slow, noisy links from inducing prohibitive spacetime overhead or becoming a global orchestration bottleneck. The resource-estimation protocol integrates hardware constraints and circuit-level error-correction performance into utility-scale algorithmic cost estimates, enabling a controlled assessment of different architectures. Using RSA-2048 factorization as a demanding benchmark, we estimate resources under experimentally anchored parameters and realistic superconducting hardware constraints. Compared with a large, ideal monolithic baseline, the resulting distributed architecture requires only modest additional resource overhead in both qubit count and execution time. More importantly, the overhead is nearly scale-invariant across a broad module-capacity window, decoupling chip size from global performance. This decoupling turns module capacity from a finely tuned architectural parameter into a flexible engineering degree of freedom, allowing chip sizes to be set by manufacturability and control-packaging constraints rather than architectural fine-tuning. These results establish a viable route to distributed fault-tolerant superconducting quantum computation that scales without prohibitive resource growth or heavy orchestration burden.

The Vlasov–Poisson system provides the fundamental kinetic description of plasma and plays a central role in understanding collective phenomena such as Landau damping and wave–particle interactions. Efficient numerical simulation of these dynamics remains challenging because of the high dimensionality of phase space. In this work, we studied magnetized and non-magnetized plasma using a quantum simulation framework for the linearized Vlasov-Poisson equation by reformulating the discretized system as a Hermitian Hamiltonian suitable for gate-based quantum computation. The time evolution is implemented using first, second and fourth-order Trotter–Suzuki product formulas and the recently proposed Time-Resolved Interaction Framework (THRIFT). The performance of the different simulation methods is systematically evaluated through electric field evolution, state fidelity, convergence behavior, energy conservation, entanglement entropy and quantum resource requirements, including circuit depth and two-qubit gate complexity, for both magnetized and non-magnetized plasma models. To further reduce finite time step errors without increasing circuit depth, Richardson extrapolation is incorporated as a error-mitigation technique. The results provide a comprehensive comparison of Trotter and THRIFT approaches and establish practical guidelines for accurate and resource-efficient quantum simulation of plasma dynamics on gate-based quantum computers.

We study a hybrid proposal–correction architecture for reduced-order PDE forecasting in which a pure-state quantum reservoir computer (QRC) predicts latent coefficient dynamics and a PINN-based physics-informed corrector (PIC) refines local rollout windows. The method is evaluated on Burgers and Kuramoto–Sivashinsky (KS), with KS as the primary chaotic benchmark. On KS, QRC+PIC consistently improves over QRC alone in RMSE, NRMSE, and PDE residual, while Burgers highlights a regime in which simple baselines remain strong. These results suggest that QRC proposals with local physics-informed correction are a viable benchmark-dependent reduced-order forecasting strategy.

This study presents a detailed proposal for an optomechanical system consisting of two mechanical oscillators coupled to a common cavity, aimed at generating pure and entangled two-mode squeezed mechanical steady states. We found that the violation of Bell's measurement may not occur where the entanglement is maximum; rather, nonlocality can be observed for lower entangled states. A central result is that optomechanical coupling imperfections can enhance mechanical entanglement while simultaneously suppressing Bell nonlocality by reducing the purity of the mechanical state. To mitigate this trade-off, we introduce phase-dependent phonon hopping between the mechanical oscillators and show that Bell nonlocality can be selectively enhanced in specific dark-mode configurations, even when the overall entanglement is reduced. We trace this contrasting behaviorto changes in state purity associated with the imbalance of the Bogoliubov-mode occupations. Compatible with existing microwave cavity optomechanical platforms, the proposed architecture provides an experimentally accessible route for controlling nonlocal quantum correlations in multimode mechanical systems. Our proposed scheme serves as an attractive platform for the deployment of continuous-variable teleportation and high-fidelity quantum communication.

We introduce a novel physics-guided linear mapper (PGLM) for quantum error mitigation that uses seven distinct interpretable features derived from circuit complexity and device calibration data. The goal is to provide a data-efficient, interpretable, and low-latency alternative to the black-box machine learning for quantum error mitigation in noisy-intermediate scale quantum devices. Evaluated on 52 simulated benchmark circuits (1–4 qubits), PGLM demonstrates strong performance in noise-accumulation regimes: 50.1% RMSE reduction on 3-qubit circuits and 32.3% on 4-qubit circuits, while single-qubit circuits show degraded performance. A circuit-size-aware deployment policy achieves 32.6% aggregate improvement. Sub-millisecond inference enables integration into variational algorithms, and analysis of learned coefficients reveals that circuit depth and CNOT count dominate error prediction, consistent with decoherence mechanisms. Results are simulator-based with idealized noise models; hardware validation remains essential future work.

We study parameter recovery in the Caldeira–Leggett (quantum Brownian) oscillator from partial moment traces. Our model is a moment-level PINN that predicts the five first/second moments and enforces the linear CL/HPZ ODEs by automatic differentiation. Physical structure is imposed through a PSD (Cholesky) covariance head, high-temperature CL assumptions with $D_{xp}\approx0$, and fluctuation–dissipation ties between $D_{pp}$ and $\gamma$. On synthetic CL data with channels ${\mu_x,\sigma_{xx},\sigma_{xp}}$, the constrained variant recovers $(\omega,\gamma)$ accurately, stabilizes $D_{pp}$, and achieves low rollout error compared to finite differences and Kalman–EM (expectation–maximization) with exact Van Loan discretization. Fisher-style checks confirm that diffusion needs at least one variance observable, and sparse $\sigma_{pp}$ “anchors” restore conditioning. We also show that the same PINN can learn time-varying HPZ coefficients.

The notation $\psi(x,t)$ is among the first pieces of quantum mechanics that students learn. It is also among the easiest to over-interpret. Because $x$ and $t$ occur as arguments of the same function, students may ask whether they have the same mathematical status. They may also ask whether $\psi(t)$ should require a generalized bra $\bra{t}$ in the same way that $\psi(x)=\braket{x}{\psi}$ is often written. A related question is whether the absence of a universal time operator follows simply from Pauli's argument. These questions mix several structures that are usually introduced in different parts of the curriculum. We present a unified pedagogical treatment built around two maps hidden in $\psi(x,t)$. Time evolution selects a state along a trajectory in Hilbert space. A spectral representation then maps that state to amplitudes labelled by outcomes of a chosen observable. We formulate the position representation without generalized eigenkets. We recover Dirac's $\ket{x}$ notation as a controlled continuum shorthand and use a finite-grid limit to show where delta normalization enters. We distinguish background coordinates, translation parameters, observables, spectral labels, and physical records. We also clarify the Stone-theorem analogy, compare prescribed-time position measurements with arrival-time measurements, state what the strong form of Pauli's argument excludes, and exhibit an exactly solvable boundary case in which a canonical self-adjoint time observable exists. Spin, circuit-QED, and optical-clock examples provide experimentally grounded checks. The aim is not a new interpretation of time. It is a reusable teaching framework for separating mathematical role from notation.

Accurate identification of unknown quantum systems is essential for quantum computing, sensing, and control because the Hamiltonian governs quantum state evolution. This work proposes a QNN based framework for black box Hamiltonian learning and quantum system emulation using full density matrix trajectory learning. Unlike approaches based only on final states or selected observables, the method exploits the complete temporal evolution of the density matrix under Lindblad dynamics. A synthetic dataset of physically admissible Hamiltonians and dissipation parameters is generated to emulate experimental measurements. The QNN learns a nonlinear mapping from control inputs to a 32-dimensional Hamiltonian coefficient vector, enabling reconstruction and differentiable emulation of the unknown system. Chirped excitation and randomized initial quantum states are incorporated to improve robustness and provide richer dynamical information. Performance is evaluated using trajectory density loss, quantum-state fidelity, and trace distance. Randomized initialization improves state-level reconstruction, increasing fidelity to 0.929 for the single qubit benchmark and 0.787 for the unknown system, while reducing trace distance to 0.124 and 0.316, respectively. In contrast, chirped excitation primarily improves optimization by accelerating convergence and reducing trajectory density loss. Finally, the learned Hamiltonian is mapped onto a physical two qubit bus resonator architecture in the dispersive regime, yielding key circuit parameters including transmon capacitances, Josephson inductances, qubit separation, and bus-resonator length. The framework therefore establishes a data-driven pathway from black box quantum system identification to physical quantum emulation, with potential applications in quantum digital twin modeling.

Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal $\Theta(\sqrt{N})$ oracle and non-oracle gate counts. For spatial search on $d$-dimension grids, it recovers the $O(\sqrt{N})$ time for $d\geq3$ and the $O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)$ bound of Aaronson and Ambainis for $d=2$. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.

At a deeper level the principle of least action is interpreted as the law of least entropy increase consistent with Prigogine's principle of minimum entropy production, and the implications of quasistatic information quantization rule (Proc. R. Soc. A 480: 20240024), are explored for the conjectured second law of quantum complexity. It is thus shown that the conjectured second law of complexity is derivable from the information quantization rule such that long after heat-death the quantum state complexity evolves as $C(t)=C_{\rm max}\exp(-1/t)$, increasing with time to a saturation value $C_{\rm max}$ that is exponential in the equilibrium entropy $S_{\rm max}$. For the out-of-equilibrium circumstances, however, the quantum complexity can decrease with the time, asymptotically tending to a minimum determined by the distance from equilibrium.

We study estimation of the trace polynomial $\operatorname{tr} p(P\rho P)$ from global classical shadows, where $\rho$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$, the quadratic degenerate term has order $s^2/N$ under batching and $s^2/N^2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s^2N\log^2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.

Topological photonic lattices are commonly designed using short-range Hamiltonians, yet realistic nanophotonic structures are governed by geometry- and wavelength-dependent long-range electromagnetic interactions. Here we introduce a geometry-calibrated quantum equilibrium-propagation framework for inference and inverse design in finite nonlocal plasmonic Su-Schrieffer-Heeger lattices. A two-qubit equilibrium sensor is trained in effective-coupling space to distinguish boundary-localized from trivial finite-lattice responses. Because labels inherited from the nearest-neighbor SSH model become unreliable in the presence of nonlocal hopping, each sample is independently relabeled using nonlocal winding numbers and a finite-gap criterion. On this physics-verified evaluation set, the sensor achieves 99.8% sensitivity and 98.1% specificity. A physics-gated robustness score then ranks verified configurations by boundary response, response contrast, gap stability and compatibility with the selected nonlocality regime. Full-wave extinction spectra of isolated and paired gold nanoparticles establish a geometry-to-coupling calibration linking particle size and separation to wavelength-resolved pair couplings. Projecting this calibration onto the verified coupling landscape identifies a finite plasmonic geometry supporting spectrally distinct corner- and edge-dominated responses at 546 and 642 nm, with sector-to-bulk intensity contrasts of $2.4 \times 10^{4}$ and $1.5 \times 10^{3}$, respectively. The framework links realistic electromagnetic geometry to nonlocal topological design, moving beyond nearest-neighbor design rules with physics-verified, geometry-resolved inference.

Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.

Fock-space variational Monte Carlo (FS-VMC) evaluates variational quantities by sampling discrete many-body configurations. In molecular applications, concentrated Born distributions can hinder Markov-chain mixing, while Monte Carlo estimators can also exhibit large variance. We introduce importance-reweighted FS-VMC (IR-FS-VMC), which combines Metropolis–Hastings sampling, an evaluation kernel, and self-normalized importance sampling to estimate the Born-distribution energy, gradient, and stochastic reconfiguration (SR) matrix. Controlled comparisons on equilibrium H2O and Fe2S2 show that the acceptance rate and local-energy variance can respond differently to the sampling and estimation procedure. A single production protocol yields accurate variational energies for H2O dissociation, 36-site hydrogen lattices, and Fe2S2 and Fe4S4 active spaces without system-specific sampling parameters.

The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann–Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous $\pi$ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family—the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann–Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.

Quantum states with conserved total spins, or spin eigenfunctions, are important for studying quantum chemistry and quantum manybody physics problems. A typical class of spin eigenfunctions are Dicke states, which attain maximal spins. While we already have many efficient quantum algorithms to prepare Dicke states, it is not yet clear if we could do so for arbitrary spin eigenfunctions deterministically. Generalizing Bärtschi and Eidenbenz's elegant algorithms for Dicke state preparation, we successfully prepare arbitrary spin eigenfunctions characterized by branching paths and binary spin trees. As a byproduct, we also develop the corresponding classical algorithms to reconstruct all these spin states.

Multi-photon interference can be observed using independently generated photons as input. In the most simple case, these photons meet up at a beam splitter, resulting in quantum interference between transmission and reflection of the photons. Here, we show that non-local multi-photon interference can be implemented by using a mode swap operation to generate entanglement between the photons detected in the outputs of two spatially separated interferometers. The spatial separation of the output photons makes this implementation of multi-photon interference particularly suitable for quantum protocols that distribute quantum information to different parties.

Fault-tolerant architectures implement non-Clifford T gates through magic-state distillation, so the T-count of a synthesized circuit dominates its physical cost. The Solovay-Kitaev algorithm approximates any single-qubit unitary from a finite gate set with a sequence length that grows only polylogarithmically in the inverse target error, but it optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see. We report a measurement of what diagrammatic post-processing recovers from that redundancy. Twelve hundred random single-qubit targets, spanning the three Pauli rotation families and the general gate U(theta, phi, lambda), are synthesized over Clifford+T at three recursion depths, translated into graph-like ZX-diagrams, simplified by automated rewriting, and extracted back to circuits. Post-processing removes 26.6-30.1% of the total gate count and 18.5-22.2% of the T-count. The absolute saving grows with recursion depth, from about 60 to about 1600 gates, while the fractional saving does not: it rises slightly from the shallowest setting and is then flat across a twenty-five-fold change in circuit length, and by the deepest setting the four target families are no longer distinguishable from one another. Because the rewrite rules preserve the implemented linear map, the approximation error is unchanged. The compile-time cost of the rewriting layer, by contrast, grows sharply with depth and comes to dominate the synthesis itself.

Gate synthesis and circuit optimization are usually studied separately, and evidence on their interaction is contradictory: ZX-calculus rewriting removes a stable fraction of Solovay-Kitaev circuits, yet almost nothing from number-theoretically synthesized circuits. We show both behaviours follow from a single bound. For any optimizer that preserves the implemented element exactly–including all sound ZX rewriting with extraction–the achievable T-count is bounded below by the denominator exponent of the synthesized ring element. This exactness barrier is computable per instance and separates exact post-processing from approximation-aware resynthesis by a certified factor reaching 101x at recursion depth five. The two behaviours are then the barrier operating at different distances from the floor. For Solovay-Kitaev circuits we prove that the local ZX simplification layer (spider fusion and identity removal) computes exactly the free-product normal form of Z_2 * Z_8, giving exact per-instance compression and, under a calibrated ergodicity hypothesis, a depth-independent limit law confirmed on two independently constructed nets. For number-theoretically synthesized circuits the floor is already saturated: on single-qubit words automated ZX simplification attains it exactly, via a closed-form formula for minimal T-count in terms of phase linkage through the Z-axis normalizer. At two qubits and beyond the same valuation yields unconditional rigidity certificates, which on the quantum-Shannon-decomposition plus gridsynth pipeline certify 99.4-99.9% of the synthesized T-count as incompressible, with rigidity strengthening as accuracy tightens. This explains, and predicts the size of, the near-null optimization recently reported for that pipeline.

We study spectral density estimation for the normalized adjacency matrix of an unweighted graph under local access model. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ local queries to the graph. In this paper, we prove that every constant-success estimator with Wasserstein–$1$ error at most $\eps$ requires $2^{\Omega(1/\eps)}$ queries, showing that the Cohen-Steiner algorithm is optimal up to constant in the exponent. This resolves the open problem left by previous researches Jin et al. [COLT 2023] and Peng et al. [COLT 2026]. We then turn to quantum local access model. We give an $\widetilde O(\eps^{-3})$-query algorithm estimating the spectral density with Wasserstein-1 error at most $\eps$. Finally, we prove a $\widetilde\Omega(\eps^{-4/3})$ quantum lower bound when the graph is sufficiently large. As a result, quantum local access model changes the dependence on $\eps$ from exponential to polynomial.

Everettian quantum mechanics is often said to lack coherent probability because every possible measurement result occurs. If all of them exist globally, what is probability supposed to be a probability of? This paper defends an axiomatic answer: normalized Born measure is postulated as objective probability over the records borne by an observer's future continuations. Collapse interpretations likewise postulate the Born probabilities, but instead assign them to possible global outcomes. Neither theory derives the probabilities from its ontology. The contribution is an explicit record-level probability model that separates probabilities for future local records from certainty about their global existence. I use this model to answer objections to the axiomatic approach and argue that neither global selection nor a further theory of personal persistence is required. At the level of recorded results, the Everettian law supports the same predictions and empirical inferences as the collapse law, despite the theories' different ontologies.

Nonlinear waveguide QED serves as a platform for studying correlated photon dynamics, where its interplay with topological bound states can give rise to unconventional phenomena. We investigate a triangular ladder waveguide in which a pair of emitters forms a doublon bound state through strong nearest-neighbor interactions. By mapping the doublon dynamics onto an effective SSH chain, we show that the system is equivalent to an effective emitter coupled to a topological doublon bath. Using the vacancy-like dressed state approach, we demonstrate the existence of a chiral bound state with a node at the coupling site and an exponentially localized wave function inherited from the SSH edge states. Extending this picture to two emitter pairs, we derive an effective four-body interaction whose strength is determined by the relative chirality and the separation parity of the bound states. When the two bound states face each other, coherent Rabi oscillations emerge; when they are arranged back-to-back or have an even separation, the interaction vanishes. Our results provide a route toward engineering tunable many-body interactions between correlated photon pairs and realizing nonlinear quantum networks based on flying doublons.

In this paper, we investigate the characterization of multipartite entanglement based on the unified $(q,s)$-entropy framework. For bipartite quantum states, we propose an entanglement measure $E_{q,s}^{A|B}(\rho)$ based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ for quantifying $k$-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters $q$ and $s$ are in certain ranges, $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures $A^k_{q,s}(\rho)$ and $G^k_{q,s}(\rho)$ with respect to the parameters $q$ and $s$ for fixed quantum states. In addition, for fixed $q$ and $s$, we also give their order relations for any two pure states.

The ability to tailor photon-number statistics is central to the generation and control of nonclassical states of light. Here, we investigate how Fock-state filtering of squeezed coherent states can be used to engineer their photon statistics. We consider the selective removal of the $m=0$ and $m=1$ Fock components and show that this process can enhance the sub-Poissonian character of the field, depending on the coherent amplitude. We then analyze the evolution of these engineered states in a thermal environment. Although all states eventually relax toward thermal statistics, the signatures of the filtering process may persist over a significant part of the evolution, while the purity of the filtered states is rapidly degraded by incoherent population redistribution. Our results illustrate both the potential and the limitations of photon-number engineering in realistic, dissipative settings.

Cooperative emission from three-level ensembles is conventionally diagnosed through the radiated intensity alone, which cannot say which channels carry the photons at each instant. We promote every transition of the cooperative-decay graph into a time-dependent edge flux and show that node-wise conservation of this flux is an exact restatement of the Pauli master equation. Suppose the graph carries a depth function that decreases by one along every edge. Then the total cooperative rate is the descent speed of the mean graph depth, and its energy-weighted form gives the radiated intensity as $I(t)=-\dd\langle E\rangle/\dd t$ for arbitrary level spacings. The same ordering triangularizes the generator. This delivers the relaxation spectrum from the state exit rates, together with a generalized Bateman closed form for every population on an arbitrary finite directed acyclic graph, in which all exit-rate degeneracies are absorbed into a polynomial-times-exponential recursion.

The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables – the change of an expectation value <A(t)> – have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of <A> in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = |<A(T)> - <A(0)>|; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T (<H> - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope – tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states – show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.

Uncertainty relations constrain the fluctuations of incompatible observables, but most familiar bounds depend on the quantum state. State-independent uncertainty relations instead ask how much fluctuation remains unavoidable for every quantum state. For observables generated by a continuous symmetry, sharp state-independent bounds are known when the symmetry representation is irreducible. Multipartite collective systems, however, generally appear as reducible tensor-product representations of a symmetry algebra, which raises the question of how to determine their total uncertainty. We resolve this problem for multipartite quantum systems with a compact semisimple symmetry algebra $\mathfrak g$. Exploiting the symmetry structure, we formulate a general framework for state-independent uncertainty based on representation theory. The total variance admits an exact decomposition into intrinsic fluctuations within irreducible sectors and a nonnegative dispersion between sectors. This yields the sharp state-independent bound for the total variance $\Delta_\rho^2(\mathfrak g)$ on the multipartite Hilbert space $\mathcal H$ \[ \min_{\rho}\Delta_\rho^2(\mathfrak g) = \min_{\lambda\in\Lambda(\mathcal H)} 2\langle\lambda,\delta\rangle, \] where $\rho$ is any density operator on $\mathcal H$, $\Lambda(\mathcal H)$ is the set of highest weights $\lambda$ labeling those sectors, and $\delta$ is the Weyl vector. This demonstrates that the ultimate uncertainty is completely controlled by its intrinsic symmetry structure. As an application of the general theory, we confirm our previous conjecture [JPA 59, 155302 (2026)] that the total uncertainty floor of collective spin-$1/2$ systems depends only on the parity of the particle number. We further illustrate the framework for multipartite spin-$1$ systems, demonstrating that the same symmetry-sector mechanism persists beyond spin-$1/2$.

In particle physics the virtual N$^3$LO contribution to gluon-fusion Higgs production in the Standard Model requires four-loop multi-scale QCD amplitudes that in dimensional regularization are dominated by singular integral reduction and difficult master-integral evaluations. In this note we show that a nonlocal UV completion makes each fixed-order four-loop Feynman diagram fully convergent in four dimensions while keeping the analytic structure needed for Lehmann-Symanzik-Zimmermann (LSZ) reduction formula. For gluon-fusion $gg\!\to\!H$ we derive a closed form Schwinger-parameter representation where all loop-momentum integrals are performed analytically and thus leaving finite parameter integrals viable for direct numerical evaluation without needing the usual UV subtractions. This method heps to isolate the collider-hard contribution as a convergent four-loop form factor. The strong coupling then is defined by a finite normalization condition in the entire-function regularization scheme and it is related to the $\overline{\mathrm{MS}}$ coupling by finite matching where after this matching, the regulator-dependent corrections decouple in the local limit $E_M\!\to\!\infty$.

Recently, Wooltorton et al. [Phys. Rev. Lett. 132, 210802 (2024)] and Farkas [Phys. Rev. Lett. 132, 210803 (2024)] have exhibited the mismatch between Bell inequality violations and their cryptographic application in device-independent quantum key distribution, by showing that arbitrarily close to the set of local behaviours there exist quantum correlations guaranteeing a constant rate of secret key. While these results require correlations attaining the maximum quantum value of a suitable Bell observable (aka the Tsirelson bound) and rely on a kind of ideal self-testing of a maximally entangled state and associated Bell measurement, here we show that the effect is robust: for every one of the Bell inequalities considered by Wooltorton et al., a constant rate of secret key ensues if the observed Bell violation is sufficiently close to the respective Tsirelson bound. For these and also the Bell inequalities of Farkas, we furthermore present numerical results based on semidefinite relaxations of the minimum min-entropy consistent with a certain Bell violation, which demonstrate that small but nonzero key rates can be guaranteed (in principle) by practical and efficient means.

In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.