Papers

Hybrid classical quantum learning is often bottlenecked by communication overhead and approximation error from generic variational ansatzes. In this study, we introduce Neural Native Quantum Arithmetic (NNQA), which compiles classically learned nonlinear representations into precise quantum arithmetic composed of native unitary blocks. Theoretically, we prove that the universal approximation of quantum polynomial arithmetic can be realized by transforming a classical neural network into a quantum circuit, with the resulting error arising solely from measurement shot noise, thereby extending classical operator-level estimation guarantees into the quantum regime. Empirical validation on IBM Quantum Heron3 and IonQ Forte processors shows performance limited primarily by device noise without variational fine tuning: we achieve over 99.5% accuracy for polynomials up to degree 35 and demonstrate scalability on IonQ hardware up to 36 qubits and circuit depths of 70, reaching a negligible RMSE of 0.005. Overall, NNQA establishes a new paradigm of synthesizing quantum arithmetic for native quantum computation.

Quantum Key Distribution (QKD) provides information-theoretic security by exploiting the principles of quantum mechanics. Among QKD protocols, the BB84 scheme remains the most widely adopted for both theoretical research and practical implementation. A critical parameter determining the reliability and security of BB84 is the Quantum Bit Error Rate (QBER), which quantifies errors in the sifted key arising from channel noise or potential eavesdropping. This paper presents a systematic review and analysis of QBER within the BB84 protocol, examining its calculation, statistical estimation methods, and role in detecting eavesdropping activity. Simulation results, corroborated by reported experimental observations, reveal a near-linear relationship between eavesdropping intensity and QBER, with values approaching 25% under full intercept-resend attacks. Four confidence interval estimation methods, Wald, Wilson, Clopper-Pearson, and Hoeffding's inequality, are compared for robust QBER analysis in finite-key scenarios. Protocol enhancements, including decoy-state methods, hybrid cryptographic models, and quantum-resistant authentication, are discussed as mechanisms to mitigate errors and strengthen resilience across fiber, free-space, underwater, and satellite QKD systems. Open challenges in distinguishing noise-induced errors from malicious eavesdropping, and the role of adaptive error correction and machine-learning-assisted QBER estimation in future quantum networks, are identified as key directions for further research.

Foundation models have recently improved electrocardiogram (ECG) representation learning, but their deployment can be limited by computational cost and latency constraints. In this work, we fine-tune ECGFounder as a high-capacity teacher for binary ECG classification on PTB-XL and the MIT-BIH Arrhythmia Database, and investigate whether knowledge distillation can transfer its predictive behavior to compact students. We evaluate two classical 1D students (ResNet-1D and a lightweight CNN-1D) and a quantum-ready pipeline that combines a convolutional autoencoder, which compresses 256-sample ECG windows into a low-dimensional latent representation, with a 6-qubit variational quantum circuit implemented in Qiskit and executed in a simulated backend. Across both datasets, the teacher provides the strongest overall performance, while distillation yields competitive students under a considerable reduction in trainable parameters. We further analyze the sensitivity of student performance to distillation settings, highlighting consistent accuracy–efficiency trade-offs when compressing a foundation ECG model into classical and quantum-ready learners under a unified evaluation protocol.

Qubit controllers are essential for scaling superconducting quantum processors, but implementing them at the 10 mK stage of a dilution refrigerator remains challenging due to stringent cooling constraints. Here we report an ultralow-power qubit controller using adiabatic quantum-flux-parametron (AQFP) logic, termed an AQFP-multiplexed qubit controller with virtual Z gates (AQFP QC-VZ). The AQFP QC-VZ generates multi-tone microwave pulses for qubit control with an ultralow power dissipation of 111 pW per qubit. By combining microwave and time-division multiplexing, the AQFP QC-VZ enables parallel application of X and virtual Z gates to multiple qubits using only a few control lines from room temperature. We demonstrate coherent single-qubit gates at the 10 mK stage using an AQFP mixer, a core component of the AQFP QC-VZ, without observable degradation in coherence.

In the framework of linearized quantum gravity, we investigate the quantum gravitational interaction induced by the gravitodiamagnetic coupling of two massive objects to vacuum fluctuations of the gravitational field. Starting from the Lagrangian of a particle in a gravitational field and employing the formalism of Weyl gravitoelectromagnetism, we derive the interaction Hamiltonian associated with gravitodiamagnetic coupling. Unlike the linear couplings that arise in gravitoelectric and gravitomagnetic interactions, the gravitodiamagnetic coupling depends quadratically on the gravitomagnetic field. Based on this Hamiltonian, we show that, for a spherically symmetric gravitational hydrogen-like system in its ground state, the induced quadrupole moment has the opposite sign to the applied gravitomagnetic field, which is the defining signature of gravitodiamagnetism. Using leading-order perturbation theory, we further obtain an explicit expression for the resulting interaction potential, which is attractive and scales as $r^{-11}$ at all separations, where $r$ denotes the distance between the two objects.

We investigate nonclassical photon-bundle correlations in the quantum Rabi model and its extended cases, using the quantum dressed master equation. By tuning the light–matter coupling strength at finite temperature, the quantum Rabi model exhibits controllable nonclassical transitions between two-photon bundle bunching and antibunching, allowing for the two-photon bundle emission and statistics. We further introduce anisotropic coupling and nonlinear Stark interactions, which enrich the photon statistical behaviors and provide additional tunability of photon-bundle correlations. Extreme correlation behaviors are found to be closely linked to excited-state quantum phase transitions, suggesting a potential pathway for predicting and exploiting excited-state phenomena. These effects can be controlled solely by tuning intrinsic system parameters, without the need for an external modulating field. The quantum Rabi model family thus provides a flexible and experimentally feasible platform for high-purity photon bundle generation and controllable multi-photon sources.

Critical phenomena of quantum systems offer a promising strategy to improve measurement precision. So far, many criticality-enhanced quantum metrological schemes have been proposed by using the adiabatically evolved photonic states of composite systems involving a qubit and a field interacting with each other. These schemes focus on the measurement of the system's inherent frequencies. We here propose a criticality-enhanced quantum sensing protocol, aiming to estimate the amplitude of an external signal field with the interacting qubit-photon system. The signal field is coupled to the photonic mode, so that the composite system has a unique dark state, where the photonic mode follows a squeezed vacuum state. The information about the signal field amplitude is encoded in photon number or one quadrature of the quantized photonic mode, which exhibits a divergent behavior near the critical point. The measurement precision can approach the Heisenberg limit with respect to the time to encode the signal and the photon number of the field mode.

We present a new class of quantum neural networks (QNNs) whose states are solutions of $p$-adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of $p$-adic cellular neural networks (CNNs). The CNNs are continuous limits of discrete hierarchical neural networks (NNs). The CNNs are bio-inspired by the Wilson-Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new $p$-adic Schrödinger equations, which allows the construction of new QNNs on simple graphs. We also conduct detailed numerical simulations, offering a clear insight into the functioning of the new QNNs. At a mathematical level, we show the existence of local solutions for the new $p$ -adic Schrödinger equations.

Neural quantum states are powerful variational wavefunctions, but it remains unclear which many-body states can be represented efficiently by modern additive architectures. We introduce Walsh complexity, a basis-dependent measure of how broadly a wavefunction is spread over parity patterns. States with an almost uniform Walsh spectrum require exponentially large Walsh complexity from any good approximant. We show that shallow additive feed-forward networks cannot generate such complexity in the tame regime, e.g. polynomial activations with subexponential parameter scaling. As a concrete example, we construct a simple dimerized state prepared by a single layer of disjoint controlled-$Z$ gates. Although it has only short-range entanglement and a simple tensor-network description, its Walsh complexity is maximal. Full-cube fits across system size and depth are consistent with the complexity bound: for polynomial activations, successful fitting appears only once depth reaches a logarithmic scale in $N$, whereas activation saturation in $\tanh$ produces a sharp threshold-like jump already at depth $3$. Walsh complexity therefore provides an expressibility axis complementary to entanglement and clarifies when depth becomes an essential resource for additive neural quantum states.

Schrodinger path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.

The Tensor Network Quantum Virtual Machine (TNQVM) is a high-performance classical circuit simulation backend for the eXtreme-scale ACCelerator (XACC) framework that leverages the Intelligent Tensor (ITensor) library for tensor network–based quantum circuit simulation. However, TNQVM's original C++ ITensor backend is tied to an older integrated release, limiting access to newer tensor network algorithms, diagnostics, and performance improvements available in the actively developed Julia-based ITensors ecosystem. We introduce JuliaITensorTNQVM, an interoperability layer that bridges TNQVM's C++ visitor infrastructure and the Julia-ITensors runtime through a C-compatible application binary interface. This design preserves the existing XACC/TNQVM programming model while enabling access to modern tensor network capabilities, including entanglement entropy diagnostics exposed directly to XACC. We evaluate the implementation through two studies: a Page-curve verification protocol using Haar-random states, and QAOA MaxCut simulations on 3-regular graphs. Within these tested regimes, results are consistent with expected entanglement behavior and established scaling trends, supporting JuliaITensorTNQVM as a practical modernization path for tensor network simulation in TNQVM.

Thin-film lithium niobate (TFLN) has emerged as a leading platform for large-scale programmable photonic circuits for quantum and classical applications. As circuits scale in complexity, low-loss routing of broadband pump and signal fields becomes essential. Here, we present closed-form analytical models and experimentally demonstrate compact, fast-quasi-adiabatic driving-optimized wavelength combiners and filters operating at the fundamental harmonic (FH, 1550 nm) and second-harmonic (SH, 775 nm) wavelengths. Our designs achieve ultra-low loss below 0.06 dB across a 90 nm bandwidth at FH, while maintaining extinction ratios exceeding 25 dB. At SH, the loss remains below 0.12 dB over a 45 nm bandwidth with extinction ratios greater than 19 dB. Devices fabricated on a 300-nm TFLN platform exhibit added loss below 0.1 dB across 1550 - 1600 nm, with minimum values of 0.04 dB around 1580 nm and 0.021 dB at 775 nm. Combined with recent advances in on-chip quantum state generation, low-loss interferometers, and detection, these results enable high-fidelity quantum photonic circuits on the TFLN platform.

While remarkable progress has been achieved in engineering nontrivial Hamiltonians across a wide range of physical platforms, preparing their corresponding nontrivial ground states remains a major experimental challenge. The commonly used strategy for state preparation relies on adiabatic protocols. However, when a trivial initial state is unitarily driven toward nontrivial regimes, the dynamics must cross gap-closing critical points, rendering the process intrinsically nonadiabatic, and the state remains topologically trivial. Here, we present a nonunitary method for dynamically preparing nontrivial states in two-dimensional topological insulators. By introducing dephasing noise into a slowly driven unitary evolution, we demonstrate that the topological number of the resulting dephased states can coincide with that of the target nontrivial Hamiltonian. This nearly adiabatic nonunitary state-preparation protocol provides a powerful alternative to conventional adiabatic approaches for accessing topological states.

Fault-tolerant measurement-based quantum computing (MBQC) provides a compelling framework for fault-tolerant quantum computation, in which quantum information is processed through single-qubit measurements on a three-dimensional entangled resource known as cluster state. To date, this resource has been predominantly studied on Euclidean lattices, most notably in the Raussendorf-Harrington-Goyal (RHG) construction, which underlies topological fault tolerance in MBQC. In this work, we introduce the hyperbolic cluster state, a generalization of the three-dimensional cluster state to negatively curved geometries, obtained via the foliation of periodic hyperbolic lattices. We present an explicit construction of hyperbolic cluster states and investigate their fault-tolerant properties under a realistic circuit-level depolarizing noise model. Using large-scale numerical simulations, we perform memory experiments to characterize their logical error rates and decoding performance. Our results demonstrate that hyperbolic cluster states exhibit a fault-tolerance threshold comparable to that of the Euclidean RHG cluster state, while simultaneously supporting a constant encoding rate in the thermodynamic limit. This represents a substantial improvement in qubit overhead relative to conventional cluster-state constructions. These findings establish hyperbolic geometry as a powerful and experimentally relevant resource for scalable, fault-tolerant MBQC and open new avenues for leveraging negative curvature in quantum information processing.

The standard Caldeira-Leggett model addresses the problem of Brownian motion in a thermal equilibrium environment. Here, we look for generalizations of the Caldeira-Leggett model to account for thermal gradients in the environment. We devise two types of models, and discuss the advantages and limitations of each one. From both models, we find signatures of thermophoresis, i.e., particle transport due to a thermal gradient. In principle, our models can be employed to describe thermophoresis in quantum Brownian particles, an open problem so far.

Measurements can be used to monitor the evolution of quantum systems and can give rise to quantized return statistics. It is known that the mean return time is quantized for strong monitoring through the winding number of the monitored quantum state. We discuss that under coherent weak monitoring, implemented via ancilla coupling, the mean return time of a quantum walk obeys a scaling relation with respect to the measurement strength. An analog scaling relation was previously found for random-time monitoring, indicating that weak and random-time monitoring have similar effects. We discuss how weak monitoring via ancilla coupling is linked to the unitary evolution, and how this connection can be controlled by a convergent perturbation theory.

Constructing models for cavity quantum materials requires a careful treatment of the light-matter coupling. In general, one must specify matrix elements constructed from the material wavefunctions, which are often unknown in a tight-binding framework. The Peierls substitution is frequently used to avoid introducing these additional parameters in the multi-center dipole (or Peierls) gauge, under the assumption that contributions from intraband and interband dipole moments can be neglected. We present the derivation of the Peierls gauge description, including these dipole moment terms, in the passive view of canonical transformations. We construct a toy model for a multi-band system with two sites, which we couple to a uniform field in the Coulomb, dipole, and Peierls gauges. We find that all polarization field terms are required to describe multi-band coupling in the Peierls gauge. The Peierls substitution can only be justified under restriction to a single band in one dimension, provided one also ignores self-polarization corrections arising from bands outside the retained subspace. However, these corrections are frequently non-negligible. More generally, the Coulomb, dipole, and Peierls gauges define distinct partitions of the composite system into the light and matter subsystems. We illustrate the implications of this subsystem relativity for observables such as the photon number and on the performance of orbital truncations in each gauge.

This paper considers two challenges faced by practical quantum networks: the bootstrapping of seedless Quantum Random Number Generators (QRNGs) and the resilient combination of Post-Quantum Cryptography (PQC) and Quantum Key Distribution (QKD) keys. These issues are addressed using universal hash functions as strong seeded extractors, with security foundations provided by the Quantum Leftover Hash Lemma (QLHL). First, the 'randomness loop' in QRNGs – the requirement of an initial random seed to generate further randomness – is resolved by proposing a bootstrapping method using raw data from two independent sources of entropy, given by seedless QRNG sources. Second, it is argued that strong seeded extractors are an alternative to XOR-based key combining that presents different characteristics. Unlike XORing, our method ensures that if the combined output and one initial key are compromised, the remaining key material retains quantifiable smooth min-entropy and remains secure in exchange of longer keys. Furthermore, the proposed method allows binding of transcript information with key material in a natural way, providing a tool to replace computationally based combiners to extend ITS security of the initial key material to the final combined output. By modeling PQC keys as having HILL (Hastad, Impagliazzo, Levin and Luby) entropy, the framework is extended to hybrid PQC-QKD systems. This unified approach provides a mathematically rigorous and future-proof mechanism for both randomness generation and secure key management against quantum adversaries.

Quantum computers promise massive computational speedup for problems in many critical domains, such as physics, chemistry, cryptanalysis, healthcare, etc. However, despite decades of research, they remain far from entering an era of utility. The lack of mature software, architecture, and systems solutions capable of translating quantum-mechanical properties of algorithms into physical state transformations on qubit devices remains a key factor underlying the slow pace of technological progress. The problem worsens due to significant reliance on domain-specific expertise, especially for software developers, computer architects, and systems engineers. To address these limitations and accelerate large-scale high-performance quantum system design, we ask: Can large language models (LLMs) help with solving quantum software, architecture, and systems problems? In this work, we present a case study assessing the performance of LLMs on quantum system reasoning tasks. We evaluate nine frontier LLMs and compare their performance to graduate UT Austin students on a set of quantum computing problems. Finally, we recommend several directions along which research and engineering development efforts must be pursued.

The anharmonicity of a transmon qubit, defined as the difference in energy level spacing, is a key design parameter. In transmons built from hybrid superconductor-semiconductor Josephson elements, the anharmonicity is tunable with gate voltages that control both the Josephson energy and the weak link transparency. In Sn-InAs nanowire transmons, we use two-tone microwave spectroscopy to extract anharmonicity ranging in absolute value from the transmon charging energy $E_c$ to values smaller than $E_c/10$. This behavior contrasts with the predictions of the multi-channel short-junction model, which sets a lower limit on anharmonicity at $E_c/4$. Coherent operation of the qubit is still possible at the point of the lowest anharmonicity. These findings demonstrate the potential of quantum circuits that benefit from widely electrically tunable anharmonicity.

We describe a method for obtaining the scattering matrix for nuclear or chemical reactions on a finite lattice. Aside from the preparation of the initial and final states as wave packets, the only other operation required is unitary time evolution, making this approach ideal for simulations on quantum hardware. The central quantity is a time-dependent overlap between incoming and outgoing wave packets whose Fourier transform corresponds to the scattering matrix at fixed energy, from which one can calculate elastic and inelastic cross sections for reactions involving two interacting clusters. Working in Cartesian coordinates enables an efficient encoding of the problem on quantum hardware via the first quantization mapping, with favorable qubit scaling for describing asymptotic scattering states. Within this framework, we describe a quantum algorithm for probing the scattering amplitude through different angles, including the forward direction, which provides access to the total cross section via the optical theorem. We demonstrate our methods through a series of numerical examples, for both elastic and inelastic processes, comparing against exact calculations. The techniques we describe can more readily be extended to a large number of constituent particles than other existing approaches, once fault-tolerant quantum hardware becomes available.

The set $\mathcal{Q}$ of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an $(n, 2, d)$ Bell operator decomposes into a direct sum of $(k, 2, d-1)$ Bell operators which we call subset games. We then derive three key insights: (1) in the $(n, 2, 2)$ case, if $p_0$ is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of $\mathcal{Q}$ is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in $\mathcal{Q}(D)$ unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation.

We argue that near-term experiments with neutral atoms in movable optical tweezers can simulate circuits that mimic the Trotterized time-evolution of simple matrix models in quantum mechanics. As a cartoon of this proposal, we study Floquet Clifford circuits which exhibit a number of signatures of fast scrambling. One such illustration is a simplified Hayden-Preskill recovery protocol, in which stabilizer quantum error correction replaces postselection.

Thermalization and its breakdown in interacting quantum many-body systems are governed by mid-spectrum eigenstates, which are typically accessible only in small system sizes amenable to exact diagonalization. Here we demonstrate that the density-matrix renormalization group (DMRG) effective Hamiltonian, an object routinely used to variationally approximate ground states, encodes detailed information about the dynamics far from equilibrium. In the random-field XXZ spin chain, the spectrum of the effective Hamiltonian is shown to capture the transition from thermal to many-body localized regimes, including spatially resolved probes of ergodic bubbles. Furthermore, the same approach also captures weak ergodicity breaking associated with quantum many-body scars. Our results establish the DMRG effective Hamiltonian as a versatile spectral probe of quantum thermalization and its breakdown in large systems beyond exact diagonalization.

A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation \(E \propto |p|^{j}\), where \(j = N - 1\). Starting from the generalized Minkowski distance in \(L^j\)-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent \(j\)-th order Schrödinger equation. The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries. While plane-wave solutions and translational invariance are preserved, the spectral structure is modified, with bound-state energies scaling as \((2n+1)^{j}\), leading to cubic and quartic growth in higher geometries. The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity. A generalized probability framework based on \(j\)-fold conjugation is introduced, ensuring a real-valued probability density and consistent expectation values. Despite these generalizations, the Heisenberg uncertainty principle is preserved. The formulation presents quantum mechanics as a geometry-dependent theory in which dispersion relations, spectral properties, and probabilistic structure emerge from the underlying spatial metric.

Modern quantum devices require high-precision Hamiltonian dynamics, but environmental noise can cause calibrated Hamiltonian parameters to drift over time, necessitating expensive recalibration. Detecting when recalibration is needed is challenging, especially since the very gates required for sophisticated verification protocols may themselves be miscalibrated. While cloud quantum computing services implement heuristic routines for triggering recalibration, the fundamental limits of optimal recalibration are not yet known. We develop efficient Hamiltonian certification and changepoint detection protocols in the autonomous setting, where we cannot rely on an external noiseless device and use only single-qubit gates and measurements, making the protocols robust to the calibration issues for multi-qubit operations they aim to detect. For unknown $n$-qubit Hamiltonians $H$ and $H_0$ with operator norm bounded by $M$, our certification protocol distinguishes whether $\|H-H_0\|_F\geq\epsilon$ or $\|H-H_0\|_F\leq O(\epsilon/\sqrt{n})$ with sample complexity $O(nM^2\ln(1/\delta)/\epsilon^2)$ and total evolution time $O(nM\ln(1/\delta)/\epsilon^2)$. We achieve this by evolving random stabilizer product states and performing adaptive single-qubit measurements based on a classically simulable hypothesis state. Extending this to continuous monitoring, we develop an online changepoint detection algorithm using the CUSUM procedure that achieves a detection delay time bound of $O(nM\ln(M\mathbb{E}_\infty[T])/\epsilon^2)$, matching the known asymptotically optimal scaling with respect to false alarm run time $\mathbb{E}_\infty[T]$. Our approach enables quantum devices to autonomously monitor their own calibration status without requiring ancillary systems, entangling operations, or a trusted reference device, offering a practical solution for robust quantum computing with contemporary noisy devices.

High-performance single-photon sources at telecom C-band wavelentghs are key building blocks for applications in long-distance quantum communication. Here, we report the generation of highly indistinguishable, single photons at a clock-rate of 2.5 GHz. This is achieved by coherently driving the biexciton transition ($T_1^\mathrm{XX}=64(1)\,$ps) of a semiconductor quantum dot embedded in a microcavity with strong asymmetric Purcell enhancement. Employing pulsed two-photon resonant excitation, strong multiphoton suppression with $g^{(2)}(0) < 4\%$ and high two-photon-interference visibility of $V_\mathrm{raw}> 85\%$ is observed. The observed photon indistinguishability is close to the theoretical limit expected for the photonically engineered radiative cascade and matches values obtained at lower repetition rates. Our results show a substantial advancement towards interference-based quantum information protocols at unprecedented data rates in the telecom C-Band.

We investigate how a localized curvature affects the dynamics of massless Dirac fermions in a curved surface. We consider a smooth bump with axial symmetry, adopting two specific geometric models, namely a Gaussian and a volcano-like bumps. By considering a minimal coupling between the spinor and the surface geometry, described by the vielbeins and the spin connection, we study the behavior of the wave function over the surface. By using appropriate numerical methods, we find a linear discrete energy spectrum for the Dirac fermions and its corresponding wavefunctions when the Fermi velocity is considered. It turns out that, since the curvature vanishes asymptotically, the electron states are free waves far from the bumps, but around the curved points, the wave function increases its probability density.

Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in high-dimensional regimes or at fine resolution. Nonlinear phenomena such as turbulence are notoriously difficult to predict because of their extreme sensitivity to small variations in initial conditions, except when certain stability conditions are fulfilled. Indeed, stability allows us to achieve reliable approximate dynamics, since it determines whether small perturbations remain bounded or are amplified, potentially leading to markedly different long-term behavior. Here, we investigate the stability of dissipative partial differential equations with second-order nonlinearities. By analyzing the time evolution of solution norms in Sobolev spaces, we establish a sufficient condition for stability that links the characteristics of the linear dissipative operator, the quadratic nonlinear term, and the external forcing. The resulting criterion is expressed as an explicit inequality that guarantees stability for a wide range of initial conditions. As an illustration, we apply the framework to fluid-dynamical models governed by nonlinear partial differential equations. In particular, for the Burgers equation, the condition admits a natural interpretation in terms of the Reynolds number, thereby directly linking the stability threshold to the competition between viscous dissipation and inertial advection. We further demonstrate the scope of the approach by extending the analysis to the KPP-Fisher and Kuramoto-Sivashinsky equations.

A recently developed notion, known as spread complexity, captures how a quantum state spreads in Krylov space under unitary evolution. Related Krylov-space approaches are increasingly being investigated for applications in quantum control, simulation, and metrology, as well as are being considered for implementation in near-term quantum platforms. However, its connection to fundamental quantum resources such as entanglement and quantum coherence remains unclear. We show that the dynamics of entanglement is constrained by the extent of spreading in Krylov space. For multipartite systems, we relate state delocalization in the Krylov basis, quantified by the inverse participation ratio, to geometric measures of multipartite quantum correlations. Furthermore, we derive analytical relations between the quantum coherence of the initial state in the energy eigenbasis and spread complexity for qubit and qutrit systems. Our results make progress towards understanding the subtle interplay between the dynamical spreading in Krylov space and fundamental quantum resources.