Search Results (204)

Variational quantum algorithms (VQAs) are a leading framework for realizing quantum advantages in the Noisy Intermediate-Scale Quantum (NISQ) era, with applications spanning discrete combinatorial problems and continuous optimization. While the topologies of parameterized quantum circuits (ansatzes) fundamentally govern both expressibility and trainability in continuous landscapes, existing reviews predominantly focus on static algorithmic classifications or discrete settings, leaving the structural evolution and practical limitations of ansatz architectures insufficiently explored. To address this gap, this review presents a systematic analysis of variational ansatz architectures, tracing their progression from static, pre-defined topologies to adaptive growth mechanisms. Beyond traditional gradient-driven and architecture-search paradigms, we evaluate supplementary strategies such as layerwise training and noise-adaptive construction, revealing inherent vulnerabilities such as local minima entrapment and the compilation overhead induced by calibration drift. The mathematical foundations of VQAs are outlined, and representative fixed ansatz architectures, including hardware-efficient, physics-inspired, and problem-specific designs, are characterized within continuous-domain mappings. Intrinsic limitations arising from barren plateaus (BPs) and noise-induced barren plateaus (NIBPs) are analyzed, revealing the fundamental coupling between circuit depth, parameter scaling, and trainability degradation. Furthermore, adaptive construction strategies and recent advances in automated variational quantum architecture search (VQAS) are examined. Through the synthesis of intrinsic limitations (BPs, NIBPs, and hardware-algorithm coupling) and the evaluation of standardized benchmarking protocols, this review rigorously assesses the resource trade-offs of current VQA frameworks. Ultimately, next-generation ansatz design will adopt hardware–software co-design principles grounded in physical constraints, enabling scalable and noise-resilient quantum optimization. Full article

Bandstructure methods occupy a central place in the physics of nanostructures, and the multiband $\mathit{k}·\mathit{p}$ theory of Luttinger, Kohn, and Kane has served as one of the most widely used computational frameworks for modelling electronic states and energies in low-dimensional semiconductor systems for several decades. Despite its broad success, the theory harbours a fundamental mathematical difficulty that has been largely overlooked: the multiband Luttinger–Kohn Hamiltonians are non-elliptic partial differential operators for the overwhelming majority of common III–V and III-nitride crystalline materials, a fact that violates the axiomatic requirements of quantum mechanics and is the root cause of the long-standing problem of spurious solutions. In this paper, we derive ellipticity conditions rigorously for the $6\times 6$, $8\times 8$, and $14\times 14$ zinc-blende Hamiltonians, demonstrating that non-ellipticity affects a substantially larger class of materials than previously reported. We develop and justify a systematic parameter rescaling procedure for the $8\times 8$ Kane Hamiltonian and obtain admissible parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN. The inversion-asymmetry parameter B is shown to play an essential and previously unrecognized role in maintaining ellipticity, and it is used to optimize the bandstructure fit of the rescaled parameter sets. Analysis of several known $14\times 14$ models reveals structural sources of non-ellipticity, pointing to the need for a revision of perturbative assumptions regarding out-of-basis band contributions. The consistent parametrization framework developed here provides the rigorous mathematical foundation required by inverse design methodologies, AI-enhanced electronic structure calculations, and data-driven multifidelity approaches in nanoscience and nanotechnology. Full article

This article offers a new argument concerning the relationships between mathematics and physical reality in quantum theory. While, as dedicated to the centenary of Heisenberg’s invention of quantum mechanics, the article contains a historical discussion, it is not historical, but physical and philosophical. Heisenberg’s thinking leading him to QM and beyond is only part of the genealogy of this argument, which also differs from that of Heisenberg and builds on the previous work of the present author. Among the contributions in the article are a new conception of the ultimate reality responsible for quantum phenomena; a new understanding, based on this conception, of the role of mathematics in physics in quantum theory; a critique of Heisenberg’s and all Platonism in quantum theory and beyond; a new perspective on the role of symmetries in quantum field theory; and a new understanding of the relationships between continuity and discontinuity in quantum physics. Full article

The increasing pace of Internet of Things (IoT) and Industrial Internet of Things (IIoT) applications has exacerbated the security challenges in resource-constrained environments, where traditional cryptographic protocols incur prohibitively high computational and energy costs. These constraints are also worsened by the advent of quantum computing, which poses a long-term security risk to popular crypto-key cryptographic-based efforts. To overcome these difficulties, this paper proposes an Energy-Efficient Cryptographic Protocol Framework (EECPF) that provides mutual optimization between energy consumption, security level, and communication latency to achieve sustainable IoT security. The presented framework proposes an adaptive encryption selection mechanism that dynamically chooses cryptographic algorithms depending on device capabilities, network conditions, and threat levels derived from intrusion detection outputs. EECPF combines privacy-preserving federated learning for distributed intrusion detection with collaborative threat intelligence sharing, eliminating centralized data sharing. In addition, lattice-based post-quantum cryptography primitives are added and combined with lightweight blockchain-enforced identity management to ensure long-term authentication resilience. The models on which the framework is based are mathematically based, modeling the consumption of energy, the robustness of security, and latency, providing principled multi-objective optimization under resource constraints. The publicly available Edge-IIoTset dataset was subjected to extensive experimental assessment under realistic IIoT and IoT attack scenarios. Experiments show that EECPF can reach an intrusion detection rate of 94.7%, while reducing energy consumption by 47.3% and latency by 23.8% compared with other commonly used lightweight cryptographic methods. These were continually noticed across different heterogeneous devices and deployment environments. In general, EECPF offers an energy-aware, quantum-resilient, and scalable security solution that can be used for next-generation IoT systems, such as smart healthcare, industrial automation, and smart city infrastructures. Full article

An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms. Full article

We derive the charged-lepton mass chain from the Recognition Composition Law (RCL) together with normalization, curvature normalization, and standard regularity. Through the theorem chain Tr1–Tr8, these postulates fix the golden ratio $\phi = \left(1+\sqrt{5}\right)/2$, the minimal period T_min = 8, the selected dimension D = 3, and the cube integers entering the master mass law. The charged-lepton formula is then assembled from the coherence scale, the lepton-sector baseline, the charge correction, and the derived generation steps. All parameters are discrete structural inputs, integers from cube geometry, named symmetry factors, and one external mathematical constant, rather than continuously adjustable dials. The construction is a structural constraint on the effective charged-lepton flavor pattern, not a replacement for the electroweak Higgs mechanism or for the full Standard Model quantum field theory. At the conversion stage to the International System of Units (SI), the electron fixes the single calibration anchor ${\tau }_0$, while the fine-structure constant $\alpha$ enters only as a fixed external dimensionless constant in the refinement layer. The phrase “zero continuously adjustable parameters” refers to the dimensionless content of the framework: the anchor ${\tau }_0$ is a unit-scale calibration fixed by the measured electron mass and cancels identically from every charged-lepton mass ratio. With that one anchor set, the remaining charged leptons become forward predictions: $m_{\mu }\in \left(\mathrm{105.5,105.9}\right) \mathrm{M}\mathrm{e}\mathrm{V}$ and $m_{\tau }\in \left(\mathrm{1774,1779}\right) \mathrm{M}\mathrm{e}\mathrm{V}$, with relative errors below $0.3\%$ and $0.2\%$, respectively. Floating-point evaluation gives $m_{\mu }\approx 105.658 \mathrm{M}\mathrm{e}\mathrm{V}$ and $m_{\tau }\approx 1776.71 \mathrm{M}\mathrm{e}\mathrm{V}$. Full article

The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group $SU\left(2\right)\otimes U\left(1\right)$ using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates. Full article

The rapid growth of digital communication and large-scale data exchange has increased the demand for advanced cryptographic techniques capable of resisting emerging computational threats. Conventional encryption methods primarily rely on mathematical complexity, which may become vulnerable with the advancement of high-performance computing and future quantum technologies. Biological molecules such as deoxyribonucleic acid (DNA) and RiboNucleic Acid (RNA) provide unique properties, including extremely high storage density, massive parallelism, and complex nucleotide structures that can inspire novel cryptographic mechanisms. This study proposes a bio-inspired cryptographic framework that integrates DNA encoding and RNA-based transformations to enhance data security. In the proposed framework, digital information is first converted into binary format and mapped to nucleotide sequences using a predefined encoding scheme. The encryption process incorporates multiple molecular transformations, including complementary base pairing, sequence permutation, and transcription-inspired DNA-to-RNA conversion to generate a highly randomized ciphertext. Decryption reverses these transformations to reconstruct the original plaintext. Security evaluation demonstrates that the proposed framework produces high entropy outputs, a substantially large key space, and enhanced resistance to statistical and brute-force attacks. The results indicate that DNA and RNA-inspired cryptographic systems can substantially enhance encryption complexity while maintaining reliable data recovery. This research highlights the potential of molecular cryptography as a promising interdisciplinary approach for future secure communication and biological data storage systems. Full article

Negative capacitance (NC) has been reported across a wide range of physical systems, yet its interpretation has remained fragmented due to the lack of a unified conceptual framework. Existing explanations—spanning ferroelectric free-energy curvature, tunneling transport, plasmonic resonances, and electronic compressibility—have often been treated as unrelated or even contradictory. This review resolves these inconsistencies by showing that all manifestations of NC arise from non-synchronization between external excitation and internal response. We classify NC into three fundamental categories: temporal mismatch, originating from delays or inertia in charge or polarization dynamics; spatial mismatch, caused by nonuniform field or mode distributions; and quantitative mismatch, resulting from intrinsic parameter reversal such as negative curvature or negative compressibility. Despite their diverse physical origins, these mechanisms share the same mathematical signature (${\mathrm{C}}_{eff}=\partial \mathrm{Q}/\partial \mathrm{V}<0$). Organizing NC within this unified framework clarifies long-standing ambiguities, connects previously isolated research fields, and establishes a systematic foundation for engineering NC in electronic, photonic, and quantum devices. The framework further highlights tunnel-current-induced NC as a representative single-particle mechanism within the temporal mismatch category, expanding the scope of NC beyond ferroelectricity and collective modes. Overall, this work positions NC not as a singular anomaly but as a universal response class emerging from the interplay between excitation and internal dynamics. Full article

This paper investigates a new class of mixed impulsive fractional boundary value problems (BVPs) in which the mixing occurs both in the governing fractional differential equations—through the combined presence of $\psi$-Caputo and quantum (q-difference) fractional derivatives—and in the boundary conditions formulated via fractional integral constraints. By incorporating two distinct operators within the same dynamical framework, the proposed model is capable of capturing both memory effects and discrete-scale behaviors inherent in complex hybrid systems. Using the Banach contraction mapping principle and the Leray–Schauder nonlinear alternative, sufficient conditions ensuring the existence and uniqueness of solutions are established. The theoretical results unify and extend several known fractional models. Owing to its flexible structure, the proposed framework may serve as a useful mathematical tool for modeling impulsive phenomena in systems where non-local memory and scale-transition mechanisms coexist, such as in engineering, physics, and applied sciences. Finally, numerical examples are provided to illustrate the applicability and qualitative behavior of the solutions. Full article

In this paper, a new class of special polynomials, called the Sheffer-type general-$\lambda$-matrix polynomials, is introduced within the framework of the monomiality principle. This family is obtained by combining the structure of Sheffer sequences with the theory of general-$\lambda$ matrix polynomials, which leads to a unified formulation encompassing several polynomial families. Fundamental properties of the proposed polynomials are established, including their generating function, explicit series representation, summation formulas, quasi-monomial structure, differential relations, and determinant representation. The proposed framework addresses an important problem in the theory of special functions: the systematic construction of matrix-valued polynomial families that simultaneously generalize both classical scalar polynomials and existing matrix polynomial hierarchies. Such a unified structure is of broad significance, with applications in quantum mechanics (wave function expansions), mathematical physics (matrix differential equations and spectral problems), approximation theory, and the study of special functions in the matrix domain. Several hybrid forms of the proposed family are derived through appropriate choices of the defining functions, which yield polynomial subclasses related to classical families such as Hermite, Laguerre, Bessel, and Poisson–Charlier polynomials. These subclasses illustrate how the proposed framework provides a systematic approach for constructing and studying generalized polynomial structures. In each case, the matrix parameter L introduces a new layer of structural richness not present in the scalar setting, enabling the modelling of phenomena governed by matrix-valued spectral data. Furthermore, a numerical and graphical investigation of selected hybrid forms is carried out using Mathematica (version 14.3, 2025; Wolfram Research, Inc.). Surface plots, distributions of complex zeros, and real-zero patterns are presented for different parameter values, highlighting the influence of the parameters on the behavior and structural characteristics of the polynomials. Full article

Post-quantum cryptographic implementations in Internet-of-Things (IoT) devices are significantly threatened by physical side-channel attacks, where practical attack risks are increased by physical accessibility and resource limitations. In particular, recent work has shown that belief propagation-based attacks can recover secret keys from lattice-based digital signatures using only a single side-channel trace of the Number Theoretic Transform (NTT). This work introduces the Quantum-Randomized Number Theoretic Transform (QR-NTT), an implementation-level defense mechanism that integrates quantum-derived entropy directly into the execution flow of lattice-based signature algorithms. Rather than treating randomness as a static input, QR-NTT uses quantum entropy to introduce controlled variability in execution ordering, arithmetic factor usage, and memory access behavior while preserving mathematical correctness and constant-time execution. The proposed framework is designed for embedded platforms and remains compatible with existing post-quantum cryptographic standards and IoT communication protocols. A complete implementation on an ARM Cortex-M4 platform, coupled with commercial quantum random number generator (QRNG) hardware, demonstrates that QR-NTT significantly degrades the effectiveness of template matching and belief propagation attacks. Experimental evaluation shows a reduction in single-trace attack success rates from over 90% to below 3% and an increase of approximately two orders of magnitude in the number of traces required for successful key recovery. These security gains are achieved with moderate overheads of 18.3% in execution time and 1.8 KB of additional memory while remaining well within practical IoT constraints. The results indicate that quantum-derived entropy can be leveraged as a practical implementation-level defense against physical attacks, complementing algorithmic post-quantum security. QR-NTT demonstrates a viable path toward strengthening the real-world resilience of post-quantum IoT systems without sacrificing deployability. Full article

The aim of the present work is to give a mathematical underpinning for the use of quasi-probabilities and pseudo-metrics in infinite-dimensional Banach manifolds. The notion of a continuous binary structure is introduced. It is a triple consisting of a continuous symmetric bilinear form together with a pair of closed linear subspaces of a Banach space. Such binary structures are abundant in Hilbert spaces. In order to confirm their existence in arbitrary Banach spaces, the auxiliary notion is introduced of subspaces that are positive with respect to a given symmetric bilinear form. It is shown that any subspace which is maximally positive with respect to the bilinear form induces a continuous binary structure on the Banach space. The Wigner function of a system of quantum mechanical particles is treated as an example. Full article

Non-convex stochastic optimization presents fundamental mathematical challenges across machine learning, wireless networks, data center resource allocation, and optical wireless communication systems, where complex loss landscapes with multiple local minima and saddle points impede classical variational inference methods. This paper introduces the Quantum-Inspired Variational Inference (QIVI) framework, which systematically integrates quantum mechanical principles (superposition, entanglement, and measurement operators) into classical variational inference through rigorous mathematical formulations grounded in Hilbert space theory and operator algebras. We develop a unified optimization framework that encodes classical parameters as quantum-inspired states within finite-dimensional complex Hilbert spaces, employing unitary evolution operators and adaptive basis selection governed by gradient covariance eigendecomposition. The core mathematical contribution establishes that QIVI achieves a convergence rate of $\mathit{O}({log}^{2}T/T^{1/2})$ for $\sigma$-strongly non-convex functions, provably improving upon the classical $\mathit{O}\left(T^{-1/4}\right)$ rate, yielding a theoretical speedup factor of $1.85$–$1.96\times$. Comprehensive experiments across synthetic benchmarks, Bayesian neural networks, and real-world applications in network optimization and financial portfolio management demonstrate 23–$47\%$ faster convergence, 15–$35\%$ superior objective values, and 28–$46\%$ improved uncertainty calibration. The principal contributions include: (i) a rigorous Hilbert space-based mathematical framework for quantum-inspired variational inference grounded in operator algebras, (ii) a novel hybrid quantum–classical algorithm (QIVI) with adaptive basis selection via gradient covariance eigendecomposition, (iii) formal convergence proofs establishing provable improvement over classical methods, (iv) comprehensive empirical validation across diverse problem domains relevant to machine learning and network optimization, and (v) demonstration of the framework’s applicability to optimization problems arising in wireless networks, data center resource allocation, and network system design. Statistical validation using the Friedman test (${\chi }^2=847.3$, $p<0.001$) and post hoc Wilcoxon signed-rank tests with Holm–Bonferroni correction confirm that QIVI’s improvements over all baseline methods are statistically significant at the $\alpha =0.05$ level across all benchmark categories. The framework discovers $18.1$ out of 20 true modes in multimodal distributions versus $9.1$ for classical methods, demonstrating the potential of quantum-inspired optimization approaches for challenging stochastic problems arising in machine learning, wireless communication, and network optimization. Full article

Consciousness presents a structural puzzle: a unified, context-sensitive, globally integrated mode of experience emerging from distributed neural dynamics. While classical neuroscience has mapped synaptic, oscillatory, and network-level mechanisms with increasing precision, debate persists as to whether classical formalisms fully capture the integrative and contextual features of conscious processing. This review examines whether quantum principles offer explanatory leverage in two distinct senses: as formal mathematical frameworks for modeling contextual cognition, and as mechanistic hypotheses proposing biologically instantiated non-classical states. We surveyed empirical and theoretical developments spanning zero-quantum-coherence in MRI signals, entanglement-structured learning paradigms, quantum-inspired computational models, and proposed neural substrates, including microtubules, nuclear spins, and photonic architectures. Although certain findings have been interpreted as consistent with a non-classical structure, no study to date has demonstrated entanglement, long-lived coherence, or collapse dynamics in neural tissue under operational criteria comparable to those used in controlled quantum systems. Replication remains limited, biological entanglement witnesses are not yet established, and nonlinear classical dynamics can reproduce many putative quantum signatures. Accordingly, the decisive question is not whether the brain is quantum, but whether its dynamics exceed the explanatory reach of rigorously defined classical models. Progress hinges on replication, adversarial scrutiny, and operational criteria precise enough to discriminate genuine non-classical correlations from classical complexity. Whether quantum mechanisms ultimately prove necessary or refined classical models remain sufficient, this inquiry compels a deeper understanding of integration, contextuality, and the physical constraints shaping conscious experience. Full article