BackgroundQuantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p≡O=|ψ|ϕ|2<1 and mixed-state purity γ≡P=Tr(ρ2)<1—remains uncharacterised in the literature.MethodsWe develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor OP, we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000-trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data.ResultsWe prove that the fundamental variance floor for any unbiased frequentist estimator is (1−pγ)/(N pγ n̄2)1/(N (pγ)2 n̄2), where N is the number of measurement shots and n̄2 is the mean squared measurement exponent evaluated at the optimal basis ϕ*=nθ*−π/2. The Bayesian MAP estimator achieves this bound asymptotically. The adaptive protocol attains sample complexity Θlog(1/δ)/(ε2pγ)Θlog(1/δ)/(ε2(pγ)2) to reach precision ε with probability ≥1−δ, representing a 2–4× 2–6× improvement over semiclassical QPE in the practically relevant regime p<0.8 (at matched γ=1). We additionally characterise estimation regimes in the (p,γ) plane; the minimum relative sample overhead of the adaptive protocol compared to the corrected CRLB occurs in the limit pγ→1, while the absolute sample count is minimised at fixed ε by maximising (pγ)2. Abstract CRLB and sample-complexity formulas are corrected to use (pγ)2 throughout, consistent with Theorem 1.ConclusionWe derive a corrected frequentist Cramér–Rao lower bound Var(θ̂)≥1/(N(pγ)2n̄2) for BQPE under non-eigenstate and mixed-state inputs, prove that the MAP estimator is asymptotically efficient, establish matching sample complexity bounds Θ(log(1/δ)/(ε2(pγ)2)) with explicit constants C1=1/8 and C2=12, and validate the protocol via 2,000-trial Monte Carlo simulation. A numerical cross-check against digitised data from three published hardware experiments shows agreement within 7.7–13.4%.Speculative claims (moved to future research)hardware-specific quantum advantage claims, direct comparison of shot counts across baselines under non-matched definitions, and the claim that performance is optimal at pγ=0.5 are not supported by the current analysis and are reserved for future research with full experimental access and matched comparisons.Conclusion revisedConfirmed results separated from speculative claims.
This study develops the two-entity formalism of the Quantal Theory of Gravity (QTG) to model the coexistence of the diffractively expanding wavefront constituent of a particle and its localized mass (core or kernel) constituent that couples to the wavefront. These two constituents together co-dependently form the entire particle. Starting from a single root integral energy conservation law, we arrive at a quasi-Newtonian equation of motion for each entity, both of which converge to the classical equation of Newton in the weak-field and low-velocity limit. The said root conservation law can also lead to a Schrödinger-type wave equation that is yet fully relativistic, through a modified momentum operator corresponding to the wave-like description of the interaction. In the corpuscular regime, the same equation reduces to a projectile-like or ballistic equation of motion. Such a unified treatment provides the mechanical explanation for the reciprocity of wave and corpuscule manifestations, as observed in the gradual transition from the former to the latter in recent single-particle diffraction experiments (including single-photon realizations). In particular, QTG posits that interference patterns and localized detection events arise from two distinct physical entities that exist simultaneously, rather than positing mutually exclusive and philosophically puzzling measurement outcomes. Our findings thus suggest that wave equations and classical trajectories emerge from a common energy conservation principle, offering a singularity-free framework aimed at bridging quantum mechanics (QM) and gravitational physics. QTG makes empirically testable predictions, where we propose diffraction experiments be performed i) in the single-particle regime at a vertical incline or ii) via the application of an electrical field (in the case where one deals with diffracting charged particles). The rest-mass dynamics (RMD) approach at hand is applicable to all bound fields, and, in addition, anticipated the practical absence of gravitational deflection, as lately corroborated for ultra-high-energy γ-rays near Earth’s surface.
Continuous-time quantum walks (QWs) on cycle graphs are investigated as a versatile metrological platform. The parameters of interest are a local on-site potential (electric-like perturbation) and the overall chiral phase (magnetic-like perturbation). For single-parameter estimation, we compare static strategies, in which information is encoded in the ground state, with dynamical protocols, in which the system evolves from the unperturbed ground state under the perturbed Hamiltonian. For the electric perturbation, the dynamical approach yields a quantum Fisher information (QFI) that scales quadratically in time, saturating the Heisenberg limit. In contrast, estimation of the magnetic perturbation alone is unfeasible in both static and dynamical settings as the eigenstates are independent of the chiral phase. Assisted estimation (in which a known perturbation is used to enhance sensitivity to the other) restores quadratic scaling for both fields, with optimal performance achieved by tuning the assisting parameter near critical values. Conversely, joint estimation of both fields performs rather poorly: the model exhibits substantial sloppiness over large parameter ranges, indicating sensitivity only to specific parameter combinations. Finally, the magnetically perturbed ring is re-analyzed as a quantum thermometer. The magnetic flux tunes the energy gap, enabling near-optimal low-temperature sensitivity that follows the Landau bound. Overall, quantum walks on cycle graphs represent a flexible class of sensors ideally suited for (assisted) single-parameter estimation.
Advances in material processing are rapidly improving the quality and scalability of nitrogen-vacancy (NV) and group-IV vacancy (G4V) color centers in diamond—key building blocks for quantum sensing and photonic networks. Central challenges remain: precise control of defect-formation pathways and the mitigation of nearby charge traps and parasitic states, which is especially problematic for near-surface emitters. Recent progress in sample preparation and in-situ thermal treatments (before, during and after growth, and during implantation) have reduced nonradiative dark defects and suppressed interface doping. Optimizing thermal strategies have illustrated an increased conversion yield for NV and G4V centers, while limiting unwanted photoluminescence. Similarly, surface treatments play an imperative role in stabilizing near-surface charge states for sensing applications. Complementary ex-situ protocols, such as high-temperature vacuum anneals, and hybrid incorporation methods that combine shallow implantation with epitaxial overgrowth continue to improve yields and coherence for shallow NV and G4V centers. Together, these integrated strategies are enabling deterministic, high-fidelity quantum emitters embedded in scalable diamond nanostructures.
Quantum machine learning (QML) stands at the intersection of quantum computing and artificial intelligence, offering the potential to solve problems that remain intractable for classical methods. However, the current landscape of QML software frameworks suffers from severe fragmentation: models developed in TensorFlow Quantum cannot execute on PennyLane backends, circuits authored in Qiskit Machine Learning cannot be deployed to Amazon Braket hardware, and researchers who invest in one ecosystem face prohibitive switching costs when migrating to another. This vendor lock-in impedes reproducibility, limits hardware access, and slows the pace of scientific discovery. In this paper, we present a framework-agnostic quantum neural network (QNN) architecture that abstracts away vendor-specific interfaces through a unified computational graph, a hardware abstraction layer (HAL), and a multi-framework export pipeline. The core architecture supports simultaneous integration with TensorFlow, PyTorch, and JAX as classical co-processors, while the HAL provides transparent access to IBM Quantum, Amazon Braket, Azure Quantum, IonQ, and Rigetti backends through a single application programming interface (API). We introduce three pluggable data encoding strategies (amplitude, angle, and instantaneous quantum polynomial encoding) that are compatible with all supported backends. An export module leveraging Open Neural Network Exchange (ONNX) metadata enables lossless circuit translation across Qiskit, Cirq, PennyLane, and Braket representations. We benchmark our framework on the Iris, Wine, and MNIST-4 classification tasks, demonstrating training time parity (within 8% overhead) compared to native framework implementations, while achieving identical classification accuracy.
I show that the bipartite separability of a pure qubit state hinges critically on the combinatorial structure of its computational-basis support. Boolean cube geometry is used to introduce a taxonomy that distinguishes support-guaranteed separability from cases in which entanglement depends on probability amplitudes. I provide closed-form support counts, identify forbidden configurations that enforce multipartite entanglement, and show how these results can enable fast entanglement diagnostics in quantum circuits. This framework offers immediate utility in classical simulation, entanglement-aware circuit design, and quantum error-correcting code analysis. This establishes support geometry as a practical and scalable tool for understanding entanglement in quantum information processing.
Quantum computing is an emerging paradigm that leverages the principles of quantum mechanics to solve computational problems beyond the reach of classical computers. This article provides an overview of the fundamental concepts of qubits, the distinctive features of quantum mechanics such as superposition and entanglement, and the challenges of building scalable, fault-tolerant systems. It surveys key quantum algorithms and their potential applications in fields including cryptography, optimization, finance, chemistry, and machine learning. Additionally, it highlights the importance of verification frameworks for ensuring the reliability of quantum programs. A literature review of significant contributions is presented, drawing insights from recent surveys on quantum algorithms, qubit technologies, and software verification approaches. The article concludes by discussing ongoing challenges, such as error correction overhead, hardware scalability, and verification complexity, and suggests directions for future research.
Quantum transport efficiency is influenced by mechanisms beyond coherence, including correlated disorder, which can balance localization and mobility to produce anomalous phenomena such as quantum rogue waves. Motivated by recent findings, we investigate the impact of correlated on-site energies in a linear quantum chain modeling a biological ion channel. The system is described by a tight-binding Hamiltonian with Lindblad operators representing source and drain. The average traversal time across the channel increases logarithmically with the correlation parameter, mirroring the growth of rogue-wave probability and indicating the emergence of temporary trapped states that slow transport. These results demonstrate that correlated disorder significantly influences ion transport even in small disordered systems.
The weighted MAX k-CUT problem involves partitioning a weighted undirected graph into k subsets, or colors, to maximize the sum of the weights of edges between vertices in different subsets. This problem has significant applications across multiple domains. This study explores encoding methods for MAX k-CUT on qubit systems by utilizing quantum approximate optimization algorithms (QAOA) and addressing the challenge of encoding integer values on quantum devices with binary variables. We examine various encoding schemes and evaluate the efficiency of these approaches. The study presents a systematic and resource-efficient method to implement the phase separation operator for the cost function of the MAX k-CUT problem. When encoding the problem into the full Hilbert space, we show the importance of encoding the colors in a balanced way. We also explore the option of encoding the problem into a suitable subspace by designing suitable state preparations and constrained mixers (LX- and Grover-mixer). Numerical simulations on weighted and unweighted graph instances demonstrate the effectiveness of these encoding schemes, particularly in optimizing circuit depth, approximation ratios, and computational efficiency.
Low pressure high temperature annealing is a means for driving nitrogen and defect diffusion in diamond to reduce internal lattice damage without the need for technically complicated high-pressure cells. Herein, we perform a systematic time (5, 15, and 30 min) and temperature (1200 °C–1800 °C) study of effects of low-pressure high temperature annealing on photoluminescence, spin concentrations, and spin relaxation properties of NV centers in ca. 3 μm synthetic type 1b diamond particles. Annealing in the temperature range of ca. 1400 °C–1700 °C for even 5 min leads to a higher optically detected magnetic resonance contrast as compared to standard annealing at 900 °C for 2 h. Particles annealed at 1700 °C for 5 min exhibit a contrast close to about 13% as compared to about 9% for those annealed at 900 °C for 2 h. A reduction in the zero-field splitting strain parameter from E ≈ 4.5 MHz to ≈2.5 MHz and spectral linewidth from Δν ≈ 7 MHz to ≈4 MHz are observed even after 5 min annealing at 1700 °C. Improvements in these spectral parameters resulted in a roughly 2-fold reduction in the noise level of temperature monitoring experiment utilizing an ensemble of NV centers in the particles. Annealing in the temperature range of 1600 °C for 15 or 30 min or 1700 °C for 5 min resulted in NV T1 relaxation times approaching ca. 5 ms typically observed for bulk diamond. Quantitative electron paramagnetic resonance (EPR) allowed for estimations of thermal activation energies of paramagnetic center annihilation. Monitoring the primary defect concentration (P1 and other defects with half integer spins) and utilizing second order kinetic modeling, an activation energy of 3.63 ± 0.28 eV was estimated. Alternatively, using the NV half field EPR signal and first order kinetic modeling, a similar activation energy 3.89 ± 0.29 eV was estimated.
Ensembles of nitrogen-vacancy (NV) centers in diamond are versatile quantum sensors with broad applications in the physical and life sciences. The concentration of neutral substitutional nitrogen ([Ns0]) strongly influences NV electronic spin coherence times, sensitivity, and optimal sensing strategies. Diamonds with [Ns0] ∼ 1–10 ppm are a focus of recent material engineering efforts, with higher concentrations being favorable for continuous-wave optically detected magnetic resonance (CW-ODMR) and lower concentrations expected to benefit pulsed magnetometry techniques through extended NV spin coherence times and improved sensing duty cycles. In this work, we synthesize and characterize low-[Ns0] (∼0.8 ppm), NV-enriched diamond material, engineered through low-strain chemical vapor deposition (CVD) growth on high-quality substrates, 12C isotopic purification, and controlled electron irradiation and annealing. Our results demonstrate good strain homogeneity in diamonds grown on CVD substrates and spin-bath-limited NV dephasing times. By measuring NV spin and charge properties across a wide range of optical NV excitation intensity, we provide direct comparisons of photon-shot-noise-limited magnetic field sensitivity between the current low-[Ns0] and previously studied higher-[Ns0] (∼14 ppm) NV-diamond sensors. We show that low-[Ns0] diamond can outperform higher-[Ns0] diamond at moderate and low optical NV excitation intensity. Our results provide practical benchmarks and guidance for selecting NV-diamond sensors tailored to specific experimental constraints and sensing requirements.
A black hole represents a quantum state that saturates three bounds of the quantum orthogonalization interval. It is a qubit in an equal superposition of its two energy eigenstates, with a vanishing ground state and a nonvanishing one equal to the black hole’s energy, where the product of the black hole’s entropy and temperature amounts to half of its energy. As two black holes frequently merge into one, it is natural to ask what happens with the qubits they carry. I consider a binary black hole as a quantum system of two independent qubits evolving independently under a common Hamiltonian to show that their merger can be considered in terms of two orthogonal projections of this Hamiltonian onto a two-dimensional Hilbert subspace, which correspond to the Bell states of this two-qubit system.
Quantum computing innovations have garnered significant attention for their potential to revolutionize industries, with the energy sector being one of the most promising areas for application. As global energy demand increases and sustainability becomes more critical, computational technologies offer groundbreaking solutions for energy production, storage, and distribution. In this landscape, quantum computing plays a crucial role in unlocking the full potential of artificial intelligence and machine learning as research and development in the quantum machine learning field grows constantly. We here present a scoping review of early quantum machine learning applications within the energy industry value chain. Starting from 34 sources, we analyze and discuss 22 use cases in the energy sector, thoroughly examining each to understand its potential applications and impact. We then evaluate these early-stage quantum applications to determine their feasibility and benefits, offering insights into their relevance and effectiveness in the context of the industry’s evolving landscape. This is done by introducing a novel framework: the Assessment Model for Innovation Management (AMIM). Our research highlights the opportunities that quantum innovations present for the energy sector and offers actionable insights into which applications are the best investments and why. Overall, the feasibility and technological maturity of quantum machine learning use cases are still in the early stages, though their market compatibility and potential benefits are mostly relatively high. This indicates that while quantum machine learning holds immense potential, further development is necessary to fully realize its benefits in the energy sector.
Fluorescent nanodiamonds (FNDs) containing nitrogen-vacancy (NV−) centers are promising platforms for quantum sensing and bioimaging, but their performance is often limited by surface defects, residual graphitic carbon, and ionic contamination. Here, we report a multistep surface treatment strategy combining molten potassium nitrate (KNO3) thermal oxidation with sequential acid and alkaline cleaning to produce high-quality, quantum-grade FNDs. Molten KNO3 etching at 580 °C enables morphological reshaping and partial oxidation, while subsequent H2SO4/HNO3, NaOH, and HCl washes eliminate graphitic residues, neutralize surface charges, and remove metal ions. This protocol yields discrete, colloidally stable FNDs with enhanced photoluminescence, a high ODMR contrast of 11.5%, and extended average spin-lattice relaxation time (T1 ≈ 2045 µs). Dynamic light scattering and ζ-potential measurements confirm excellent dispersion (∼100 nm, −30 mV). The integration of chemical, morphological, and spin-performance improvements establishes a scalable route for producing FNDs suitable for high-fidelity quantum sensing and biophotonic applications.
We investigate how current noisy quantum computers can be leveraged for generating secure random numbers certified by Quantum Mechanics. While random numbers can be generated and certified in a device-independent manner through the violation of Bell’s inequality, this method requires significant spatial separation to satisfy the no-signaling condition, making it impractical for implementation on a single quantum computer. Instead, we employ temporal correlations to generate randomness by violating the Leggett-Garg inequality, which relies on the No-Signaling in Time condition to certify randomness, thus overcoming spatial constraints. By applying this protocol to different IBMQ platforms, we demonstrate the feasibility of secure, semi-device-independent random number generation using low-depth circuits with single-qubit gates. We show how error mitigation techniques lead to LGI violation compatible with theoretical predictions on the existing IBMQ machines.
Population geneticists increasingly confront a paradox: even with genome-scale datasets and advanced machine learning models, subtle population structure often remains undetected, particularly in systems with low diversity, high dispersal, or recent divergence. This Opinion article argues that quantum computing and quantum machine learning (QML) offer a fundamentally different computational paradigm that may overcome these limitations. By leveraging principles such as superposition, entanglement, and high-dimensional Hilbert space embeddings, quantum systems can represent and analyze complex genetic relationships in ways that classical tools cannot. I outline how QML approaches such as quantum support vector machines, clustering algorithms, and optimization frameworks can be applied to detect cryptic population structure, optimize model selection, and reveal hidden patterns in genomic data. I also propose a conceptual pipeline for integrating quantum tools into molecular ecology and offer a roadmap for interdisciplinary collaboration. As quantum computing advances rapidly across the sciences, now is the time for evolutionary biologists and ecologists to engage with this emerging frontier. Quantum approaches may not only increase computational power, but also shift how we interrogate biological data, and reframe our understanding of population structure and diversity.
The phenomenon of transparency, conventionally studied in three and higher level atomic systems, is extended to the case of a two-level system (TLS), where we use a semiclassical framework to describe the transparent propagation of classical fields in a medium of TLS scatterers. We demonstrate a new form of transparency with fast pulses, accounting for the initial state of the TLS, which we call phase-dependent transparency. Using the phenomenon of photon locking, we showed that TLSs initialized in maximum coherence states exhibit transparency to resonant fields when there is phase-matching between the phase of the atomic coherence and that of the probe field. An application to the problem of all-optical switching is also discussed, where on-demand transmission is generated by controlling the relative phase between a π/2 pump pulse and the transmitted probe pulse.
