Papers

In many-body physics, an explicit wavefunction often provides the most thorough understanding, yet for monitored random Clifford circuit it has remained missing. Here we develop a framework that grants direct access to the typical output states. As every stabilizer state is local-Clifford-equivalent to a graph state, the graph adjacency matrix, a classical bit matrix, provides a complete description of the quantum state. In the large-$N$ limit, we show that these graphs converge to the Erdős–Rényi random graph $G(N,1/2)$, which allows us to resolve the open problem of Greenberger–Horne–Zeilinger (GHZ) entanglement generated by deep random Clifford circuits. We obtain analytically the mean GHZ content $\langle g_3\rangle=1.204$ for even $N$ and $1.325$ for odd $N$. For monitored Clifford circuits with a 1D layout, we uncover an emergent Erdős–Rényi subgraph $G(N_{\mathrm{sub}},1/2)$ in the output states of the volume-law phase, where $N_{\mathrm{sub}}/N\approx \sqrt{1-p/p_c}$ with $p$ the measurement rate and $p_c$ the critical point of the measurement-induced phase transition (MIPT). The output state is thus equivalent to the output of an unmonitored random Clifford circuit on $N_{\mathrm{sub}}$ qubits, weakly perturbed by the remaining $N-N_{\mathrm{sub}}$ qubits carrying little entanglement. This result directly accounts for the quantum error-correcting capability of the volume-law phase, and implies the same GHZ statistics for the whole volume-law phase. We further identify a clustering effect for qubits in the dense subgraph, which we reproduce with an infection-recovery toy model that exhibits a measurement-induced absorbing-state phase transition. Finally, a mean-field argument on the graph locates the MIPT critical point at $p_c = 0.1608$, in excellent agreement with the numerical value $p_c\approx 0.16$.

Optically addressable spin defects in hexagonal boron nitride (hBN) are well suited for near-surface quantum sensing. They offer bright, wavelength-tunable single-photon emission with high optically detected magnetic resonance (ODMR) contrast at room temperature. Here we controllably synthesise a high density of spin-complex defects in carbon-doped hBN flakes. We find that the zero-field splitting parameter D is directly correlated with the zero-phonon line of an emitter, while the ODMR contrast shows no such correlation. We further analyse an individual narrowband defect showing 68% ODMR contrast and enhance its photon collection by ~40% using a solid immersion lens. By advancing both the practical synthesis of the spin complexes and the understanding of their microscopic origin, our results move us toward the deterministic creation of ODMR-active quantum emitters

The most subradiant eigenstate of a finite subwavelength atomic chain in free space, protected by strongly suppressed radiative decay, offers a powerful resource for photon storage, quantum sensing, and many-body quantum optics. Yet its optical preparation is hindered by the simultaneous need to match a wave vector outside the light cone and a nonuniform envelope. Here, we show that a free electron can overcome these constraints: its velocity sets the imprinted wave vector, while the trajectory of the diffracting wave packet shapes the excitation envelope. This simultaneous momentum and envelope matching enables heralded preparation with near-unity conditional fidelity ($F>99.5\%$) even in a deeply subwavelength regime that is difficult to access with propagating free-space photons. We further show that a path-superposed free electron can excite an antisymmetric state in two closely spaced parallel chains, whose interchain destructive interference yields stronger subradiance than a single chain with the same total number of atoms. These results establish free electrons as quantum writers for collective excitations that are difficult to access with propagating optical fields.

Quantum algorithms based on linear-system approaches for solving differential equations demand qubit and precision resources beyond near-term capabilities. To address these challenges, this work proposes a physics-informed quantum machine learning (PIQML) framework with hard constraint embedding, specifically designed for NISQ era. Within this framework, parameterized quantum circuits serve as machine learning models, where the input variable is encoded into a high-dimensional feature space via a Fourier feature map. Subsequently, to eliminate approximation errors in critical physical conditions, the solution is constructed through a rigorously designed function mapper that analytically enforces initial conditions as hard constraints. Crucially, we compute derivatives with respect to the input variable using the parameter-shift rule—a quantum native gradient evaluation technique that avoids classical discretization. Unlike generic loss functions that target abstract data patterns, our loss function focuses on the differential equation residual and reference data. This design ensures that the trained model not only approximates the data but also intrinsically satisfies the physical constraint expressed by the DE itself. Our method is validated on several differential equations, including highly oscillatory ones, demonstrating its capability to tackle challenging nonlinear dynamics. Results demonstrate that our quantum model successfully learns the solution, showing close agreement with a high-precision classical numerical benchmark.

Photonic multi-qubit entanglement is key to optical quantum information processing, particularly universal quantum computing. Yet multi-photon sources suffer from low emission efficiency, making single-photon high-dimensional encoding an appealing alternative. Here we propose an explicit and resource-efficient high-dimensional encoding approach to achieve the target multi-qubit quantum state. The technically challenging preparation of multi-photon quantum states is replaced by single-photon operations involving high-dimensional expansion, routing, and multi-layered quantum measurement. Besides, each photon in the resource multi-photon quantum state can be used to encode multiple qubits in a distributed manner, and a larger entangled state will be constructed. We demonstrate this approach using programmable photonic integrated circuits, where multi-qubit graph states–including the Greenberger-Horne-Zeilinger state and the cluster state–are generated and characterized. We additionally demonstrate the Grover search algorithm using the single-photon cluster state. Our findings unlock a novel route towards diverse entangled state generation with photons and advance large-scale and universal photonic quantum information processing.

We propose a scheme to compute Pauli error rates in photonics-based quantum error correction using experimental observables of single photons produced from quantum emitters. We show that first-order coherence measurements and first-order cross correlations, which can be implemented using photon counting, can extract single-photon and entangled single-photon wavefunctions in the presence of imperfections due to photon distinguishability, laser noise, and photon loss. Leveraging this, we show that the wavefunction of any entangled state of noisy photons produced from a single quantum emitter can be expressed in matrix-product-state form and can be used to analytically compute the expectation value of the stabilizer generators of the corresponding entangled state. From this we obtain analytic expressions for the Pauli error probabilities in terms of the photon noise parameters. Furthermore, we calculate Pauli error maps for entangled photons after undergoing Bell- state measurements in terms of these parameters. Our work provides a method to use experimental measurements to determine the required quality of photons produced from quantum emitters for fault-tolerant fusion-based photonics quantum computing.

Rydberg atoms have emerged as a versatile platform for quantum optics due to their exaggerated properties, particularly their strong long-range interactions, which enable a new regime of light-matter interaction. By mapping the interactions between Rydberg atoms onto photons, effective photon-photon interactions can be realized at the single-photon level, thereby overcoming the intrinsic weakness of conventional optical nonlinearities. In this review, we first introduce the fundamental physical principles of Rydberg-mediated quantum optics, and then discuss some key developments, including single-photon engineering, photonic quantum gates, contactless nonlinear optics, and quantum entanglement, providing a comprehensive overview of the current state and prospects of this rapidly developing field.

We investigate interferometric phase-estimation using separable photon inputs that evolve into number-path entangled states through linear optical networks, followed by photon-number-resolving detection. A simple analytical expression for the classical Fisher information at zero phase is derived for arbitrary $N$-photon states distributed across 2$N$ optical modes, partitioned into phase-encoding and reference blocks. Among all possible photon distributions between these blocks, the balanced configuration maximizes the phase sensitivity for every photon number $N$ and uniquely exhibits a phase-independent response. The achievable sensitivity degrades monotonically with increasing asymmetry in the photon distribution. We further investigate the robustness of the protocol in the presence of realistic photon loss and extend the analysis to distributed architectures with multiple receivers. In the low photon-flux regime, vacuum fluctuations fundamentally limit local quadrature measurements, whereas nonlocal photon-number-resolving measurements exploit multiphoton interference to mitigate loss-induced sensitivity degradation. Together, these results establish a scalable framework for quantum-enhanced distributed multimode metrology.

Neural networks can be studied not only as parameterized computational models but also as probability distributions over functions. In this paper we develop a Wilsonian interpretation of criticality in the neural network-quantum field theory correspondence, treating the infinite width Gaussian-process limit as a free field fixed point and finite width corrections as perturbations that create non-Gaussian interactions. In this way, we do not see the departure from infinite width as a small approximation error but as the process by which interaction, complexity, expressivity and phase-like behaviour enter neural network function space. Width, depth, activation nonlinearity, initialization variance, and training dynamics are considered as control parameters that change the effective action of the network ensemble. The critical regime is when the higher-order connected correlation functions become non-negligible, and when the finite width operators become relevant or marginal scaling factors, and when the function distribution becomes sensitive to scale-dependent structure. There we can view overparameterization as suppressing a relationship of interacting terms and we find that the critical structure of finite neural networks is given by finite-width effects. This framework is a theoretical basis for studying trainability, generalization, and architectural universality of neural network functions through Wilsonian fixed points, perturbations, and critical surfaces.

We assess 18 chapter-specific reviews of the ten mathematical results announced by OpenAI on 1 August 2026, alongside review standards, Lean formalizations, subsequent research, and mathematical references. The article audits this review record without claiming a complete reconstruction of all ten proofs. No confirmed substantive mathematical error in a principal result remains in the examined assessments, although review depth varies and some dependencies remain partly checked. Chapter 8 presents the strongest reservation: a specialist review requests major revision of compressed analytic arguments. In Chapter 6, an apparent polarity error was withdrawn after an overbar lost during PDF extraction was recovered from the typeset source. Subsequent research independently reuses the Chapter 3 proof mechanism and confirms that Connes's rigidity conjecture is false, without independently reproducing Chapter 4's stronger infinite-family result. Among the cited follow-ups, Chapter 7 receives the strongest direct theorem-level corroboration through a stronger hardness theorem. Related equality results in Chapter 8 do not verify the analytic inequality proof. Some follow-ups disclose material AI assistance. We argue that confidence should combine formal checking, human reconstruction, independent mathematical use, and a public record supporting correction of both proofs and reviews.

The recent success of error detecting codes points toward their potential application to fault-tolerant simulations of nature. In this work, we examine the utility of sparse error detection for simulating lattice gauge theories using quantum computers. In particular, we study the time evolution of the lattice Schwinger model embedded into the Iceberg code family, $[[N+2, N, 2]]$, as well as the Hypercube code family, $[[2^N, N, 2]]$. The lattice of electrons and positrons in the axial gauge is embedded into a single code block or into multiple code blocks, and this work finds that large codeblocks are advantageous in the absence of connectivity constraints. Noisy classical simulations with realistic near-term error rates, infrequent syndrome measurements and physics-aware postselection are found to improve observable estimation. Under realistic noise rates for near-term quantum computers, this work finds that sparse error detection in quantum simulations has the potential to improve accuracy of observable estimation. Additional rounds of error detection are found to systematically drive errors in observables to the noise floor set by the code. These findings suggest that incorporating minimal implementations of fault tolerance in the near-term will enhance the performance of quantum simulations in nuclear physics and high-energy physics.

Using the framework of quantum multiparameter estimation, we study the problem of separating independent thermal optical sources mixed by an unknown passive linear transformation, which is known as blind source separation in signal processing. We propose a sensing-assisted method that iteratively estimates and suppresses optical correlations directly within the unit cells of a programmable interferometer. We show that collective quantum measurements exhibit a substantial advantage over conventional detection methods by constructing collective measurements that asymptotically achieve the Holevo Cramér–Rao bound in a unit cell of the interferometer for the respective sub-problem. By systematically arranging multiple such cells in a photonic mesh, our proposal implements an in situ Jacobi diagonalization of the input multimode correlation matrix. We contrast our method with a conventional approach where heterodyne detection is used to reconstruct the full covariance matrix and subsequently diagonalize it on a classical computer. A comparison with this heterodyne tomography approach shows that our sensing-based method is particularly advantageous in the weak-light regime.

We introduce a non-commutative Laplace transform between functionals on path space and formal series in a tensor algebra, under which a natural convolution of path functionals becomes an algebraic product of series. Applying it to a random unitary matrix-valued path development - the quantum path signature - we show that the governing planar loop equations take a non-commutative spectral form. We then extend the loop equations to a $1/N$ genus expansion, organised by topological recursion, and obtain a hierarchy of integral equations on path space for the corrections.

Spin-squeezed states enable entanglement-enhanced frequency estimation; however, the achievable performance is limited in practice by decoherence. We study the problem of parameter estimation under classical collective noise that acts along a fixed direction during signal encoding. While we restrict our analysis to Gaussian noise statistics, no assumption is made on the nature of the noise temporal correlations. Our approach captures both parallel (dephasing) and single-axis transverse noise as special cases, and covers both Markovian and non-Markovian metrological regimes. In the properly squeezed limit, where a Holstein-Primakoff description is accurate, we identify a geometric noise-immunity mechanism: for known non-parallel signal and noise axes, an appropriate one-axis-twisted input encodes the signal in a quadrature that is metric-orthogonal to the direction of noise-induced diffusion. Irrespective of the noise temporal correlations, the resulting estimation precision exhibits Heisenberg scaling in the probe number. Optimal performance is achievable by measuring a single collective spin component. Imperfect knowledge of the noise axis produces a crossover from Heisenberg scaling at moderate probe number back to the known collective-dephasing bounds asymptotically.

Quantum decoherence induced by defects remains a major limitation for solid-state quantum technologies, yet predicting decoherence in realistic materials remains computationally challenging. Complex defect populations are often approximated as homogeneous spin baths, obscuring the role of defect-specific electronic structure and spin dynamics. Here, we develop a predictive framework for decoherence in diamond by combining first-principles electronic-structure calculations, quantum many-body spin-bath simulations, and experimental validation. The framework incorporates defect-resolved spin Hamiltonians and heterogeneous spin baths containing multiple paramagnetic defect species. Using diamond nitrogen-vacancy ensembles as a model platform, we investigate mixed nitrogen-, vacancy-, and hydrogen-related defect environments. We show that decoherence depends not only on defect density but also on defect identity and bath composition, whose distinct electronic structures, hyperfine interactions, and spin dynamics produce different coherence behavior. Heterogeneous defect populations can either suppress or enhance decoherence, producing trends unexplained by homogeneous-bath models. Magnetic-field-dependent Hahn-echo measurements on samples with different defect concentrations validate the framework. The calculations reproduce the observed coherence times and stretched-exponential decay behavior across a broad magnetic-field range and identify vacancy-related defects as critical contributors beyond the conventionally assumed P1 spin bath. By linking atomistic defect properties to quantum coherence, our framework provides a predictive route for identifying hidden defect environments and optimizing decoherence in defect-based quantum materials.

Reinforcement learning is a subfield of machine learning that studies how an agent interacts with an environment in order to extract as large a reward as possible. A standard approach to study such interaction is through Markov Decision Processes (MDPs) and the task of choosing an optimal policy — a function that tells the agent which action to take. In this work, we study two types of MDPs — finite-horizon and infinite-horizon discounted — and propose new quantum algorithms for computing approximate optimal policies. Our quantum algorithms are based on a new combination of standard value iteration and quantum subroutines like quantum mean estimation and quantum maximum finding, overall enhanced with techniques from sample-optimal classical algorithms. Our resulting query complexities improve upon previous works, thus approaching already established quantum lower bounds.

Neutral-atom quantum computers provide a scalable platform for large-scale quantum computation due to their all-optical control, room-temperature operation, and flexible lattice geometry. Although the favorable energy characteristics of these systems are well recognized, the relationship between system-level energy consumption, runtime, and computational fidelity remains poorly understood, limiting practical scheduling decisions. In this work, we develop a hardware-grounded analytical model that captures how energy and runtime scale with qubit utilization in neutral-atom systems. We introduce PaQit, a fidelity-aware qubit packing framework that integrates device-level Rydberg interaction physics with system-level scheduling to jointly optimize energy, runtime, and fidelity. By translating fidelity targets into packing decisions, PaQit identifies operating regimes that maximize parallelism while respecting interaction-driven crosstalk constraints. We validate the analytical framework using simulations of QuEra's analog Aquila system and digital Gemini system, as well as real hardware executions, demonstrating close agreement between the predicted trends and observed system behavior.

We consider the problem of Coulomb drag in interactively coupled quantum circuits built of adiabatic constrictions: quantum point contacts and short quantum-wire channels. The interplay of spatial confinement and magnetic field leads to a rich oscillatory response of the drag current as a function of gate voltage and magnetic field: drag peaks track the depopulation of magnetoelectric subbands, are asymptotically periodic in inverse field, and their visibility is controlled by the competition of temperature with the field-sharpened tunneling width of the constriction. We derive a closed expression for the linear drag conductance whose interaction kernel simplifies dramatically in the experimentally relevant limit of a long thermal length compared with the range of the interwire coupling, investigate the drag in the nonlinear regime, where the drag current measures the transconductance of the drive channel at any field, and discuss physically motivated models of dissipation-induced nonreciprocity of the drag signal. Extensions accounting for Zeeman splitting, interaction renormalization of the barrier transmission, backscattering at high field, and the frequency structure of the circuit coupling delineate how each mechanism imprints itself on the temperature dependence and lineshapes of the drag oscillations.

Increasing renewable-energy penetration heightens power-system variability and complicates disturbance containment. Controlled islanding mitigates cascading failures by partitioning a stressed network to limit disrupted power transfer while preserving each island's operational integrity, but this constrained partitioning problem is NP-hard. Although QAOA offers a complementary search strategy, limited near-term qubit capacity restricts conventional formulations. This paper presents a qubit-efficient hybrid quantum framework combining a physics-informed compact encoding with Lagrangian constraint handling and classical feasibility refinement. The encoding exploits grid structure while formally preserving the original feasible solution space and objective. For a fixed island count on sparse working graphs, the formulation reduces phase-separator and per-layer gate complexity from quadratic to linear scaling with system size. Tests on eight IEEE systems ranging from 9 to 89 buses and multiple quantum-provider backends produce feasible, high-quality islanding solutions under practical circuit and sampling budgets. Factorial ablation attributes resource and runtime improvements to the complementary effects of compact encoding and qubit-efficient constraint handling. Noise analysis shows stable solution quality under tested device noise, while landscape diagnostics reveal smoother, more consistently scaled QAOA cost surfaces and improved parameter-optimization behavior. These results offer a transferable approach for scaling constrained quantum optimization toward larger real-world applications on near-term hardware.

We study the energy spectrum of the degenerate $\chi^{(2)}$ Hamiltonian describing second-harmonic generation and parametric down-conversion at large total excitation numbers. Owing to the conservation of the total excitation number, the spectral problem reduces to the diagonalization of finite-dimensional tridiagonal matrices. Following the approach of Alvarez and Alvarez-Estrada, we map this problem onto an effective one-dimensional Schrödinger equation with a double-well potential. We then derive an implicit quantization condition for the energy levels near the top of the potential barrier and obtain two explicit asymptotic formulas applicable in complementary spectral regions. The ranges of applicability of the resulting formulas are established, and their accuracy is compared with exact matrix diagonalization and with the conventional JWKB approximation. The proposed approach provides an accurate analytical description of the energy levels near the center of the spectrum, where the conventional approximation loses accuracy.

Image formation in transmission electron microscopy (TEM) is governed by the coherent evolution of the electron wavefunction through the specimen and the objective lens. This physics maps naturally onto the gate model of quantum computation. We present QuScope, an open-source Python framework that expresses the complete TEM image-formation pipeline as quantum circuits. The $N\times N$ electron wavefunction is amplitude-encoded in $2\log_2 N$ qubits, and every optical element, including phase-grating transmission, Fresnel propagation between specimen slices, and the aberrated objective lens, is implemented as a diagonal unitary conjugated by quantum Fourier transforms. On this foundation, QuScope v0.2.0 implements validated imaging pipelines, covering conventional TEM under the phase-object approximation, full multislice CTEM and STEM for thick specimens. All quantum results reported here come from exact, noise-free statevector simulation of the circuits on classical hardware, and every result is validated against a classical twin implementation. Quantum and classical multislice exit waves agree to unit fidelity, and all physical constants are verified against standard references. We provide transpiled quantum-resource estimates for each algorithm, an analysis of the diagonal-synthesis bottleneck that governs near-term hardware execution, and a fully tested, documented, and pip-installable package. QuScope v0.2.0 is available at https://github.com/QuScope/QuScope.

Suppressing errors is the central challenge for useful large-scale quantum computing. While quantum error correction promises a viable solution to this challenge, existing codes typically suffer from trade-offs among encoding efficiency, error threshold, and hardware feasibility. Here, we introduce Cornucopia codes, a family of practical, hardware-efficient quantum low-density parity-check codes that achieve an ultra-high encoding rate exceeding $1/2$ while maintaining a pseudo-threshold exceeding $0.4\%$ under the standard circuit-level noise model. Inspired by recent affine-permutation-based code constructions and the long-range connectivity available in reconfigurable neutral-atom arrays, we adopt a structured code geometry in which the code layout, atom rearrangement, and syndrome-extraction schedule are co-designed. This structure enables nonlocal syndrome measurements through simple, parallel atom rearrangements. A complete syndrome extraction cycle measures all $X$- and $Z$-type checks in parallel with $12$ entangling layers, independent of the code size. The resulting threshold is comparable to those of the surface code and bivariate bicycle codes. In particular, a single code block $[[2844,1426,18]]$ encodes $1{,}426$ distance-$18$ logical qubits, achieving an extrapolated logical error rate of $2.6\times10^{-16}$ ($1.9\times10^{-31}$) per logical qubit per cycle, assuming the physical error rate of $0.1\%$ ($0.01\%$). By comparison, a bivariate bicycle code implementation would require more than $68{,}000$ physical qubits to encode the same number of logical qubits at a comparable logical error rate. These results bring demonstrations of ultra-low-overhead quantum error correction within the reach of near-term quantum processors.

Gauge invariance is central to modern physics and underpins quantum simulations of lattice gauge theories (LGTs). Existing quantum simulation approaches employ Gauss's law either to energetically suppress gauge-violating processes in analog platforms or to detect and discard gauge-violating outcomes in digital devices. Here we introduce a third paradigm, in which Gauss's law is used to dynamically generate the gauge theory itself from a substantially simpler Hamiltonian. Starting from a readily programmable three-body XXX model, we employ experimentally efficient single-qubit U(1) gauge symmetry-generator terms that induce the dynamics of a U(1) LGT. We implement this approach using 101 qubits on a 156-qubit IBM quantum processor and observe real-time dynamics in quantitative agreement with the target LGT while reducing the entangling-gate depth per Trotter step by a factor of five compared with a direct implementation. Our results establish gauge protection as a resource for Hamiltonian engineering rather than merely symmetry preservation, opening a scalable resource-efficient route towards digital quantum simulations of increasingly complex gauge theories in higher spatial dimensions.

Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law $N_c\sim\ln D$, where $D$ is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent $\alpha$, the onset becomes algebraic, $N_c\sim D^{\alpha/3}$. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for $\alpha<2$, in which the criticality threshold equation depends only on the system aspect ratio $N_c/D$. At $\alpha=2$ and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.

While the full non-stabilizerness (magic) of a quantum state contains local, basis-dependent contributions, a non-local formulation based on a minimization over local unitaries isolates the component associated with genuinely non-local correlations. Within such a framework, the Schmidt-gauge formulation of non-local magic provides a direct connection between genuinely non-local non-stabilizer correlations and the entanglement spectrum of quantum many-body states. Building on the exact Walsh–Hadamard representation introduced in our accompanying Letter, we develop the mathematical theory associated with this formulation. We extend the formalism to arbitrary bipartitions, prove the exactness of the Schmidt gauge for arbitrary $1\times N$ bipartitions, and derive general analytical properties of Schmidt-gauge non-local magic, including entanglement bounds, selection rules, and exact relations with the moments of the normalized Walsh spectrum. This representation also provides an interpretation of Schmidt-gauge non-local magic as a logarithmic inverse participation ratio normalized by a universal harmonic baseline, thereby relating it to excess delocalization in Walsh space. These results demonstrate that the Walsh–Hadamard representation reveals an underlying discrete harmonic structure that is hidden in the original spectral formulation and provides considerably more than an equivalent expression for Schmidt-gauge non-local magic. Rather, it furnishes the natural mathematical framework for its analytical investigation, placing the theory within the broader context of discrete harmonic analysis.

The eigenstate thermalization hypothesis (ETH) provides the prevailing framework for understanding quantum thermalization and ergodicity in isolated many-body systems. Yet, no general theory describes the continuous onset of ETH violation between the conventional ETH and its complete breakdown. Here, we introduce a two-parameter ansatz for the ETH violation that unifies and distinguishes two mechanisms: fading ergodicity and trapped ergodicity. While fading ergodicity captures the established route to ergodicity breaking, trapped ergodicity describes a distinct scenario in which ETH is violated in finite systems but restored in the thermodynamic limit. We test this framework in the spin-1/2 $J_1$-$J_2$ chains with on-site disorder and linear potential. In both cases, we find that the observed ETH violation is consistent with trapped ergodicity.

Universal hallmarks of quantum chaos—such as the ramp and plateau in the spectral form factor—and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models—including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons—can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects—such as baby-universe and wormhole condensates—which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-$d$ polynomial series to approximate $e^{-i\tau x}$ on $[-1,1]$ within error $\epsilon$. While $d\in\tilde{\mathcal{O}}(\tau)$ is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of $\approx e/2$.

High-frequency gravitational waves (HFGWs) provide a unique window into high-energy and early-universe physics, yet they evade traditional macroscopic interferometry. To bridge this detection gap, we propose a novel quantum-sensing paradigm utilizing superconducting transmon qubits embedded in resonant microwave cavities. Through the inverse Gertsenshtein effect, HFGWs propagating in a static magnetic field resonantly excite a cavity mode. By leveraging the characteristic spin-2 quadrupolar pattern of the induced electromagnetic field, we position qubits directly at the electric-field hot spots of the $\mathrm{TE}_{212}$ mode to act as localized sensors. Crucially, configuring this array as an entangled quantum register via symmetric Dicke states unlocks a fundamental scaling advantage: the signal probability scales quadratically with the qubit number, translating to a $h_{\min} \propto n_q^{-3/4}$ strain sensitivity scaling. We demonstrate that an idealized global register of 800 qubits reaches a strain sensitivity that surpasses standard macroscopic cavity-power limits by five orders of magnitude. Benchmarked against representative axion-haloscope parameters, this collective quantum enhancement decisively mitigates the profound Planck-scale suppression inherent to gravitational interactions, establishing a transformative framework for next-generation HFGW searches in the GHz band.

Equipped with the recently recognized symmetry structure of mixed states of matter and "strong-weak spontaneous symmetry breaking" (SW-SSB), we develop a quantum particle-vortex duality of emergent model-F and model-A hydrodynamics of 2d boson/rotor systems. This duality relation demonstrates that classical hydrodynamics of 2d bosons can be described in terms of the charge symmetry $U(1)_c$, but also equivalently in terms of the dual (emergent) 1-form symmetry $U(1)^{(1)}_e$, as well as $U(1)_v$ associated with the conservation of vortices. This duality provides a bridge between the hydrodynamics of matter and magnetohydrodynamics.