Papers

We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not seen any efficient implementation on quantum computers due to its dissipative nature and hence non-unitary dynamics. However, a recent work has shown an efficient implementation for certain cases on quantum computers using the Taylor series. Unlike previous work, our work provides a completely different algorithm for doing Laplace Transform using Quantum Eigenvalue Transformation and Lap-LCHS, very efficiently at points which form an arithmetic progression. Our algorithm can implement $N \times N$ discrete Laplace transform in gate complexity that grows as $O((log\,N)^3)$, ignoring the state preparation cost, where $N=2^n$ and $n$ is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity $O(N\cdot log\,N)$ for the same cases. Also, the circuit width grows as $O(log\,N)$. Quantum Laplace Transform (QLT) may enable new Quantum algorithms for cases like solving differential equations in the Laplace domain, developing an inverse Laplace transform algorithm on quantum computers, imaginary time evolution in the resolvent domain for calculating ground state energy, and spectral estimation of non-Hermitian matrices.

Many optical measurement techniques, such as light scattering from wavelength-scale particles or detecting motion from a surface with an optical lever, encode information in a complex radiation pattern. Extracting all available information is essential for many quantum-enhanced sensing protocols but is often impractical, as it requires many channels to spatially resolve the scattered signal. We present a new method to visualize how efficiently a practical measurement scheme captures the information available in the scattered light by mapping out the local contribution to the detection efficiency on the detector surface. We use this tool to experimentally optimize the free space measurement of the amplitude of motion of an optomechanical resonator with a quadrant photodiode. We show that blocking sections of the photodetector enhances sensitivity, counterintuitively yielding a significant improvement in detecting higher-order mechanical modes in the system. We also show how our method can be applied to light scattering measurements of small particles.

Neural Quantum States (NQS) are powerful variational representations of quantum many-body wavefunctions, yet their performance depends sensitively on the chosen basis. Using an exactly solvable one-dimensional Ising model, we show that local basis rotations leave the minimization landscape unchanged while relocating the exact ground state in parameter space. This provides a controlled framework to disentangle representational limitations from optimization-induced trainability effects. This geometric displacement, quantified through information-geometric measures, can steer optimization of shallow architectures toward saddle points and high-curvature regions. As a result, low energy errors may coexist with an incorrect wavefunction structure. By comparing energy and infidelity optimization within the same variational architectures, we show that optimization failure can persist even when the rotated target state remains representable. Our results identify a geometric mechanism contributing to basis dependence in NQS and motivate landscape-aware variational design.

Realizing and controlling the unconventional pairing featured by topological superconductors remains a central challenge. We introduce a cavity QED quantum simulator that engineers competing chiral $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ orders by tailoring cavity-mediated couplings between atomic pseudospins that emulate momentum-dependent pairing channels. The desired spatially inhomogeneous cavity-mediated couplings can be engineered in a 2D optical lattice using incommensurate cavity-lattice wavelengths naturally occurring in cavity QED systems. This minimal and fully tunable platform enables controlled state preparation and continuous measurement of superconducting order parameters. To probe this competition, the platform enables selective excitation and direct time-domain detection of Bardasis-Schrieffer modes—collective excitations of the subdominant pairing channel. The same competing channel can have an even stronger impact out of equilibrium, where a sudden quench can drive its growth and abruptly reshape the dominant superconducting order, despite the stability of the dominant order in equilibrium. Together, these results establish cavity QED as a controlled setting for revealing hidden competing orders and exploring nonlinear dynamics of topological superconductivity that remain elusive in solid-state and ultracold-atom systems.

We introduce the two-point propagation field (TPPF), a real-valued, phase-sensitive quantity defined as the functional derivative of the single-photon detection probability with respect to an infinitesimal opaque perturbation placed between the source and detection slits. The TPPF is analytically derived and shown to exhibit a stable, high-frequency sinusoidal structure with periods of 4~7 nm near the X-ray detection slit. This structure enables shot-noise-limited displacement detection with $\sim200 pm$ precision for 6 keV X-rays, using total photon counts on the order of $1\times10^{7}$ and detector photon counting as low as 287. Beyond displacement detection, the TPPF physically performs a Fourier-Radon transformation of the projection data, providing a pathway to non-iterative frequency-domain tomography. Two conceptual strategies, a central blocker and off-axis multi-slit arrays, are estimated to lower the required incident photon budget by more than one order of magnitude each, yielding combined reductions of two to three orders of magnitude with near-term detector development. The TPPF concept, originally developed in a perturbative study of single-particle propagation, bridges quantum measurement questions with practical high-resolution X-ray physics. This work provides the foundational physics required for future discrete sampling and 3D numerical reconstruction algorithms.

We introduce an entanglement witness that identifies off-diagonal long-range order (ODLRO) – a distinctive form of entanglement – in systems containing both fermionic and bosonic particles. By analyzing the particle-hole reduced density matrices of each subsystem, the approach detects ODLRO independently in both fermionic and bosonic sectors and identifies when long-range order develops across the entire mixed-particle system. The witness also quantifies the magnitude of ODLRO within each particle type, revealing how fermionic and bosonic correlations combine to form the total entanglement of the system, including a bosonic condensation of particle-hole pairs driven by many-body correlations rather than particle statistics. Using the Lipkin-Meshkov-Glick spin model, we show how the transition from ODLRO localized to one particle type to ODLRO shared by both particle types captures the onset of collective entanglement in a mixed-particle environment, providing new insight into systems where fermionic and bosonic correlations coexist.

Dissipative coupling refers to the effect where two systems interact with each other mediated by dissipation channels. Recent advances in controlling light-matter systems have opened new avenues to explore non-Hermitian effects arising from dissipative coupling, such as level attraction and anomalous dispersions. In this work, we perform a parametric study of these effects in a polariton system, i.e., a light-matter superposition, under both dissipative and coherent coupling. We characterize the effects of different sources of non-Hermitian behavior and analytically identify the conditions for the emergence of negative effective mass, exceptional points, and bound states in the continuum as a function of the light-matter detuning, the coherent-to-dissipative coupling ratio, and the relative decay rate of the non-interacting subsystems. We also analyze the classical limit of the polariton system within a non-Hermitian framework, employing coherent states.

Ge/SiGe heterostructures represent a promising platform for hosting various quantum devices such as hole spin qubits and Andreev spin qubits. However, the compatibility of such heterostructures with high-quality-factor microwave superconducting cavities remains a challenge due to defects in the material stack. In this work, we present an approach to enhance the coherence of cavity modes on a reverse-graded Ge/SiGe heterostructure, which consists of etching the full $\sim 1.6~\mathrm{\mu m}$-thick Ge/SiGe stack down to its starting high-resistivity Si substrate, in order to pattern superconducting cavities directly on it. We engineer the mesa step to be tapered, so that it can be easily climbed by the superconducting cavities to reach the quantum devices potentially hosted in the Ge quantum well. Using this approach, we observe internal quality factors of $Q_\mathrm{i} \approx 10000-20000$ for high-impedance frequency-tunable Josephson junction array resonators, limited by the junctions' fabrication, and $Q_\mathrm{i} \approx 100000$ for $50~\mathrm{\Omega}$ coplanar waveguide Nb lift-off resonators. These $Q_\mathrm{i}$ are preserved despite the overlap with the mesa structure in the climbing region, and are comparable to the ones obtained for identical resonators fabricated on a high-resistivity Si wafer reference. Thereby, this work paves a practical path toward superconductor-semiconductor hybrid devices, immediately applicable to emerging technologies on planar Ge.

Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, 22 (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.

The Hawking effect – the amplification of fluctuations at the horizon – has been detected in quantum fluids through real-space density correlations. However, real-space observables integrate over frequency, mixing distinct scattering channels into a single interference pattern and obscuring the spectral and entanglement structure of the emission. Here, we numerically compute momentum-space two-point correlations in a transcritical quantum fluid, using the truncated Wigner approximation applied to a realistic, driven-dissipative polariton system. We spectrally resolve the Hawking–partner as well as greybody factor channels and find that both carry comparable correlation strength, demonstrating that the well-known real-space “moustache” is an interference between these spectrally distinct contributions. The Hawking–partner channel is dominated by negative correlations, a direct signature of quantum-vacuum pair creation. All features reach amplitudes detectable in state-of-the-art experiments. Our results establish that the full three-mode output state must be considered for entanglement characterization, and provide a general framework applicable to any quantum fluid.

Designing parameterized quantum circuits (PQCs) that are expressive, trainable, and robust to hardware noise is a central challenge for quantum machine learning (QML) on noisy intermediate-scale quantum (NISQ) devices. We present a Domain-Aware Quantum Circuit (DAQC) that leverages image priors to guide locality-preserving encoding and entanglement via non-overlapping DCT-style zigzag windows. The design employs interleaved encode-entangle-train cycles, where entanglement is applied among qubits hosting neighboring pixels, aligned to device connectivity. This staged, locality-preserving information flow expands the effective receptive field without deep global mixing, enabling efficient use of limited depth and qubits. The design concentrates representational capacity on short-range correlations, reduces long-range two-qubit operations, and encourages stable optimization, thereby mitigating depth-induced and globally entangled barren-plateau effects. We evaluate DAQC on MNIST, FashionMNIST, and PneumoniaMNIST datasets. On quantum hardware, DAQC achieves performance competitive with strong classical baselines (e.g., ResNet-18/50, DenseNet-121, EfficientNet-B0) and substantially outperforming Quantum Circuit Search (QCS) baselines. To the best of our knowledge, DAQC, which uses a quantum feature extractor with only a linear classical readout (no deep classical backbone), currently achieves the best reported performance on real quantum hardware for QML-based image classification tasks. Code and pretrained models are available at: https://github.com/gurinder-hub/DAQC.

Non-Gaussian states of light are a critical resource for fault-tolerant quantum computing and enhanced metrology, but are typically faint and often obtained via post-selection. Here, we demonstrate the deterministic generation of a bright non-Gaussian state by introducing a Kerr nonlinearity to a macroscopic state of light called bright squeezed vacuum (BSV). To characterize the resulting state, we use a single-shot f-2f interferometer to sample its Husimi function. We observe a clear transformation from a 2D Gaussian distribution to an 'S'-shaped non-Gaussian profile, which is the direct statistical evidence of the intensity-dependent nonlinear phase. The negativity of the Wigner function, which is an intrinsic property of any pure non-Gaussian state, cannot be observed because BSV is a mixed state even under minute optical loss. However, we show that BSV can be considered as a mixture of pure squeezed coherent states, for some of which Kerr-induced Wigner-function negativity is quite tolerant to loss. This work bridges the gap between quantum optics and ultrafast nonlinear optics, opening a path to quantum applications that require high photon flux.

We present numerical calculations, and simulations performed on a Rydberg atom quantum simulator, of the adiabatic evolution of many-body quantum systems around a quantum phase transition. We demonstrate that the end-to-end transfer error, for a given process duration and dissipative losses, can be suppressed by adopting smooth initial and final scheduling functions for the Hamiltonian. We consider a one-dimensional mixed-field Ising model, as well as a chain of Rydberg atoms, and compare numerical calculations and experimental results for the end-to-end transfer error with different schedule functions. We show, in particular, that if the time dependent Hamiltonian is $n$ times differentiable with vanishing $1^{st}$ to $n^{th}$ order derivatives in the beginning and in the end, the infidelity with respect to the final adiabatic eigenstate scales as $1/T^{n+1}$ when evolving for time $T$.

Quantum computers are believed to solve a class of computational problems that are based on modular arithmetic faster than classical computers. Among the arithmetic building blocks, comparison of integer pairs is a primitive. Here we report its demonstration in the Reimei quantum computer at RIKEN, whose trapped-ion architecture provides all-to-all qubit connectivity together with high gate fidelities. We observe high success probabilities for bit widths n = 3, 5, 7, and 9: Under a conventional output-only success criterion we obtain 95% at n=9; under a stricter criterion additionally requiring the ancilla to be correct, the success is 69% at n=9. These results demonstrate reliable quantum comparison at scales far beyond those previously achieved experimentally, not only for comparators but also in the broader context of quantum arithmetic circuits.

The standard mean-field treatment of low-temperature superconductors leads to a square-root divergent density of states at the gap value. This feature can lead to unphysical logarithmic divergences in various quantities, such as currents and qubit transition rates. We revisit their possible regularization based on the proximity effect between two superconducting films with different gaps. We derive analytical approximations for the density of states in each superconducting film. We find that the smearing of the density of states grows with the gap asymmetry. As a concrete example, we discuss the regularization of transition rates in qubits with frequency close to resonance with the gap asymmetry between the two films, and the consequent smoothening of the jump discontinuity in the qubit frequency shift. Our results could also aid the development of superconducting detectors and related devices based on superconducting bilayers.

We derive the continuous spacetime limit of the one dimensional lazy discrete time quantum walk, obtaining explicit macroscopic evolution equations for a three state model in the presence of decoherence. While continuum limits of two state quantum walks are well established, an explicit continuous spacetime formulation for the lazy three state walk, particularly including noise, has not previously been constructed. Using an SU(3) representation of a Grover type coin together with a Lindblad formulation of decoherence acting either on the coin or the spatial subspace, we systematically expand the discrete dynamics in both space and time to obtain continuum master equations governing the coarse grained evolution. The resulting generators yield a genuine partial differential equation description of the walk, going beyond purely probabilistic or spectral correspondences. We show that the unitary limit is governed by a Dirac-type SU(3) Hamiltonian describing ballistic advection of left and right moving modes coupled by local symmetric mixing, with the rest state acting as an additional internal degree of freedom. Coin dephasing selectively damps internal coherences while preserving coherent spatial transport, whereas spatial dephasing suppresses long range spatial interference and rapidly drives the dynamics toward classical behaviour. This continuum framework clarifies how internal symmetry, rest state coupling, and distinct decoherence channels shape large scale transport in lazy open quantum walks, and provides a foundation for future extensions toward multichannel quantum transport models and quantum-inspired algorithms.

The rise of quantum computing/technology potentially introduces significant security challenges to cloud computing, necessitating quantum-resistant encryption strategies as well as protection schemes and methods for cloud infrastructures offering quantum computing time and services (i.e. quantum clouds). This research explores various options for securing quantum clouds and ensuring privacy, especially focussing on the integration of homomorphic encryption (HE) into Eclipse Qrisp, a high-level quantum computing framework, to enhance the security of quantum cloud platforms. The study addresses the technical feasibility of integrating HE with Qrisp, evaluates performance trade-offs, and assesses the potential impact on future quantum cloud architectures. The successful implementation and Qrisp integration of three post-quantum cryptographic (PQC) algorithms demonstrates the feasibility of integrating HE with quantum computing frameworks. The findings indicate that while the Quantum One-Time Pad (QOTP) offers simplicity and low overhead, other algorithms like Chen and Gentry-Sahai-Waters (GSW) present performance trade-offs in terms of runtime and memory consumption. The study results in an overall set of recommendations for securing quantum clouds, e.g. implementing HE at data storage and processing levels, developing Quantum Key Distribution (QKD), and enforcing stringent access control and authentication mechanisms as well as participating in PQC standardization efforts.

We investigate how optical second-order cross correlations witness the quantum features of a prototype donor-acceptor light-harvesting unit. By considering a pair of detuned two-level emitters electronically coupled and incoherently driven to a non-equilibrium steady-state, we gain insight into how electronic quantum properties such as exciton eigenstate delocalization, coherent energy transfer and steady-state electronic coherence, are manifested in the joint probability of emission or optical second-order cross correlation. Specifically, we show that the frequency associated with oscillations present in time-resolved second-order cross correlation functions quantifies not only the time scale of coherent energy transfer but also the degree of delocalization of the exciton eigenstates. Furthermore, we show that time-resolved cross correlations directly witness steady-state electronic coherence. Our work strengthens the idea that measurements of the intensity quantum cross correlations can provide distinctive signatures of the quantum behavior of biophysical emitters.

Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods are solved numerically, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a framework for extracting rigorous certified bounds on non-commutative optimization problems from inexact numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.

Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one $k$-outcome measurement, as well as the case of four dichotomic qubit measurements.

Digital-analog quantum computing with two-level systems is a computational paradigm that combines an analog Hamiltonian with single-qubit gates to achieve universality. We extend this framework to $d$-level systems by conjugating an analog Hamiltonian block with single-qudit gates drawn from the Weyl-Heisenberg basis, which provides a natural set of operations for qudit architectures. More specifically, we propose a protocol to simulate arbitrary two-body Hamiltonians with at most $O(d^4 n^2)$ analog blocks. The power of this approach is illustrated by the simulation of many-body qudit spin Hamiltonians including magnetic quadrupolar terms.

Entanglement is central to quantum physics, yet detecting and exploiting it in non-Gaussian systems remains a major challenge. In continuous variable platforms, standard inseparability criteria based on Gaussian statistics-such as the Duan-Simon criterion-fail when quantum correlations are encoded in higher moments of the field quadratures. Here we introduce a framework for detecting non-Gaussian entanglement using higher-order quadrature cumulants. In the Gaussian limit, the lowest order condition reduces to the Duan criterion, while higher-order violations reveal entanglement inaccessible to second-order methods. We experimentally demonstrate a tomography-free certification of inseparability in photon-subtracted squeezed states of light, where Gaussian witnesses fail despite the presence of entanglement. We further show that such higher-order inseparability enables enhanced quantum teleportation of Wigner negativity as compared to Gaussian protocols. These results identify higher-order cumulants as natural observables for non-Gaussian entanglement and open new routes to harnessing non-Gaussian resources in continuous variable quantum technologies.

The advent of high-intensity ultrafast laser pulses has opened new opportunities for controlling and designing quantum materials. In particular, terahertz (THz) pulses can resonantly drive optical phonon modes, enabling dynamic manipulation of lattice degrees of freedom. In this work, we investigate the ultrafast quantum thermodynamics of optical phonon mode driven by a THz pulse by treating the phonon as an open quantum system coupled to a thermal environment within a Caldeira-Leggett-type framework. We derive the quantum heat current between the phonon and the bath and analyze its behavior under realistic pulse protocols. Our results demonstrate that ultrafast laser driving can reveal and even induce significant deviations from the commonly adopted Markovian approximation, thereby providing a pathway to probe and control non-Markovian dissipation in driven solid-state systems.

Use cases for emerging quantum computing platforms become economically relevant as the efficiency of processing and availability of quantum computers increase. We assess the performance of Restricted Boltzmann Machines (RBM) assisted by quantum computing, running on real quantum hardware and simulators, using a real dataset containing 145 million transactions provided by Stone, a leading Brazilian fintech, for credit card fraud detection. The results suggest that the quantum-assisted RBM method is able to achieve superior performance in most figures of merit in comparison to classical approaches, even using current noisy quantum annealers. Our study paves the way for implementing quantum-assisted RBMs for general fault detection in financial systems.

We demonstrate the implementation of $\Lambda$-enhanced gray molasses cooling on the D$_2$ line of $^{85}$Rb atoms in an optical tweezer array. This technique yields lower atomic temperatures of 4.0(2) $\mu$K compared to red-detuned polarization gradient cooling, and consequently extends the $T_2^*$ coherence time of the hyperfine clock qubit by a factor of 1.5. The method is alignment-free and can be readily implemented on laser beams used for magneto-optical trapping, as it only requires frequency and phase modulation control. Our experimental observations are corroborated by a numerical model based on a semi-classical force approach extended to a four-level system, including two hyperfine states of the upper manifold that are 120 MHz apart.

In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength $D$ show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever $D \gg \hbar^{\frac43}$, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition $D \gg \hbar^{2}$ always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling $D \sim \hbar^{\frac43}$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an $\hbar$-independent quantum-classical discrepancy at the Ehrenfest time whenever $D \ll \hbar^{\frac43}$, even for $\hbar$-independent "macroscopic" smooth observables.

We investigate how interactions affect the quantum state of scalar perturbations during inflation and the quantum correlations they may exhibit. Focusing on the case of scalar perturbations in single-field inflation, we model interactions using a Lindblad equation with a non-unitary contribution quadratic in the scalar perturbations, and of parametrisable amplitude and time dependence. We compute the quantum state of these interacting perturbations, which is fully described by its purity and squeezing parameters. First, we show that, in most of the parameter space, not only the purity but also the squeezing parameter is significantly reduced by interactions. Second, we show that this de-squeezing induced by the interactions, on top of the purity loss, causes a further suppression of quantum correlations. We thus emphasise that the quantum or classical character of the correlations exhibited by the perturbations cannot be correctly determined by computing the effect of interactions on the purity alone. Since the phenomenological framework adopted in this paper encompasses a wide class of possible interactions, our results provide general insights into the nature of decoherence processes in primordial fluctuations.

This paper presents an on-chip nonreciprocal space-time-periodic Josephson frequency multiplier that up-converts a single low-frequency input tone into a comb of high-order harmonics while enforcing directional signal propagation along the bus. We show that the harmonic content and bandwidth of the generated comb can be controlled through the applied DC and RF flux bias, and that individual output channels can be selectively addressed by tuning the bus operating point. We characterize the resulting nonreciprocal isolation between channels and discuss the associated design trade-offs, including the influence of modulation depth on harmonic-generation efficiency and back-reflection. The proposed device delivers multiple mutually isolated frequency channels from a single low-frequency source and it is well suited to applications requiring multiplexed delivery of tones to spatially distributed, frequency-addressable loads; we discuss multiplexed control of superconducting qubits as one such potential application area. Master-equation simulations of a four-qubit array driven by the harmonic comb, together with a one-dimensional line model of the directional conversion, quantify the resulting crosstalk and directional asymmetry; all results are simulations, and no experimental demonstration is claimed. The proposed architecture offers a compact, integrated route toward on-chip frequency multiplexing and directional isolation for cryogenic microwave systems.

This study investigates quantum computing approaches for solving the windfarm layout optimization (WFLO) problems formulated as a quadratic unconstrained binary optimization (QUBO) problem. We investigate two encoding methods that require fewer than one qubit per grid point: the previously developed Pauli correlation encoding (PCE) and a novel single-qubit operator encoding (SQOE). These methods are tested on three windfarm configurations - two from prior WFLO scaling studies and a new real-world model based on an existing windfarm in Wales. The improved encoding methods allow us to solve WFLO problems on $9\times 9$ grids using up to 20 qubits on a quantum computer simulator. The results show that both encoding methods perform competitively and demonstrate favorable scaling characteristics across the tested systems.

We solve the stationary Schrödinger equation for a particle confined to a 3D spherical wedge – the region $\{(r,\theta,\phi): 0 \leq r \leq R,\, 0 \leq \theta \leq \pi,\, 0 \leq \phi \leq \Phi\}$ with Dirichlet BCs on all surfaces. This exactly solvable constrained-domain model exhibits spectral reorganisation under symmetry-breaking BCs and provides an operator-domain viewpoint on angular momentum quantisation. We obtain three main results. First, the stationary states are standing waves in the azimuthal coordinate and consequently are not eigenstates of $\hat{L}_z$; we prove $\langle L_z \rangle = 0$ with $\Delta L_z = \hbar n_\phi\pi/\Phi \neq 0$, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number $\mu = n_\phi\pi/\Phi$ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter $\nu$ to satisfy $\nu - \mu \in \mathbb{Z}_{\geq 0}$. This regularity constraint yields a hierarchy: sectoral solutions ($\nu = \mu$, satisfying the first-order highest-weight condition) exist for any real $\mu > 0$, while tesseral and zonal solutions require integer steps, appearing only when $\mu$ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification $\phi \sim \phi + 2\pi$ that defines the full-sphere Hilbert space domain; modified boundary conditions yield a reorganised spectrum with non-integer effective angular momentum. The model clarifies the distinct roles of single-valuedness (selecting integer $m$ via azimuthal topology) and polar regularity (selecting integer $\ell \geq |m|$ via analytic constraints) in the standard quantisation of orbital angular momentum.