Papers

We study rank-one delta interactions supported by small normal deformations of an \(n\)-dimensional hypersphere \(S_R^n\subset\mathbb{R}^{n+1}\). The interaction is defined by the normalized surface measure on the support and therefore corresponds to the rotationally invariant sector of the hyperspherical shell problem. For the undeformed hypersphere, we derive the bound-state equation for \(E=-\nu^2\) in terms of the modified Bessel product \[ I_{\frac{n-1}{2}}(\nu R)K_{\frac{n-1}{2}}(\nu R). \] We then consider an inward normal deformation \[ X_\varepsilon(\omega)=(R-\varepsilon h(\omega))\omega, \qquad \omega\in S^n, \qquad 0<\varepsilon\ll1. \] Our main result is that, to first order in \(\varepsilon\), the bound-state energy depends only on the average normal displacement \[ \langle h\rangle = \frac1{|S^n|}\int_{S^n}h(\omega)\,d\Omega_n(\omega). \] Equivalently, the deformed hypersphere is spectrally equivalent, up to \(O(\varepsilon^2)\), to a round hypersphere with effective radius \(R-\varepsilon\langle h\rangle\). In particular, mean-zero deformations do not change the rank-one bound-state energy at first order.

The preparation of ground states of Hamiltonians with symmetries that are fundamentally different than those of the underlying device is a central challenge for quantum technologies. Here, we address this challenge with an analog protocol for the iconic generalized cluster Hamiltonian with the translational symmetry resultant from periodic boundary conditions based on a magnetization-preserving interaction on a linear geometry with open boundary conditions. Our results show that this goal can be achieved in a control duration that scales only moderately with the number of spins and Hamiltonian interaction complexity.

Optical quantum dots (QDs) are central to photonic quantum technologies, with fabrication approaches tailored to different spectral ranges. A key challenge, however, is the realization of a unified platform$\unicode{x2014}$a single growth method combined with a host material offering a designable architecture$\unicode{x2014}$enabling wavelength tunability across the full emission range and co-integration of multiple quantum dots. Here, we introduce AlGaAs nanowires as a universal host for GaAs, InGaAs, and InAs QDs. Building on our previous demonstration of high-quality GaAs QDs, we realize InGaAs QDs with tunable emission by varying the growth duration from 2 to 5 s, achieving emission at 780 and 920 nm. We further showcase the platform's versatility for multi-quantum-dot devices by co-integrating two InGaAs QDs, as well as GaAs and InGaAs QDs within a single nanowire. Finally, we initiate a first step toward pure InAs QDs by growing a pristine InAs segment on AlGaAs nanowires, demonstrating material compatibility.

In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $\gamma$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}{\gamma}$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.

Suppose we can apply the unitary $U=e^{i H}$ for some Hamiltonian $H$, and are given access to a unitary that prepares a guiding state promised to have overlap at least $\gamma >0$ with the ground space of $H$. Our goal is to estimate the ground-state energy of $H$ within additive error $\delta > 0$ and success probability at least $1-\varepsilon$, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound $O(\log(1/\varepsilon)\log(1/\gamma)/\gamma\delta)$ was known, and was improved to $O(\log(1/\varepsilon)/\gamma\delta)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $\delta,\gamma,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $\Omega(\log(1/\varepsilon)/\gamma\delta)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/\gamma^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $\delta$ between its first and second eigenvalue; and for ground-state preparation, where $\delta$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $\Omega(\log(1/\varepsilon)/\gamma\sqrt{\delta})$ for this case.

Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for $d$-dimensional states of rank at most $r$: producing $M$ additional copies with worst-case fidelity at least $1-\varepsilon$ requires and is achievable with $N=\Theta(Mrd/\varepsilon)$ input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For $M=1$, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.

Quantum simulators based on Bose-Einstein condensates (BECs) provide a powerful platform to study relativistic quantum-field phenomena in the lab and in particular emulate relativistic analog particle detectors. However, such detectors typically couple to excitations beyond the acoustic phonon regime, so that the nonlinear Bogoliubov dispersion introduces corrections that cause the effective relativistic description to break down. We propose to overcome this limitation using an intrinsically momentum-selective Unruh-DeWitt detector model in which a tweezer-trapped atom couples to the condensate through a multimode optical cavity. Although the atom is effectively pointlike, the cavity-mediated interaction kernel causes it to sample density fluctuations over a controllable finite region of the BEC equivalent to a finite-size detector. The effective detector size is tunable via the cavity mode structure determining a controllable momentum cutoff that filters out nonphononic excitations. We analytically calculate the response of such a detector for uniform linear acceleration and uniform circular motion and show that this momentum selectivity can enhance the signal at experimentally accessible accelerations. Compatible with existing cavity-QED platforms our scheme thus provides a versatile setting for analog quantum-field measurements or relativistic quantum-information protocols.

We develop a class of amplitude estimation algorithms with tunable circuit depth $M$ and circuit repetitions $N$, requiring neither ancilla qubits nor controlled Grover operations. For additive error $\epsilon$ in the Grover angle, they attain the nearly optimal query-depth tradeoff $M^2N\in\widetilde{O}(\epsilon^{-2})$ uniformly over $\lambda\in[0,\pi/2]$, spanning the full range from classical sampling at $M=1$ to the Heisenberg limit at $M=\Theta(\epsilon^{-1})$. While prior depth-tunable work establishes comparable angle-accuracy guarantees only away from the boundaries $\lambda\to0$ or $\pi/2$ or at discrete depth tradeoffs, our guarantee extends to both boundaries, so the quantum speedup persists there rather than degrading to classical sampling. Numerical experiments confirm the predicted uniform angle accuracy and show low overhead in practice, making them strong candidates for practical amplitude estimation in the early fault-tolerant regime, with applications such as overlap certification, trial-state verification, and Monte Carlo methods.

For photonic quantum technology applications, sources capable of emitting photons with indistinguishability close to unity are essential. Ideally, these sources should not require demanding cooling systems. Here, we present temperature-dependent two-photon-interference measurements on photons produced by the radiative decay of the negative trion in a low-noise GaAs quantum dot, which are in quantitative agreement with theoretical calculations accounting for carrier-phonon interactions and coupling to excited states. While at at the lowest explored temperatures the emission linewidth reaches values only 6(2) % above the Fourier limit and the indistinguishability I between subsequently emitted photons reaches 0.966(6), the latter drops to 0.05(4) at 55 K. We show that this loss can be explained with the coupling with energetically close excited trion states and suggest that the photon indistinguishability at elevated temperatures can be increased by employing Purcell enhancement of the emission rate or by increasing the energy separation of the excited states. Using cavity-enhanced emission, we experimentally verify the first route and demonstrate an improvement in photon indistinguishability from 0.314(25) to 0.80(3) at 32 K, which - to our knowledge - is the highest reported value at such temperature.

Recent years have seen a rapid growth in literature presenting new methods for simulating non-Clifford quantum circuits with classical hardware. These methods span a range of approaches, including stabiliser decomposition and tensor contraction techniques, varying in efficiency depending on circuit class, depth, non-Clifford gate count, and other metrics. A notable limitation of this literature is the lack of a standardised approach to benchmarking, with each new paper outlining its own specification, simulating its own set of circuits on the authors' own hardware. This paper seeks to address this issue by presenting a standardised and canonical benchmark suite and infrastructure for quantifying the efficiency of non-Clifford classical simulators, with a consistent dataset of circuits and providing consistent (virtual) hardware, thereby enabling a fair comparison of results.

Gate teleportation allows us to implement a nonlocal unitary using local operations, classical communication (LOCC), and a shared entangled state. Known deterministic teleportation protocols consume at least one full ebit and achieve optimal entanglement consumption only for Clifford gates. Here, we present a deterministic LOCC protocol for implementing the two-qubit controlled-phase gate $U_{\phi}=\mathrm{diag}(1,1,1,e^{i \phi})$ with $\phi \in [0,\pi]$ whose entanglement consumption is close to optimal for every $\phi$. In particular, vanishing rotation angles require vanishing entanglement.

We present QisMC, the first quantum model checker dedicated to debugging Qiskit programs. On the theoretical side, we introduce the notion of quantum-classical transition system and a quantum computation tree logic (qCTL) grounded in Birkhoff-von Neumann logic for modeling the behaviors and specifying the properties of Qiskit programs, respectively. On the implementation side, QisMC provides an end-to-end framework encompassing transition system generation, logical formula representation, and model checking algorithms, and it efficiently performs image computation based on decision diagrams. These methods and design principles give QisMC advantages over previous Qiskit program debuggers, including a unified property specification language, a fully automated and exhaustive verification process, and the ability to generate counterexamples. Extensive illustrative examples and benchmark evaluations demonstrate the practicality, efficiency, and scalability of QisMC in verifying realistic quantum programs.

In distributed fault-tolerant quantum computing, entanglement and magic are essential resources for quantum communication and universal fault-tolerant computation, respectively. Although they are usually treated as distinct resource currencies, whether they admit a unified resource-theoretic description remains an open question. Here, we introduce the distributed resource theory of entanglement and magic (DREAM). In this framework, the free states are convex mixtures of product local stabilizer states, and the free operations are local stabilizer circuits assisted by classical communication (LSCC). We show that DREAM contains nontrivial resource states that are neither entanglement nor magic, so it is strictly richer than treating the two resources independently. Surprisingly, such resources can enable the teleportation of magic states between distant parties without consuming entanglement, revealing a counterintuitive form of resource teleportation mediated entirely by separable states. We further generalize this result to quantum networks and investigate general quantum-state teleportation under LSCC. We show that the shared resource under DREAM is closely related to the teleportation capability, quantified by the magic of the teleported state. Our work establish a systematic framework for studying distributed quantum resources and uncover intrinsic relations among distinct resources within a unified resource theory.

We investigate the cluster Ising chain with an additional nearest-neighbor interaction using tensor-network methods and a weak-coupling field theory. Without the interaction, the Jordan-Wigner transformation decomposes the model into a triplet of identical Majorana chains related by an exact $O(3)$ flavor symmetry. Their common mass vanishes at the $SU(2)_2$ Wess-Zumino-Novikov-Witten critical point with central charge $c = 3/2$ separating the symmetry-protected topological cluster phase from a ferromagnet. The interaction reduces $O(3)$ to its cyclic subgroup $C_3$, splitting the triplet into a singlet and a doublet whose gaps close separately, producing a critical bifurcation into Ising ($c = 1/2$) and Gaussian ($c = 1$) critical lines. A second ferromagnetic phase opens between these critical lines for repulsive interactions and a disordered phase for attractive ones. The Gaussian line then separates two Landau-incompatible ferromagnets, realizing a deconfined quantum critical line with emergent $O(2)$ symmetry, along which our independently extracted exponents vary continuously yet satisfy the parameter-free relation $\beta = (2\nu - 1)/4$ of the eight-vertex weak universality class. At stronger repulsion this line opens into an extended gapless floating phase with incommensurate algebraic correlations, entered through Berezinskii-Kosterlitz-Thouless transitions. All of these phases and the transitions between them are captured by the weak-coupling theory. Beyond its regime of validity, our simulations reveal a translation-symmetry-breaking antiferromagnet reached through first-order transitions.

In this paper, we study the effect of a which-path memory ancilla on the three-slit interference pattern within the framework of the bright–dark state description [Phys. Rev. Lett. 134, 133603 (2025)]. Whereas two slits give one bright mode, a photonic mode that couples to the detector atoms, and one dark mode, which does not couple to the detector atoms, three slits lead to one detector-coupled bright mode and a two-dimensional dark-mode subspace in the three-dimensional path space. We first discuss the classical three-slit interference pattern in terms of probability leakage into the dark subspace. We then study the von Neumann entropy $S_D$ of the dark subspace and the coherence measures of the reduced photonic state obtained by tracing out the memory: dark-sector coherence $C_D$ and bright–dark coherence $C_{BD}$. These quantities bring out the internal quantum structure of the two-dimensional dark subspace. We show that, whereas $C_D$ is not by itself a basis-independent physical observable, $C_{BD}$ is basis-invariant. We establish that, for two paths, $C_{BD}$ is fixed entirely by the populations and the single pairwise coherence, whereas for multiple paths, $M\geq 3$, it acquires a genuinely multipath contribution generated by asymmetry among the pairwise which-path overlaps. Finally, we distinguish the part of the path-coherence loss that is recoverable through a suitable measurement of the memory from the irreducible coherence deficit imposed by an uncontrolled environment.

Quantum noise is expected to degrade quantum machine learning by driving circuits away from their noiseless implementations. Yet recent studies show moderate noise can reduce testing error, a behavior unexplained by weak-noise perturbative error accumulation or strong-noise trainability collapse. Here we develop a statistical learning theory connecting microscopic noise processes to macroscopic learning performance. At its heart is a noise-order purity parameter, derived from a surrogate model analysis, that predicts the noise-induced reduction in model complexity and the consequent reduction in the generalization gap. Noise simultaneously increases prediction bias. Their competition explains the intermediate-noise regime left open between these limits. It produces a finite-noise optimum whose location depends on the learning setup and can disappear in the large-sample limit. Numerical experiments validate these predictions. Noise programming can move a model towards this optimum. These results make the non-monotonic effect of noise predictable and provide a route to harness it.

In this letter we introduce a theoretical framework for the simulation of ultrafast transient absorption pump-probe spectra of molecular polaritons. We derive and implement the cavity quantum electrodynamics (QED) evolution equations of polaritonic states within the framework of the quasi-classical doorway-Window pproximation, hereto referred cQUEDA /sikeda/. This framework uses outputs from mixed quantum-classical dynamics simulations in the absence of the cavity. Then by including cavity parameters accounting for cavity rate loss, coupling strength, and frequency detuning, we simulate transient absorption pump-probe spectra. Consequently, we address one of the main standing problems of cavity QED, the simulation of polaritons ultrafast dynamics and nonlinear optical properties. We demonstrate the performance of our method by computing the ground-state bleach (GSB), stimulated emission (SE), and excited-state absorption (ESA) contributions of transient absorption pump-probe spectra of pyrazine strongly coupled to a cavity. The cQUEDA is an on-the-fly computationally efficient framework with low computer requirements: the pyrazine calculations, for example, took minutes on modern laptops. cQUEDA offers wide-ranging applicability and can be generalized to model diverse nonlinear spectroscopic signals and quantum optics responses.

The Quantum Internet operates on entanglement, a non-local, non-copyable, stateful network resource, which motivates protocol organization beyond classical layering. We present a first executable specialization of the Dynamic Kernel/QPacket logic from the beyond-layering protocol suite, targeting entanglement distribution over a linear quantum repeater chain. The implementation builds on Q2NS, an ns-3-based quantum-network simulation module available through the ns-3 App Store. It realizes QPacket meta-headers with service intent and append-only action-commit stamps processed by node-local Dynamic Kernels organized as a Planner–Executor–Engine pipeline, while being deliberately scoped to an analytically verifiable service and policy. Within this scoped setting, we study node heterogeneity through a link-preparation policy that accounts for pre-distributed entanglement and uneven entanglement-generation support across nodes, including delegation via QPacket forwarding. Simulations verify analytical link-resolvability models and expose signaling load, forwarding behavior, and QPacket meta-header growth. Results show that QPacket overhead is shaped by more than just encoding, including policy choices and available network resources. Overall, this study demonstrates how the Q2NS/ns-3 substrate can support reproducible, policy-specific evaluation of quantum-native protocol-suite concepts.

Nonlinear terms present a fundamental challenge for quantum computational fluid dynamics, as their implementation on inherently linear quantum hardware typically requires resource-intensive workarounds that limit scalability to large-scale simulations. We present a hybrid quantum-classical tensor network algorithm that addresses this bottleneck by combining variational time-stepping with quantum tensor programming to efficiently compile operators and time-dependent fields into quantum circuits. Within a probabilistic framework, we replace prior state-based nonlinear implementations with tensor-based block encodings, stabilizing success probabilities that otherwise decay exponentially with system size. Benchmarking on turbulent flow fields demonstrates that the algorithm maintains high success probabilities and moderate measurement overhead across increasing Reynolds numbers and grid resolutions. Compared to fully classical tensor network solvers, our hybrid approach yields substantial reductions in both memory footprint and computational cost, establishing a scalable pathway toward practical quantum advantage in scale-resolving CFD simulations.

Quantum program analysis remains challenging due to the exponentially large state space of quantum programs and the difficulty of precisely characterizing their execution behavior. In particular, non-Clifford operations introduce additional complexity that limits the applicability of stabilizer-based techniques. Although stabilizer representations provide compact descriptions for Clifford circuits, their limited expressiveness prevents them from directly supporting general quantum program analysis. In this work, we propose StabQ, a symbolic execution framework for quantum program analysis based on stabilizer representations. StabQ extends stabilizer-based symbolic execution beyond Clifford-only programs by introducing a symbolic state representation that captures and propagates quantum state evolution while preserving execution semantics. Based on this representation, StabQ constructs a Tableau Chain that represents the evolution of intermediate symbolic states throughout program execution and enables reusable analysis of quantum program executions. Furthermore, StabQ incorporates tableau consolidation and global-phase recovery mechanisms to mitigate symbolic state growth during execution. Building upon the Tableau Chain, StabQ supports multiple quantum program analysis tasks, including quantum state reconstruction, entanglement analysis, and Clifford-property detection. We evaluate StabQ on three benchmark suites—Algorithms, MQT Bench, and QASMBench. The results demonstrate that StabQ constructs semantically consistent symbolic models, accurately preserves quantum state evolution, and effectively supports downstream analysis tasks across diverse quantum programs.

We experimentally demonstrate an entanglement-based Quantum-Secure Time Transfer (QSTT) system in an emulated low Earth orbit satellite-to-ground channel using a type-0 Sagnac-based entangled-photon source. Our new QSTT system delivers a finite Quantum Key Distribution (QKD) key rate of approximately $3~$bits$~\text{s}^{-1}$ with a $10^{-10}$ security parameter, providing a $30$-fold increase in the QKD key rate compared to the state-of-the-art entanglement-based QKD system delivered by the Micius satellite. Beyond this high key rate outcome, novel to our QSTT system is a GPS-free clock synchronization, optimized use of QKD, embedded post-quantum security, and obfuscation of the system configuration via use of a pre-shared key. Collectively, these enhancements deliver the most efficient and secure deployment of QSTT to date, and point the way forward to high-accuracy ultra-secure time transfer in space.

Limited qubit connectivity is a central architectural constraint in superconducting quantum processors, whose planar layouts require additional gates to mediate interactions between distant qubits. Here, we use the AshN control scheme, where rich two-qubit control on every nearest-neighbour pair allows a logical interaction and the required qubit routing to be merged into a single native operation, effectively transforming a sparse hardware graph into a more connected computational architecture. For the benchmark instances studied, the resulting synthesis capability enables reliable execution on constrained one- and two-dimensional lattices, with compiled two-qubit gate counts approaching those of an all-to-all-connected reference. Across seven benchmark circuits on one- and two-dimensional topologies, the AshN-based implementation achieves geometric-mean reductions of $45.2\%$ and $43.7\%$ in two-qubit gate count compared with controlled-Z-based compilation, respectively. Using AshN gates, we prepare an eight-qubit two-excitation Dicke state with a fidelity of $0.736$ and certify its genuine multipartite entanglement using a fully positive-partial-transpose witness, whereas the same witness does not certify entanglement for the CZ-based implementation. The state fidelity and entanglement certification remain robust across the tested lattice configurations, including those with up to three connectivity defects. Our work establishes native-gate engineering as a practical approach to mitigating connectivity constraints.

Determining whether a mixed quantum state is uniquely determined among all states by its k-body marginals (k-UDA) is a fundamental problem in quantum system certification. We develop a range-based approach to this problem by analyzing the structure of the range of the global state. For three-qubit states, we show that states with GHZ-SLOCC-free ranges are 2-UDA at ranks one, three, and four. We derive a necessary and sufficient range criterion for rank-two 2-UDA states and reduce it to a finite quadratic-form test. To cover the remaining range configurations, we formulate an exact range-restricted semidefinite programming criterion and extend it to arbitrary finite-dimensional tripartite states. We also show that every three-qubit state of rank at least five is not 2-UDA, and further extend high-rank obstructions to multipartite systems. For a channel-based multipartite family, we characterize exactly when a state is $(n-1)$-UDA and show that lower-order marginals never suffice. Finally, we apply these results to the certification of genuine multipartite entanglement.

The usual WKB analysis for quantum tunneling applies when the tunneling action is large, as it is for tall, wide potential barriers. In contrast we analyze tunneling when the action is small, as it is for tunneling across a tall, thin barrier. We develop a perturbative analysis where the control parameter is the inverse of the area under the potential barrier and apply our technique to several examples in $1+1$ dimensions. In resonant situations for bound particles we find that the tunneling probability grows with time as $\propto t^2$, while in non-resonant situations it grows linearly with time. We evaluate not only the tunneling probability but also the time-dependent tunneling wavefunction for a particle that escapes to infinity, i.e. from a quasi-bound state to the continuum.

BosonSampling is among the most prominent candidates for demonstrating quantum advantage. However, while the hardness of BosonSampling relies on photon-number-resolving detection, many experimentally relevant settings and applications instead use binary readout based on threshold or parity measurements, whose computational complexity has not yet been rigorously characterized. In this work, we investigate the computational complexity of BosonSampling with threshold and parity measurements in the linear-mode regime, where the number of modes scales linearly with the number of photons and is most relevant to current experiments. In particular, we establish average-case #P-hardness of estimating typical output probabilities in threshold and parity BosonSampling, a crucial ingredient in proving the classical hardness of the corresponding sampling problems. The resulting imprecision bounds match those obtained in prior hardness results for standard photon-number-resolving BosonSampling in the linear-mode regime. The key technical ingredient is a Fourier-coefficient extraction method, induced by coherent beam-splitter rotations, that extracts hidden hard components within coarse-grained output probabilities. These results indicate that the hard output-probability structure of photon-number-resolving BosonSampling can persist under natural binary coarse-grainings, even in collision-dominant regimes.

In this work, we study quantum CSS codes with transversal $T$ gates. Here, $T$ gate transversality is meant in the strongest sense; the application of physical $T$ to every physical qubit yields logical $T$ on every logical qubit, without Clifford corrections. Despite the importance of the $T$ gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal $T$ gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal $T$ gate. These are also the first codes achieving $\gamma \to 0$ for a code with a transversal $T$ gate, where $\gamma$ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.

The position–momentum correspondence combines several mathematically distinct identifications that are often conflated. For the additive group $(\mathbb R,+)$, every continuous character has the form $x\mapsto \mathrm e^{ikx}$. After choosing a coordinate $p=\alpha k$ on the dual group, one obtains $\mathrm e^{ipx/\alpha}$, so Pontryagin duality alone does not identify a particular dual coordinate with physical momentum. For the corresponding unitary Fourier transform, multiplication in position space is transported to convolution in momentum space, while the diagonal distribution is transported to the specific composition with the addition map $(p,q)\mapsto p+q$; it is not transported to pointwise multiplication in momentum space. The family $\hat P_\alpha=-i\alpha\,d/dx$ has commutator $[\hat X,\hat P_\alpha]=i\alpha I$ on $\mathcal S(\mathbb R)$, and the standard quantum-mechanical normalization $[\hat X,\hat P]=i\hbar I$ therefore selects $\alpha=\hbar$ within this family. Equivalently, the standard normalization of the translation generator gives $\hat P=-i\hbar\,d/dx$. The resulting characters $\mathrm e^{ipx/\hbar}$ have spatial period $h/|p|$ for $p\neq0$. All distributional statements are formulated in the Schwartz rigging and no product or pullback of arbitrary tempered distributions is used.

Nitrogen-vacancy (NV) centers in diamond can serve as highly sensitive solid-state quantum sensors for high-sensitivity magnetometry. However, in the noisy intermediate-scale quantum (NISQ) era, extracting reliable information from noisy, finite-shot, and measurement-limited sensing data remains a considerable challenge. Whereas, quantum machine learning (QML) offers a potential path to improve parameter estimation by learning nonlinear relationships between quantum-sensing data and the underlying physical signal. In this work, we investigate the role of QML in magnetic-field estimation within an NV center-inspired magnetometry setting. We formulated magnetic field sensing as a supervised regression task. We compared the performance of several classical machine learning models trained on measurement-based classical data with that of quantum kernel-based models trained on pre-measurement coherent quantum states. Our objective is to isolate the impact of measurement-induced information loss and therefore provide a theoretical upper bound on the sensing performance. The upper bound is achievable only when coherent quantum information is directly available to the learning model. Our results show that QML-based sensing performance improves significantly with coherent quantum-state information, and not much with changes in model complexity or learning paradigm. This observation underscores the importance of learning pipelines that tightly integrate quantum sensors and QML models to enhance magnetic field sensing under realistic constraints.

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

A circularly polarized far field must be dark somewhere: its amplitude is a section of a twisted line bundle. We prove this exacts a joint coverage-rate cost that cannot vanish faster than the inverse fourth power of the mode order, and we construct an explicit family of fields attaining that exponent. At lowest order the global optimum is the spin-1 anticoherent state, independently known as optimal for rotations about an unknown axis. The same floor caps photon collection in single-helicity quantum links; the opposite helicity removes it at a cost set by the inverse mode count.