Papers

Quantum coherence is an important quantum resource which plays a pivotal role in the field of quantum information. Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively, and illustrate the effectiveness of them by explicit examples.

We initiate an investigation into a notion of state complexity for discrete-variable quantum systems. Specifically, we propose an information-theoretic quantifier for the complexity of quantum states within the stabilizer formalism of quantum computation. This is achieved by leveraging the symmetric Jordan product (associated with classicality) and the skew-symmetric Lie product (linked to quantumness) between the square root of the quantum state and the Heisenberg-Weyl displacement operators. We establish the fundamental properties of this quantifier and demonstrate that state complexity is closely related to the nonstabilizerness of quantum states via the $L^4$-norm of their characteristic functions.

Higher-order networks with multiway interactions can exhibit collective dynamical phenomena that are absent in traditional pairwise network models. However, analyzing such dynamics becomes computationally prohibitive as their state space grows combinatorially in the multiway interaction order. Here we develop quantum algorithms for two central tasks – synchronization estimation and certification of the no-phase-locking regime – in the simplicial Kuramoto model. This model is a higher-order generalization of the celebrated Kuramoto model for coupled oscillators on graph-based networks. Under explicit assumptions on data access and types, and simplicial structure, we derive end-to-end quantum gate complexities and identify regimes with polynomial quantum advantage for synchronization estimation and super-polynomial quantum advantage for no-phase-locking certification over classical methods. More broadly, these results extend quantum algorithms for higher-order networks from structural analysis to nonlinear dynamical diagnostics, easing a major computational bottleneck and opening a route to quantum methods for probing higher-order phenomena beyond the reach of direct classical approaches.

Continuous-variable (CV) quantum teleportation is usually benchmarked by average fidelity, but when the teleportation is repeatedly used within optical networks or measurement-based architectures, uniformity across the input ensemble becomes equally important. We analyze this issue using two complementary figures of merit: the average fidelity and the fidelity deviation, which quantifies the input dependence of the single-shot teleportation fidelity. We prove that any deterministic unity-gain teleportation channel that is displacement covariant has vanishing fidelity deviation for coherent-state benchmarking, irrespective of whether the shared entangled resource is Gaussian or non-Gaussian. Nonzero deviation therefore diagnoses covariance breaking rather than non-Gaussianity. We then show that when a protocol raises the average fidelity through input-selective conditioning, the deviation generically increases in tandem, giving a quantitative universality cost. As a concrete example, we study teleportation enhanced by the so-called measurement-based noiseless linear amplification, where a heralded filter acts on the Bell-measurement record. The resulting trade-off among average fidelity, fidelity deviation, and success probability shows that stronger filtering can improve the conditional fidelity only by concentrating the successful events in favored regions of phase space, thereby suppressing the success probability and reducing input uniformity. Our results provide an operational framework for distinguishing genuine channel improvement from selectivity-driven post-selected advantage and suggest that the probabilistic CV teleportation should be assessed with average quality, universality, and heralding rate treated on an equal footing.

We propose a slack-free penalty formulation for constrained binary optimization that eliminates auxiliary slack variables and preserves the feasibility structure of the original problem. The proposed approach introduces a nonlinear custom penalty function to enforce inequality constraints directly in the objective function. To address the computational challenges associated with evaluating nonlinear penalties in variational quantum algorithms, we employ the finite-sampling method that avoids the exponential complexity required by exact expectation computation. Furthermore, we integrate the Conditional Value-at-Risk (CVaR) objective to improve optimization robustness and guide the search toward high-quality solutions. The proposed framework is evaluated on instances of the multi-dimensional knapsack problem, a classical benchmark in combinatorial optimization. We showcase that the proposed custom-penalty formulation combined with CVaR sampling achieves improved optimality gaps and more consistent performance compared with conventional slack-based QUBO formulations. We also tested our method on the Quantinuum H2 quantum hardware, demonstrating the possibilities of training small-sized VQE on noisy trapped-ion quantum computers. The results suggest that careful penalty design can play a critical role in enabling quantum and hybrid quantum-classical algorithms for constrained optimization problems that arise in operations research.

Nitrogen-vacancy (NV) centers in diamond are of great interest for nano- and macro-scale magnetic field sensing. Most sensing protocols rely on conventional optical readout, which is limited by photon shot noise. The recently developed photoelectrical (PE) readout of the NV center electron spin state promises to overcome these limitations. However, the noise of the PE readout and its influence on readout efficiency have not been thoroughly studied. In this work, we perform magnetic field sensing and estimate the sensitivity using optical and PE readout with a single and an ensemble of NV centers in diamond. We investigate the electronic noise associated with the photoelectric detection and estimate the readout efficiency, using Gaussian statistics. Our quantitative analysis shows that the Johnson-Nyquist noise-limited photoelectric magnetic field sensitivity could outperform optical measurements by an order of magnitude. This work is an essential step towards the development of on-chip magnetometers using photoelectrical detection in diamond.

We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.

Early modular fault tolerant quantum computers remain constrained by costly inter-module communication and limited magic state factory service. Understanding such bottlenecks and investigating compiler optimizations most close the gap between algorithm requirements and hardware capabilities is a concrete and practically urgent systems problem. We study the modular architectures based on Bivariate Bicycle codes and identify the dominant bottleneck: inter-module communication induced by non-Clifford operations. We build a compilation pipeline to fill the missing parts of prior works and propose compiler optimizations: synthesizing arbitrary-angle rotations at the factory (syn@fac), transvection based Clifford deferral, and Clifford insertion for critical path duration reduction. We extend the evaluation scope of the prior work to 40+ benchmark categories drawn from PennyLane and MQTBench, including quantum algorithms and Hamiltonian simulations with varying sizes. Under the present instruction cost, syn@fac reduces estimated circuit failure probability by a factor of 9.0 on average across non-Clifford benchmarks. The robustness persists across sweeps of instruction cost ratios, LPU count, and factory count. Besides, transvection reduces Clifford deferral compile time by 77.04%, while Clifford insertion reduces end-to-end circuit duration by 11.54% on average on MQTBench, with smaller gains on Hamiltonian simulations. We hope this work inspires the studies on compiler optimizations for early modular FTQC systems.

We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order $\alpha$ influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $\alpha = 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.

Higher-dimensional quantum systems, such as qudits, offer architectural and algorithmic advantages over qubits, but their increased spectral crowding and limited controllability render high-fidelity quantum gates particularly challenging. We propose a hybrid optimization framework that integrates optimal control theory methods with contextual deep reinforcement learning to achieve robust controlled-phase gates on two qutrits. Optimal control is first used to design high-fidelity control pulses for a nominal system model. Reinforcement learning is then employed as a calibration stage that learns small residual corrections to these pulses in the presence of static model mismatch, thereby preserving good gate performance under realistic parameter uncertainties. By learning structured, low-dimensional residual corrections conditioned on device-specific parameter variations, reinforcement learning enhances the transfer robustness of nominally optimal but parameter-sensitive control solutions across ensembles of devices. Crucially, the reinforcement learning step in our framework does not compete with the optimal control step but provides the adaptability required for realistic hardware, systematically reducing the sensitivity to parameter fluctuations. Our results establish reinforcement learning as a practical and scalable ingredient for robust calibration of quantum gates in high-dimensional systems.

Stationary states of quantum many-body Hamiltonians are invariant under the Hamiltonian evolution. Besides ground and thermal states, this class includes microcanonical ensembles that are of fundamental importance in statistical physics. We consider the preparation of general stationary states by leveraging recent advances in the field of open-system dynamics. In particular, constructions based on exact KMS-detailed balance with respect to Gibbs states of noncommuting Hamiltonians have only recently been proposed as a tool for their efficient preparation and, by extension to small temperatures, for ground state preparation. We extend these constructions to the problem of stationary state preparation, providing general criteria that characterize when such states have efficient implementations, along with specific results on the approximation of microcanonical ensembles. An interesting application of our work are tests of conjectured ensemble equivalences for local observables between microcanonical and Gibbs ensembles.

We present ab initio calculations using the $R$-matrix with time dependence (RMT) method for high-order harmonic generation (HHG) in argon in a short, intense pulse regime. The calculations employ a $6$-cycle $\sin^2$ pulse at $850$ nm with peak intensity $2.3\times 10^{14}$ W/cm$^2$ and, for comparison with the experiment by Guo et al. [J. Phys. B: At. Mol. Opt. Phys. 51, 034006 (2018)], a Gaussian pulse with the same frequency and peak intensity. Both pulse shapes yield the expected harmonic structure in the region above the ionization threshold (approximately $15.82$ eV in $LS$-coupling). The spectra exhibit strong carrier-envelope-phase (CEP) sensitivity. The energy region leading up to the ionization threshold contains spectral features arising from residual coherent dipole oscillations (free-induction decay) that strongly depend on spectral windowing and the post-pulse propagation time. We show that the HHG spectrum, particularly below the ionization threshold, is a defined quantity that depends on analysis choices rather than being a uniquely determined observable. Comparison between theoretical predictions and experimental observations in this energy regime, therefore, requires explicit specification of these parameters.

We present a search for small Kochen-Specker (KS) sets in dimension 3, specifically targeting extensions of the 13-ray Yu-Oh set, which has been proven to be the minimal witness to state-independent contextuality. To enable this search, we introduce a novel SAT-based orderly generation framework integrating recursive canonical labeling (RCL) with the graph isomorphism tool NAUTY. We demonstrate that previous SAT approaches relying on lexicographical canonicity suffer from exponential scaling on canonical graphs. This limitation renders them intractable on the large instances (25 to 33 vertices) encountered in our search, whereas our RCL check maintains consistent millisecond-level performance, effectively eliminating the bottleneck. Overcoming this bottleneck allows us to perform the first exhaustive enumeration of all KS sets with up to 33 rays containing the complete 25-ray state-independent contextuality (SI-C) set obtained by rigid extensions of the Yu-Oh set in 1,641 CPU hours. We found and verified that the 33-ray set discovered by Schütte is the smallest three-dimensional KS set containing the complete 25-ray SI-C set. All non-existence results are backed by independently verifiable proof certificates via an extension of the DRAT proof format.

Based on a consistent quantum microscopic approach, we investigate the peculiarities of collective polyatomic effects in atomic ensembles placed in a waveguide, caused by the presence of evanescent modes of electromagnetic field. We analyze the influence of these modes on the process of cooperative spontaneous decay, as well as on the nature of radiation transfer in the ensembles under consideration. We show that under certain conditions, their influence can be dominant compared to the role of radiation modes, and the mechanism for this influence is the modification of dipole-dipole interatomic interaction.

We present a hitherto unexplored semi-device-independent (SDI) self-testing protocol designed to certify unitary operations within a variant of prepare-measure framework. We consider a communication game which we refer to as a variant of $3$-bit prepare-measure random access code (PMRAC) involving two parties, Alice and Bob, who share a prior two-qubit quantum state. Alice encodes her message by applying unitary operations on her subsystem and sends it to Bob. To decode the message, Bob performs a measurement on the whole system. We demonstrate that the optimal quantum advantage of the variant of $3$-bit PMRAC over the classical bound enables the self-testing of Alice's unitary operations and Bob's measurements. The derivation of the optimal quantum success probability is fully analytical. The approach is so elegant that it can be generalized for any arbitrary $n$-bit PMRAC and may also be extended to other prepare-measure communication games.

We develop a recently introduced representation of quantum dynamics based on sampling negative Markov chain processes. By introducing particles and antiparticles, this formalism maps generic quantum dynamics onto a Markov process defined over an exponentially large configuration space. Within this framework, quantum complexity arises from the proliferation of stochastic particles, which ultimately renders classical simulation and sampling intractable beyond a certain timescale. In the presence of noise, we demonstrate that for any unitary evolution generated by a linear combination of local or pairwise interactions, there exists at least one noise channel that effectively classicalizes the system by suppressing particle growth and making Monte Carlo sampling efficient. As a corollary, we show that for this class of unitaries, the dynamics of an open quantum spin chain subject to depolarizing noise undergoes an exact transition to classical simulability once the noise strength exceeds a critical threshold which can be efficiently determined for any model.

We employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the $\mathbb{Z}_3$ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.

In classical dynamical systems, stochastic feedback can stabilize otherwise unstable periodic orbits, giving rise to distinct controlled and uncontrolled phases as the rate of control application is varied. In this work, we apply these control protocols in classical, semiclassical, and quantum regimes to the kicked top, a paradigmatic model of quantum chaos. The quantum kicked top, modeled as the dynamics of a spin-S object, naturally interpolates between these regimes with the spin size S acting as an effective Planck constant. We show that the dynamics of the kicked top in classical, semiclassical, and fully quantum limits can all be controlled using stochastic feedback protocols. Comparing the full quantum dynamics to a truncated Wigner approximation that captures quantum noise but neglects interference beyond the Ehrenfest time, we find that low-moment observables are largely accounted for semiclassically, while the remaining discrepancy in higher moments is consistent with contributions from interference and possibly nonlinearities in rare trajectories that explore the compact phase space. We also find rapid purification in the numerics studied for all rates of control considered, suggesting that control quenches the top's ability to encode a qubit of quantum information even in the uncontrolled phase.

We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

Near-term quantum workloads demand error management, yet the two lightest-weight techniques, Quantum Error Detection (QED) and Probabilistic Error Cancellation (PEC), have complementary cost profiles whose joint architectural design space remains unexplored. QED encodes logical qubits and discards error-flagged runs, filtering noise with low qubit overhead but leaving residual errors; PEC can correct these in software, but at exponential cost in noise strength. If QED efficiently reduces per-gate noise, PEC's cost savings can outweigh QED's discard overhead; realizing this, however, requires solving two system-level design challenges. First, the QED interval – how often detection cycles are inserted – is a tunable architectural parameter governing the cost-accuracy tradeoff. We derive an efficiency condition and show that the canonical one-cycle-per-gate frequency does not achieve break-even in any code we evaluate, while optimized intervals on high-rate Iceberg codes do. Second, we discover that naive PEC+QED integration degrades accuracy below the QED-only baseline. The root cause is a transient error profile in the first detection cycle that corrupts PEC's noise model. We develop steady-state extraction, a co-designed characterization protocol that isolates steady-state error behavior, reducing estimation bias by up to $10.2\times$. On a $[[6,4,2]]$ Iceberg code running QAOA ($p{=}4$–$8$) with a fixed shot budget, PEC+QED achieves $2$–$11\times$ lower absolute error and up to $31\times$ lower MSE versus PEC on physical qubits, with per-interval savings compounding over interval depth.

Quantum computers are moving into HPC centers, and the main challenge is now integration rather than pure hardware access. Many current software paths still depend on vendor-specific adapter chains between user SDKs, schedulers, and backend APIs. This pattern makes operations more complex than necessary and slows the transition from pilots to production workflows. We present a practical integration path centered on the Quantum Device Management Interface (QDMI). Using IQM superconducting systems as a hardware case study, we implement an IQM-backed QDMI layer and connect it to two software layers that HPC centers working with quantum computers already care about: Slurm-based job execution and Qiskit-facing user workflows. The implementation is publicly available at https://github.com/iqm-finland/QDMI-on-IQM. The key message is simple: integrating quantum hardware into HPC does not have to be a bespoke engineering effort for each backend. Once the software-hardware boundary is standardized, large parts of the stack become reusable across providers and deployment styles. Our results do not claim that standardization eliminates all HPCQC challenges. They show that this specific boundary can already be standardized today in a way that is practical for users, operators, and vendors.

We demonstrate that combining the positivity of density matrices with steady-state conditions yields a systematic bootstrap method for studying open quantum many-body systems governed by Lindblad master equations on infinite lattices, which exhibit absorbing phase transitions. As a concrete example, we apply this method to the quantum contact process with an absorbing state. We obtain bootstrap bounds on steady-state expectation values, the critical coupling, certain ratios of expectation values in the nontrivial steady state in the supercritical phase, and the Liouvillian spectral gap in the subcritical phase.

Obtaining a quantum advantage in practical applications over classical machines requires a fault-tolerant model of computation. However, the resource overhead of developing a fault-tolerant quantum computer (FTQC) architecture is quite high. Current state-of-the-art architecture designs promise constant-time access to logical qubits at the expense of a high qubit overhead, or qubit-dense layouts at the expense of increased latency in the execution of a quantum workload. In this paper, we propose (a) an architecture design that balances the tradeoff of execution time of workloads with the density of logical qubits by placing the surface code patches around the ancilla region, giving almost equal access of the ancilla region to all the data qubits, (b) design a novel algorithm to leverage the $T$-gate characteristics of the user workload to suggest the optimal placement strategy in the floorplan of our architecture, (c) a workload-selective fast-$Y$ optimization knob that accelerates $Y$-basis measurements only for qubits whose $Y$-gate demand justifies the additional tile cost, and (d) explore the scope of executing multiple programs concurrently, which enhances the reconfigurability of our proposed design. Through numerical evaluation, we observe that the proposed architecture design maintains the cycles per instruction time close to optimal while using up to $\sim21\%$ fewer data tiles, and up to $\sim90\%$ efficiency under a 10-way concurrency.

In this article, we prove that the quantum $f$-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical $f$-divergence between two corresponding classical states, which are called Nussbaum-Szkoła distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra $\mathbb{B}(\mathscr{H})$ of all bounded operators on a Hilbert space $\mathscr{H}$. We extend their result for normal states on any semifinite von~Neumann algebra, not only $\mathbb{B}(\mathscr{H})$.

We study charged massive scalar field perturbations on the rotating black hole (BH) background of Einstein-Maxwell-Dilaton-Axion (EMDA) theory, known as the Kerr-EMDA BH. Starting from the gauge-covariant Klein-Gordon equation (KGE), we perform a full separation of variables and obtain exact analytical solutions for both the angular and radial parts in terms of confluent Heun functions (CHFs). Unlike the earlier neutral scalar treatment by Senjaya and Ponglertsakul [Eur. Phys. J. C 85, 352 (2025)], the electromagnetic coupling $q$ fundamentally alters the structure of the Heun parameters and produces qualitatively new physics. Applying the CHF polynomial condition, we derive the resonant frequency spectrum whose imaginary parts are equispaced with $|\Delta\omega_I| = 1/(2M)$, a universal spacing determined solely by the BH mass. Via the Maggiore prescription and the first law of BH thermodynamics, this yields a parameter-dependent entropy quantum $\delta S_{\text{BH}} = 4\pi r_+/(r_+ - r_-)$, which reduces to $4\pi$ for Schwarzschild but diverges at extremality – {\color{black}in contrast to the universal $2\pi$ obtained for the rotating linear dilaton BH (RLDBH).} We construct the effective potential governing scalar wave scattering and analyze its dependence on the dilaton parameter $D$, rotation $a$, and scalar charge $q$. In the massless uncharged limit, the CHF reduces to the Gauss hypergeometric function, {\color{black}enabling us to compute the first analytical greybody factor (GF) for the Kerr-EMDA geometry; we show that this reduction extends to massless charged scalars, yielding a closed-form GF that captures superradiant amplification.} We examine how the dilaton deformation distinguishes the Kerr-EMDA spectrum from the standard Kerr and Kerr-Newman cases.

Recent advancements in neutral atom platforms have enabled exploration of early fault-tolerant (FT) architectures for applications with quantum advantage, such as quantum dynamics simulations. An efficient fault-tolerant architecture has both spatially efficient quantum error correction codes (low qubit overhead), and efficient methodologies (transversal based gates, extractor based gates, etc.) for logical computation, to minimize overall execution time. Achieving the right balance between space and time can be critical for enabling early FT demonstrations of quantum advantage. In this work, we identify bottlenecks in existing spatially efficient schemes, which tend to be very serial, and do not take advantage of unutilized space. We introduce a teleportation-based scheme that leverages the reconfigurable connectivity of neutral atoms to parallelize logical operations. Our approach achieves up to \textbf{$\mathbf{\sim 3 \times}$ speedup} over extractor architectures at no extra space cost and achieves the best spacetime performance among other viable architectures before accounting for external resource-states. To rigorously evaluate performance, we construct explicit quantum advantage benchmarks and simulate compilation to a fault-tolerant instruction set, including low-level gate scheduling and shuttling patterns, and resource-state nondeterminism. We find that our speedups still apply and report exact space-time cost along with success probabilities, identifying architectures capable of achieving quantum advantage \textbf{with as little as $\mathbf{11,495}$ atoms and a runtime of $\mathbf{\sim 15}$ hours}.

In this paper, we give a mathematical proof that bounds the number of CNOT gates required to synthesize an $n$ qubit phase polynomial with $g$ terms to be at least $O(\frac{gn}{\max (\log g, 1)})$ and at most $O(gn)$. However, when targeting restricted hardware, not all CNOTs are allowed. If we were to use SWAP-based methods to route the qubits on the architecture such that the earlier synthesized gates are natively allowed, we increase the number of CNOTs by a routing overhead factor of $O(\log n) \leq \alpha \leq O(n \log^2 n)$. However, if we only synthesize allowed gates, we do not need to route any qubits. Moreover, in that case the routing overhead factor is $1 \leq \alpha \leq 4 \simeq O(1)$. Additionally, since phase polynomials and Hadamard gates together form a universal gate set, we get qubit routing for almost free.

In quadratic gravity, with a positive Weyl squared coefficient, the extra spin-2 sector is shown to correspond to a dual inverted harmonic oscillator, instead of a ghost. Using the Wightman spectrum condition, we prove that the associated Källén–Lehmann spectral density vanishes, reflecting the absence of a normalizable ground state and the spacelike nature of the propagator pole. This uniquely fixes the propagator to a principal value form as a theorem, not a prescription. The optical theorem is satisfied, the dual IHO spin-2 is not an asymptotic state, and gives only virtual contributions at all loop orders. As a result, unitarity is preserved consistently with renormalizability.

A recent paper published in Nature [Nature,646,813(2025)] claims that quantum particles become entangled through their gravitational interaction, even when the gravitational potential is classical. Here we show that the entanglement found by the authors is a consequence of letting particles diffuse from one object to the other. When this is not allowed, the wavefunction remains factorized. In fact, noticing that entanglement is relative to a chosen partition of the Hilbert space, we show that the choice of partition implicitly made by the authors is such that entanglement arises as soon as particles diffuse, even when no interaction potential is present. Therefore, it is the free evolution, not the classical gravitational potential, which acts as the entangling mechanism in the scenario considered by the authors.

Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an $n$-regular block encoding, namely, a block encoding whose $k$-th power reproduces the $k$-th power of the encoded matrix for every $0 < k < n$. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most $n$ to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an $n$-regular one using only $O(\log n)$ ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix; for non-diagonalizable matrices this yields the full matrix function $P(A)$, whose action on each Jordan block involves the derivatives of $P$, and not merely the map $\lambda \mapsto P(\lambda)$.