In this work, we forecast the number of, and requirements on, N-body simulations needed to train hybrid effective field theory (HEFT) emulators for a range of use cases, using a hybrid of HMcode and perturbation theory as a surrogate model. Our accuracy goals, determined with careful consideration of statistical and systematic uncertainties, are $1\%$ accurate in the high-likelihood range of cosmological parameters, and $2\%$ accurate over a broader parameter space volume for $k<1 h Mpc^{-1}$ and $z<3$. Focusing in part on the 8-parameter $w_0w_a$CDM+$m_ν$ cosmological model, we find that $<225$ simulations are required to meet our error goals over our wide parameter space, including models with rapidly evolving dark energy, given our simulation and emulator recommendations. For a more restricted parameter space volume, as few as 80 simulations are sufficient. We additionally present simulation forecasts for example use cases, and make the code used in our analyses publicly available. These results offer practical guidance for efficient emulator design and simulation budgeting in future cosmological analyses.

In this work, we propose a non-standard dark matter (NSDM) model in which the equation of state (EoS) of dark matter (DM) is parameterized as $w_{\rm dm} = w_2 a^2$, and this DM model is motivated by the idea that DM must become cold dark matter (CDM) in the neighborhood of the scale factor $a = 0$, which implies that both the EoS of DM, $w_{\rm dm}$, and its derivative with respect to the scale factor, ${\rm d}w_{\rm dm}/{\rm d}a$, vanish at $a = 0$. By incorporating the latest cosmological datasets -- including the Planck2018 Cosmic Microwave Background (CMB) distance priors, the Baryon Acoustic Oscillation measurements from the Data Release 2 of the Dark Energy Spectroscopic Instrument (DESI), together with three independent Type Ia Supernova datasets, namely the Dark Energy Survey Year 5 (DESY5) compilation, the Union3 compilation, and the PantheonPlus sample -- we constrain the $Λw_2$DM, $ww_2$DM, and $w_0w_aw_2$DM models, which are constructed by replacing CDM with NSDM in the $Λ$CDM, $w$CDM, and $w_0w_a$CDM models, respectively. We find that there is a preference for a negative DM EoS at more than the $3σ$ confidence level for the data combinations CMB+DESI+Union3 and CMB+DESI+DESY5. Moreover, for all data combinations, replacing CDM with NSDM in the $w$CDM and $w_0w_a$CDM models significantly reduces the probability of violating the null energy condition. Furthermore, both $ww_2$DM and $w_0w_aw_2$DM are favored over $Λ$CDM with a significance comparable to that of the $w_0w_a$CDM model.

Spatial generalized linear mixed-effects models are popularly used to analyze spatially indexed univariate responses. However, with modern technology, it is common to observe vector-valued mixed-type responses, e.g., a combination of binary, count, or continuous types, at each location. Methods for jointly modeling such mixed-type multivariate spatial responses are rare. Using multivariate Gaussian processes (GPs) in the latent layer, we present a class of Bayesian spatial methods applicable to any combination of exponential family responses. Since multivariate GP-based methods can suffer from computational bottlenecks when the number of spatial locations is high, we further employ a computationally efficient Vecchia approximation for fast posterior inference and prediction. Key theoretical properties of the proposed model, such as identifiability and the structure of the induced covariance, are established. Our approach employs a Markov chain Monte Carlo-based inference method that uses elliptical slice sampling within a blocked Metropolis-within-Gibbs sampling framework. We illustrate the efficacy of the proposed method through simulation studies and a real-data application on joint modeling of wildfire counts and burnt areas across the United States.

The coherence and fidelity of quantum dot (QD) spin qubits are fundamentally limited by charge noise arising from electrically active trap states at oxide interfaces, heterostructure boundaries, and within the bulk semiconductor. These traps introduce electrostatic fluctuations that couple to the qubit via spin-orbit interactions or charge-sensitive confinement potentials, leading to dephasing and gate errors. In this work, we present a general trap characterization framework for identifying and quantifying the spectral signatures of these trap states using AC impedance spectroscopy, deep-level transient spectroscopy (DLTS), and conventional capacitance-voltage (C-V) analysis. While our case study focuses on strained Ge/SiGe quantum well heterostructures, the approach is broadly applicable to other material systems and qubit types. We demonstrate that each class of traps (oxide interface, quantum well interface, and bulk) exhibits distinct fingerprints across frequency- and time-domain measurements. Oxide traps dominate the low-frequency conductance peaks and appear strongly in Nyquist and transient spectra. QW interface traps, despite being nearly invisible at low densities in conventional C-V and AC impedance analysis, are clearly resolved through multi-exponential decay signatures in time-domain response. Bulk traps contribute to high-frequency admittance and steady-state leakage currents. By correlating each trap type to its characteristic time constant, spatial location, and spectral impact, we provide a diagnostic toolset for disentangling noise sources that degrade qubit performance. This unified methodology bridges traditional defect metrology with emerging qubit noise analysis and enables material- and process-level strategies for coherence optimization in scalable quantum devices.

We refine Feighn--Handel's results on subgroups of mapping tori of free groups to the special case of free-by-cyclic groups. We use these refinements to show that any finitely generated free-by-cyclic group embeds in a {finitely generated free}-by-cyclic group. When the free-by-cyclic group is hyperbolic, it embeds in a hyperbolic {finitely generated free}-by-cyclic group as a quasi-convex subgroup. Combined with a result of Hagen--Wise, this implies that all hyperbolic free-by-cyclic groups are cocompactly cubulated.

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

Using the graphs of prisms and Tutte Fragments, we construct an infinite family of hamiltonian and non-hamiltonian graphs in which Tutte's counterexample to Tait's conjecture appears in a certain sense as a minimal element. We observe that generalizations of the minimum-cardinality counterexamples of Holton and McKay to Tait's conjecture are as well contained in this family.

Optical cavities are frequently used in quantum technologies to enhance light matter interactions, with applications including single photon generation and entanglement of distant emitters. The Fabry-Pérot resonator is a popular choice for its high optical access and large emitter-mirror separation. A typical configuration, particularly for emitters that should not be placed close to the mirror surface like trapped ions and Rydberg atoms, features two spherical mirrors placed around a central emitter, but this arrangement can put demanding requirements on the mirror alignment. In contrast, plano-concave cavities are tolerant to mirror misalignment and only require the manufacture of one curved mirror, but have limited ability to focus light in the centre of the cavity. Here we show how mirror shaping can overcome this limitation of plano-concave cavities while preserving the key advantages. We demonstrate through numerical simulations that simple mirror shaping can increase coupling between a plano-concave cavity and a central emitter by an order of magnitude, even rivalling misalignment-sensitive concave-concave counterparts for achievable interaction strength. We use these observations to establish the conditions under which plano-concave cavities with shaped mirrors could improve the performance and practicality of emitter-cavity systems.

Random survival forests are widely used for estimating covariate-conditional survival functions under right-censoring. Their standard log-rank splitting criterion is typically recomputed at each candidate split. This O(M) cost per split, with M the number of distinct event times in a node, creates a bottleneck for large cohort datasets with long follow-up. We revisit approximations proposed by LeBlanc and Crowley (1995) and develop simple constant-time updates for the log-rank criterion. The method is implemented in grf for R and reduces training time on large datasets while preserving predictive accuracy.

We report the first complete next-to-leading order QCD calculation for processes involving fully charm tetraquark states, revealing that the renormalization constant of the color-singlet four charm quark operator is exactly unity at this order. We have investigated the possible quark configurations of the fully charm tetraquarks and expanded their states in the color symmetry-antisymmetry basis. By applying the transverse momentum dependent factorization formalism, large logarithms induced by soft and collinear gluon radiations are resummed to all orders in the expansion of the strong interaction coupling at the accuracy of next-to-leading logarithm. By combining LHCb data on the total cross section of the exotic hadron $X(6900)$ and CMS measurements of its spin-parity, we extracted its nonperturbative but universal long-distance matrix element. The rapidity and transverse momentum distributions of the $X(6900)$ and its spin-zero partners are also predicted, which await further experimental verification.

Existing beamforming-based full-duplex solutions for multi-antenna wireless systems often rely on explicit estimation of the self-interference channel. The pilot overhead of such estimation, however, can be prohibitively high in millimeter-wave and massive MIMO systems, thus limiting the practicality of existing solutions, especially in fast-fading conditions. In this work, we present a novel beam learning framework that bypasses explicit self-interference channel estimation by designing beam codebooks to efficiently obtain implicit channel knowledge that can then be processed by a deep learning network to synthesize transmit and receive beams for full-duplex operation. Simulation results using ray-tracing illustrate that our proposed technique can allow a full-duplex base station to craft serving beams that couple low self-interference while delivering high SNR, with 75-97% fewer measurements than would be required for explicit estimation of the self-interference channel.

Many users struggle to notice when a more efficient workflow exists in feature-rich tools like Excel. Existing AI assistants offer help only after users describe their goals or problems, which can be effortful and imprecise. We present InvisibleMentor, a system that turns screen recordings of task completion into vision-grounded reflections on tasks. It detects issues such as repetitive edits and recommends more efficient alternatives based on observed behavior. Unlike prior systems that rely on logs, APIs, or user prompts, InvisibleMentor operates directly on screen recordings. It uses a two-stage pipeline: a vision-language model reconstructs actions and context, and a language model generates structured, high-fidelity suggestions. In evaluation, InvisibleMentor accurately identified inefficient workflows, and participants found its suggestions more actionable, tailored, and more helpful for learning and improvement compared to a prompt-based spreadsheet assistant.

In Large Language Model (LLM) inference services, it is challenging to make a parallelism strategy configuration, to efficiently process the requests of variance context lengths. Requests of long context require high degree of parallelism to provide more memory for Key-Value (KV) Cache, while requests of short context prefer low degree of parallelism to increase concurrency, thus improving throughput. To maintain high throughput while supporting large context lengths on demand, we propose Amoeba, a runtime Tensor Parallel (TP) transformation for online LLM inference services, which adaptively adjusts the TP of running instances to align with the dynamics of incoming requests. Evaluations using real-world traces show that Amoeba improves throughput by 1.75x-6.57x compared to state-of-the-art solutions.

Electrogastrography is the recording of changes in electric potential caused by the stomach's pacemaker region, typically through several cutaneous sensors placed on the abdomen. It is a worthwhile technique in medical and psychological research, but also relatively niche. Here we present a tutorial on the acquisition and analysis of the human electrogastrogram. Because dedicated equipment and software can be prohibitively expensive, we demonstrate how data can be acquired using a low-cost OpenBCI Ganglion amplifier. We also present a processing pipeline that minimises attrition, which is particularly helpful for low-cost equipment but also applicable to top-of-the-line hardware. Our approach comprises outlier rejection, frequency filtering, movement filtering, and noise reduction using independent component analysis. Where traditional approaches include a subjective step in which only one channel is manually selected for further analysis, our pipeline recomposes the electrogastrogram from all recorded channels after automatic rejection of nuisance components. The main benefits of this approach are reduced attrition, retention of data from all recorded channels, and reduced influence of researcher bias. In addition to our tutorial on the method, we offer a proof-of-principle in which our approach leads to reduced data rejection compared to established methods. We aimed to describe each step in sufficient detail to be implemented in any programming language. In addition, we made an open-source Python package freely available for ease of use.

In this work, we establish a continuous version of the Fountain theorem by using the framework of the weak slope for continuous functionals, which generalizes Theorem 3.6 of Willem \cite{Wi}. Then we present an application to a semilinear problem.

We study the dynamics and phase structure of Abelian gauge theories in $d=1+1$ dimensions. These include $U(1)$ gauge theory coupled to a scalar and a fermion, as well as the two-flavour Schwinger model with different charges. Both theories exhibit a surprisingly rich phase diagram as masses are varied, with both $c=1$ and $c=1/2$ critical lines or points. We build up to the study of 2d chiral gauge theories, which hold particular interest because they provide a mechanism for symmetric mass generation, a phenomenon in which fermions become gapped without breaking chiral symmetries.

Gate geometry and bias asymmetry can be used to engineer spin dynamics in gate-defined Ge hole quantum dots by reshaping the confinement potential and driving transitions between distinct confinement regimes. In this work, we show that these transitions strongly modify wavefunction localization, heavy-hole/light-hole mixing, and the effective vertical electric field, leading to pronounced g-factor modulation and geometry-induced dephasing sweet spots where the qubit becomes first-order insensitive to vertical electric-field fluctuations. We further find that phonon-induced spin relaxation exhibits a strong dependence on device size and bias, with T1 following a magnetic-field scaling close to B-9, consistent with Rashba-dominated heavy-hole spin dynamics. These results are obtained using a comprehensive three-dimensional simulation framework for strained Si0.2Ge0.8/Ge gate-defined hole spin qubits, combining realistic electrostatics with a four-band Luttinger-Kohn Hamiltonian. Unlike simplified symmetric confinement models, this approach captures asymmetric wavefunction redistribution, g-tensor anisotropy, and the coupled electrostatic and spin response of realistic devices. Our results establish gate pattern and bias design as practical tools for optimizing spin coherence in Ge hole-spin qubits.

Top-quark decays are sensitive probes of light charged Higgs bosons ($H^\pm$) due to the sizable $t\bar t$ production cross section at the LHC in conjunction with their distinct experimental signatures. While dedicated ATLAS and CMS searches considered only $H^\pm$ decays into $τν$, $cs$, or $cb$ for $m_{H^\pm}<m_t$, the $WZ$ channel remains unexplored, despite being the dominant mode in $SU(2)_L$ triplet models. Since, top-quark pair production with $t \to H^\pm b$ and $H^\pm \to WZ$ gives rise to $t\bar{t}Z$-like signatures, we recast existing $t\bar{t}Z$ analyses to search for signs of charged Higgs bosons and set novel limits on the product of branching fractions Br$(t\to H^\pm b) \times $Br$(H^\pm\to WZ)$. These constraints turn out to be at the sub-permille level, despite the observed $2σ$ preference for a non-zero value. Interpreted within the hypercharge $Y=0$ Higgs triplet model, this translates into a stringent constraint on the triplet Higgs vacuum expectation value of $v_Δ\lesssim 2$ GeV, which is stronger than those from the $cs,τν$ modes and even surpasses electroweak precision constraints from the $ρ$ parameter. Moreover, the $2σ$ preference for a non-zero cross section further strengthens the cumulative case for a $\approx152$ GeV boson as suggested, in particular, by di-photon excesses.

We report the first constraint on time-reversal invariance violating (TRIV) effects in polarized neutron transmission through a transversely polarized $^{139}$La target. We formulate the transmission asymmetry within the density matrix formalism, explicitly incorporating the forward scattering amplitude of $^{139}$La including tensor polarization terms up to third-rank. The formalism is applied to existing transmission data originally obtained to measure the spin-dependent cross section near the $0.75$~eV $p$-wave resonance. Since these data were not optimized for P-odd/T-odd observables, the attainable sensitivity is intrinsically limited; nevertheless, they provide a useful test of the formalism on real experimental data. No statistically significant TRIV signal is observed. By analyzing the global $χ^2$ structure in the parameter space, we obtain an upper limit of $|W_T|<15~\mathrm{eV}$ at the 90\% confidence level. This corresponds to an upper limit on the resonance-averaged TRIV cross section of $|Δσ_{\not{T}\not{P}}|<8.3\times10^2~\mathrm{b}$. These results validate the present theoretical framework and provide guidance for future dedicated TRIV searches in polarized neutron transmission experiments.

We propose a practical, physics-inspired benchmarking suite to challenge both quantum and classical computers by mapping real-time quantum dynamics to a common optimization format. Using a parallel-in-time encoding, we convert the real-time propagator of an $n$-qubit, possibly non-Hermitian, Hamiltonian into quadratic unconstrained binary optimization (QUBO) instances that are executable in a solver-agnostic manner on quantum annealers and classical optimizers alike. This enables direct, like-for-like performance comparisons across fundamentally different computational paradigms.To stress-test the framework, we consider eight representative dynamical models spanning single-qubit rotations, multi-qubit entangling gates (Bell, GHZ, cluster), and PT-symmetric and other non-Hermitian generators, and evaluate success probability and time-to-solution as standard benchmarking metrics. Applying this methodology to two generations of D-Wave quantum annealers and to state-of-the-art classical solvers (Simulated Annealing and the GPU-accelerated VeloxQ), we find that Advantage2 consistently outperforms its predecessor, while VeloxQ retains the shortest absolute runtimes, reflecting the maturity of classical heuristics.We further extend the benchmarks to large-scale instances ($N \simeq 10^{5}$), establishing a demanding classical baseline for future hardware. Together, these results position the parallel-in-time QUBO framework as a versatile and physically motivated testbed for quantitatively tracking progress toward quantum-competitive simulation of dynamical systems.

For general one-matrix models in the large $N$ limit, we introduce the cap amplitude $ψ(b)$ as the expansion coefficient of the 1-form $ydx$ on the spectral curve. We find that the dilaton equation for the discrete volume $N_{g,n}$ of the moduli space of genus-$g$ Riemann surfaces with $n$ boundaries is interpreted as gluing the cap amplitude along one of the boundaries. In this process, one of the boundaries is capped and the number of boundaries decreases by one. In a similar manner, the genus-$g$ free energy $F_g$ is obtained by gluing the cap amplitude to $N_{g,1}$.

We introduce and validate a machine-learning assisted quantum sensing protocol to classify spatial and temporal correlations of classical noise affecting two ultrastrongly coupled qubits. We consider six distinct classes of Markovian and non-Markovian noise. Leveraging the sensitivity of a coherent population transfer protocol under three distinct driving conditions, the various forms of noise are discriminated by only measuring the final transfer efficiencies. Our approach achieves $\gtrsim 94\%$ accuracy in classification providing a near-perfect discrimination between Markovian and non-Markovian noise. The method requires minimal experimental resources, relying on a simple driving scheme providing three inputs to a shallow neural network with no need of measuring time-series data or real-time monitoring. The machine-learning data analysis acquires information from non-idealities of the coherent protocol highlighting how combining these techniques may significantly improve the characterization of quantum-hardware.

Gravitational physics and astronomy have developed rapidly over the last decade, driven by new observations and theoretical breakthroughs. As new as the science and technology of this field are, its greatest asset may be the body of early-career researchers actively engaged in driving it forward. With the aim of bringing together this community of enthusiastic scientists from a broad array of disciplines, the organisers of the GR-Amaldi meeting joined with the Gravitational-Wave Early Career Scientists organisation to create a three-day event-the Early Career Workshop. The Workshop aimed to provide a broad overview of the field's diverse scientific possibilities and introduce key theoretical foundations underpinning its science. To complement developing technical skills, the Workshop also sought to provide participants with transferable skills to aid them in their future careers. The Workshop emphasized networking and community building, offering participants opportunities to engage with peers and mentors. It encouraged interdisciplinary exchanges and cross-institutional collaboration, fostering connections across different research efforts. Collectively, these initiatives aimed to equip participants with a comprehensive understanding of the field's research and to build a more cohesive, collaborative community of early-career researchers. We summarise key points and conclusion from the various activities carried out as part of the Workshop.

Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter $α$, the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter $α$ where the computational subspace remains positive-definite. We identify special values of $α$ where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of $α$ is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.

Decision support systems are increasingly adopted to automate decision-making processes across industries, organizations, and governments. Decision support demands data privacy, integrity, and availability while ensuring customization, security, and verifiability of the decision process. Existing solutions fail to guarantee those properties altogether. To overcome this limitation, we propose SPARTA, an approach based on Trusted Execution Environments (TEEs) that automates decision processes. To guarantee privacy, integrity, and availability, SPARTA employs efficient cryptographic techniques on notarized data with access mediated through user-defined access policies. Our solution allows users to define decision rules, which are translated to certified software objects deployed within TEEs, thereby guaranteeing customization, verifiability, and security of the process. With experiments run on public benchmarks and synthetic data, we show our approach is scalable and adds limited overhead compared to non-cryptographically secured solutions.

Extracting useful work from quantum systems is a fundamental problem in quantum thermodynamics. In scenarios where rapid protocols are desired -- whether due to practical constraints or deliberate design choices -- a fundamental trade-off between power and efficiency is yet to be established. Here, we investigate the problem of finite-time optimal work extraction from closed quantum systems, subject to a constraint on the magnitude of the control Hamiltonian. We first reveal the trade-off relation between power and work under a general setup, showing that these fundamental performance metrics cannot be maximized simultaneously. We then identify a solvable class of finite-time optimal work-extraction problems. This class includes nontrivial many-body models such as the Heisenberg model and the SU(n)-Hubbard model. The key assumption is that the control Hamiltonian is optimized over a Lie algebra preserved by the uncontrolled dynamics. Within this class, the optimal work-extraction problem admits an exact reduction to a nonlinear self-consistent equation, circumventing extensive search over time-dependent control paths. The resulting optimal protocol turns out to be particularly simple: it suffices to use a time-independent control Hamiltonian in the interaction picture, determined by that equation. By exploiting the Lie-algebraic structure of the controllable terms, our approach is applicable to quantum many-body systems through efficient numerical computation. Our results highlight the necessity of rapid protocols to achieve the maximum power and provide an exact route to finite-time optimal work extraction in many-body quantum systems.

We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation-Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SOC-representable. While the proposed formulation may be of exponential size in general, we identify additional structural conditions that guarantee the existence of a polynomial-size SOC-representable formulation, which can be constructed in polynomial time. Under these conditions, the optimal value of the nonconvex quadratic program coincides with that of a polynomial-size second-order cone program. Our results serve as a starting point for bridging the gap between the Boolean quadric polytope of sparse problems and its continuous counterpart.

Generalized Sierpiński graphs constitute a distinctive class of fractal-like networks with recursive definition: given a graph $G$, $S_G^1=G$ while $S_G^n$ is obtained from $|V(G)|$ copies of $S_G^{n-1}$ by adding some edges in a prescribed way that reflects the structure of $G$. Many graph invariants have been studied in generalized Sierpiński graphs. In this paper, we focus on their injective colorings, both the vertex and the edge version. Given a graph $G$, a mapping $f$ that assigns an integer from $\{1,\ldots,k\}$ to each vertex (resp.\ edge) of $G$ is an injective (edge) coloring of $G$ if $f(x)=f(y)$ implies that $x$ and $y$ are not in a common triangle nor at distance $2$ for any two vertices (resp.\ edges) $x$ and $y$ in $G$. The minimum number of colors $k$ for which there exists an injective (edge) coloring of $G$ is called the injective chromatic number (resp.\ injective chromatic index) of $G$ and is denoted by $χ_i(G)$ (resp.\ $χ_i'(G)$). The vertex version of injective colorings in generalized Sierpiński graphs was studied in an earlier paper, where the authors determined the injective chromatic numbers of standard Sierpiński graphs, and asked about the values when $G$ is a cycle. We resolve this question by proving that $χ_i(S_{C_k}^n)=3$ for every $n\ge 2$ and every $k\ge 3$. Moreover, we prove an almost conclusive result that $χ_i(S_G^n)\in \{χ_i(G),χ_i(G)+1\}$ for any graph $G$ and any $n\ge 2$. For injective edge colorings we prove that $χ_i'(S_{K_3}^n)=5$ for all $n\ge 3$, while $χ_i'(S_{K_3}^2)=4$ and $χ_i'(S_{K_3}^1)=3$. Furthermore, if $G$ is a triangle-free graph, we prove that $χ_i'(S_G^n)\in \{χ_i'(S_G^3),χ_i'(S_G^3)+1\}$ for all $n\ge 4$, and provide some sufficient conditions on an injective edge coloring of the 3-dimensional Sierpiński graph over $G$, which ensure that $χ_i'(S_G^n)=χ_i'(S_G^3)$.

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the existence of suitable sets in topological groups is studied. It is proved that, for a non-separable $k_ω$-space $X$ without non-trivial convergent sequences, the $snf$-countability of $A(X)$ implies that $A(X)$ does not have a suitable set, which gives a partial answer to \cite[Problem 2.1]{TKA1997}. Moreover, the existence of suitable sets in some particular classes of linearly orderable topological groups is considered, where Theorem~\ref{t4} provides an affirmative answer to \cite[Problem 4.3]{ST2002}. Then, topological groups with an $ω^ω$-base are discussed, and every linearly orderable topological group with an $ω^ω$-base being metrizable is proved; thus it has a suitable set. Further, it follows that each topological group $G$ with an $ω^ω$-base has a suitable set whenever $G$ is a $k$-space, which gives a generalization of a well-known result in \cite{CM}. Finally, some cardinal invariant of topological groups with a suitable set are provided. Some results of this paper give some partial answers to some open problems posed in~\cite{DTA} and~\cite{TKA1997} respectively.

Error mitigation is essential for unlocking the full potential of quantum algorithms and accelerating the timeline toward quantum advantage. As quantum hardware progresses to push the boundaries of classical simulation, efficient and robust error mitigation methods are becoming increasingly important for producing accurate and reliable outputs. However, existing error-mitigation approaches face a fundamental tradeoff between practical performance and reliability: heuristic methods such as zero-noise extrapolation (ZNE) enjoy faster runtime but lack accuracy guarantees, while rigorous techniques such as probabilistic error cancellation (PEC) provide unbiased estimates at prohibitive computational cost. We introduce a characterization-based, rigorously-grounded quantum error mitigation and error suppression framework (QESEM) that resolves this tradeoff by leveraging the accuracy guarantees of quasi-probabilistic mitigation with dramatically reduced overhead. We explain the innovative methods underlying QESEM and demonstrate its capabilities in the largest utility-scale error mitigation experiment based on an unbiased method. This experiment simulates the kicked transverse field Ising model with far-from-Clifford parameters on an IBM Heron device. We further validate QESEM's versatility across arbitrary quantum circuits and devices through high-accuracy error-mitigated molecular VQE circuits executed on IBM Heron and IonQ trapped-ion devices. Compared with multiple variants of the widely used zero-noise extrapolation method, QESEM consistently achieves higher accuracy while avoiding the prohibitive runtime overhead associated with PEC. These results mark a significant step forward in accuracy and reliability for running quantum circuits on current devices across diverse applications. Finally, we provide projections of QESEM's performance on near-term devices toward quantum advantage.