Variational quantum algorithms have become a standard approach for solving a wide range of problems on near-term quantum computers. Identifying an appropriate ansatz configuration for variational algorithms, however, remains a challenging task, especially when taking into account restrictions imposed by real quantum platforms. This motivated the development of digital-analog quantum circuits, where sequences of quantum gates are alternated with natural Hamiltonian evolutions. A prominent example is the use of the controllable long-range Ising interaction induced in ion-based quantum computers. This interaction has recently been applied to develop an algorithm similar to the quantum approximate optimization algorithm (QAOA), but native to the ion hardware. The performance of this algorithm has demonstrated a strong dependence on the strengths of the individual ion-ion interactions, which serve as ansatz hyperparameters. In this work, we propose a heuristic for identifying a problem-specific ansatz configuration, which enhances the trainability of the ion native digital-analog circuit. The proposed approach is systematically applied to random instances of the Sherrington-Kirkpatrick Hamiltonian for up to 15 qubits, providing favorable cost landscapes. As the result, the developed approach identifies a well-trainable ion native ansatz, which requires a lower circuit depth to solve specific problems as compared to standard QAOA. This brings the algorithm one step closer to its large scale practical implementation.

This study focuses on statistical inference for the class of quasi-infinitely divisible (QID) distributions, which was recently introduced by Lindner, Pan and Sato (2018). The paper presents a Fourier approach, based on the analogue of the L{é}vy-Khintchine theorem with a signed spectral measure. We prove that for some subclasses of QID distributions, the considered estimates have polynomial rates of convergence. This is a remarkable fact when compared to the logarithmic convergence rates of similar methods for infinitely divisible distributions, which cannot be improved in general. We demonstrate the numerical performance of the algorithm using simulated examples.

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

We introduce Scale Factorized-Quantum Field Theory (SF-QFT), a framework performing path-integral factorization of ultraviolet and infrared momentum modes at a physical scale $Q^*$ before perturbative expansion through Effective Dynamical Renormalization (EDR) with Principle of Observable Effective Matching (POEM) constraints. This yields completely scale and scheme invariant observables. Because the two-loop $β$-function is universal, $a_{\mathrm{eff}}(Q)$ evolves with scheme-independent equations, with higher-order $β$-coefficients absorbed into Wilson coefficients. For the inclusive ratio $R_{e^{+}e^{-}}$, SF-QFT gives $R^{\mathrm{SF-QFT}}(31.6\,\mathrm{GeV}) = 1.05262 \pm 0.0005$, in excellent agreement with experiment ($1.0527 \pm 0.005$) while requiring calculations orders of magnitude simpler than conventional four-loop $\overline{\mathrm{MS}}$ approaches. SF-QFT generates universal algebraic recursion relations producing all higher-order contributions without additional Feynman diagrams. For QED, the formalism yields scheme-independent predictions for the electron anomalous magnetic moment with $a_e^{\text{theory}} = 0.001\,159\,652\,180\,61(76)$, differing from experiment by only $0.15σ$. The framework enables self-consistent extraction of $α_{\text{eff}}^{-1}(m_e) = 137.036005301$. SF-QFT represents a paradigm shift, replacing pursuit of ever-higher loop orders with a unified framework eliminating renormalization ambiguities through systematic EDR and POEM implementation.

In this paper, by utilizing the theory of matched asymptotic expansions, an efficient and accurate neural network method, named as "MAE-TransNet", is developed for solving singular perturbation problems in general dimensions, whose solutions usually change drastically in some narrow boundary layers. The TransNet is a two-layer neural network with specially pre-trained hidden-layer neurons. In the proposed MAE-TransNet, the inner and outer solutions produced from the matched asymptotic expansions are first approximated by a TransNet with nonuniform hidden-layer neurons and a TransNet with uniform hidden-layer neurons, respectively. Then, these two solutions are combined with a matching term to obtain the composite solution, which approximates the asymptotic expansion solution of the singular perturbation problem. This process enables the MAE-TransNet method to retain the precision of the matched asymptotic expansions while maintaining the efficiency and accuracy of TransNet. Meanwhile, the rescaling of the sharp region allows the same pre-trained network parameters to be applied to boundary layers with various thicknesses, thereby improving the transferability of the method. Notably, for coupled boundary layer problems, a computational framework based on MAE-TransNet is also constructed to effectively address issues resulting from the lack of relevant matched asymptotic expansion theory in such problems. Our MAE-TransNet is compared with TransNet, PINN, and Boundary-Layer PINN on various benchmark problems including 1D linear and nonlinear problems with boundary layers, the 2D Couette flow problem, a 2D coupled boundary layer problem, and the 3D Burgers vortex problem. Numerical results demonstrate that MAE-TransNet significantly outperforms other neural network methods in capturing the characteristics of boundary layers, improving the accuracy, and reducing the computational cost.

Triangle Counting (TC) is a procedure that involves enumerating the number of triangles within a graph. It has important applications in numerous fields, such as social or biological network analysis and network security. TC is a memory-bound workload that does not scale efficiently in conventional processor-centric systems due to several memory accesses across large memory regions and low data reuse. However, recent Processing-in-Memory (PIM) architectures present a promising solution to alleviate these bottlenecks. Our work presents the first TC algorithm that leverages the capabilities of the UPMEM system, the first commercially available PIM architecture, while at the same time addressing its limitations. We use a vertex coloring technique to avoid expensive communication between PIM cores and employ reservoir sampling to address the limited amount of memory available in the PIM cores' DRAM banks. In addition, our work makes use of the Misra-Gries summary to speed up counting triangles on graphs with high-degree nodes and uniform sampling of the graph edges for quicker approximate results. Our PIM implementation surpasses state-of-the-art CPU-based TC implementations when processing dynamic graphs in Coordinate List format, showcasing the effectiveness of the UPMEM architecture in addressing TC's memory-bound challenges.

Let $\mathcal{C}\subseteq[0,1]$ be a Cantor set. In the classical $\mathcal{C}\pm\mathcal{C}$ problems, modifying the ``size'' of $\mathcal{C}$ has a magnified effect on $\mathcal{C}\pm\mathcal{C}$. However, any gain in $\mathcal{C}$ necessarily results in a loss in $\mathcal{C}^c$, and vice versa. This interplay between $\mathcal{C}$ and its complement $\mathcal{C}^c$ raises interesting questions about the delicate balance between the two, particularly in how it influences the ``size'' of $\mathcal{C}^c-\mathcal{C}$. One of our main results indicates that the Lebesgue measure of $\mathcal{C}^c-\mathcal{C}$ has a greatest lower bound of $\frac{3}{2}$.

Diamond defects are among the most promising qubits. Modelling their properties through accurate quantum mechanical simulations can further their development into robust units of information. We use the recently developed capped density functional embedding theory (capped-DFET) with the multiconfigurational n-electron valence second-order perturbation theory to characterize the electronic excitation energies for different spin manifolds of the well-characterized negatively charged substitutional N defect adjacent to a vacancy (V$_C$) in diamond (N$_C$V$_C^-$). We successfully reproduce vertical excitation energies for both triplet and singlet states of N$_C$V$_C^-$ with errors < 0.1 eV. Unlike other embedding methods, capped-DFET exhibits robust predictions that are approximately independent of the embedded cluster size: it only requires a cluster to contain the defect atoms and their nearest neighbors (as small as a 40-atom capped cluster). Furthermore, our method is free from slowly converging Coulomb interactions between charged defects, and thus also only weakly dependent on supercell size.

We analyse the dimension spectrum of continued fractions expansions with coefficients restricted to infinite subsets of $ \mathbb{N}$. We prove that the set of powers $P_q=\{q^n\colon n\in \mathbb{N}\}$ has full dimension spectrum for each integer $q\geq 2$, answering a question by Chousionis, Leykekhman and Urbański. On the other hand, we show that the dimension spectrum for $P^*_q=\{q^n\colon n\in \mathbb{N}\}\cup\{1\}$ has many gaps and regions where it is nowhere dense. We also investigate the case where $A$ is generated by a monomial, $M_q=\{n^q\colon n\in\mathbb{N}\}$. For $M_q$ we prove that the dimension spectrum is full for $q\in\{1,2,3,4,5\}$, and it has a gap for each $q\geq 6$. Furthermore we show for $q\in\{6,7,8\}$ that the dimension spectrum of $M_q$ is the disjoint union of two nontrivial closed intervals, and it is the disjoint union of three nontrivial closed intervals for $q \in\{9,10\}$. For $q\geq 11$ we show that the dimension spectrum of $M_q$ consists of finitely many disjoint nontrivial closed intervals. The results concerning $M_q$ extend existing results for $q=1$ and $q=2$. In our analysis we employ Perron-Frobenius (transfer) operators, and numerical tools developed by Falk and Nussbaum that give rigorous estimates for the Hausdorff dimension for continued fractions expansions.

Clustering phenomena are common in many physical systems across multiple scales. The nuclear $α$ decay is one of the earliest observed evidences of clustering in quantum systems, yet its formation process remains poorly understood even today. In this letter, we propose a novel global odd-even staggering (OES) feature in $α$ decay, which emerges during the clustering process. To unveil its origin, we develop a Universal Four-Fermion Formation Framework (U4F), which describes the formation of any four-nucleon cluster, such as $α$ particle, from a general microscopic wave function, without assuming the preexistence of clustering or pairing. By combining U4F with the large-scale configuration-interaction approach, we demonstrate that the OES effect in $α$ decay arises from the suppression of clustering correlations due to unpaired nucleons. These findings significantly advance our understanding of cluster formation in nuclei and have important implications for the production of new elements and nuclear synthesis in the universe.

The readout speed of qubits is a major limitation for error correction in quantum information science. We show simulations of a proposed device that gives readout of a fluxonium qubit using a ballistic fluxon with an estimated readout time of less than 1 nanosecond, without the need for an input microwave tone. This contrasts the prevalent readout based on circuit quantum electrodynamics, but is related to previous studies where a fluxon moving in a single long Josephson junction (LJJ) can exhibit a time delay depending on the state of a coupled qubit. Our readout circuit contains two LJJs and a qubit coupled at their interface. We find that the device can exhibit single-shot readout of a qubit -- one qubit state leads to a single dynamical bounce at the interface and fluxon reflection, and the other qubit state leads to a couple of bounces at the interface and fluxon transmission. Dynamics are initially computed with a separate degree of freedom for all Josephson junctions of the circuit. However, a collective coordinate model reduces the dynamics to three degrees of freedom: one for the fluxonium Josephson junction and one for each LJJ. The large mass imbalance in this model allows us to simulate the mixed quantum-classical dynamics, as an approximation for the full quantum dynamics. Calculations give backaction on the qubit at $\leq 0.1\%$.

This paper examines neutron multiplicity spectra emitted from massive targets at depths of 3, 40, 210, 583, 1166, and 4000 m.w.e. The measurements, carried out between 2001 and 2024, used three experimental setups with either 14 or 60 He-3 neutron detectors and lead (Pb) targets weighing 306, 565, or 1134 kg. The total acquisition time exceeded six years. When available, the spectra obtained were compared with Monte Carlo simulations. Our data indicate potential anomalies that might limit the accuracy of modelling the muon-induced neutron multiplicity spectra with a single power-law function. Even at shallow depths, the single-power-law approach fails to account for a small but statistically significant excess of events at the highest multiplicities. This excess resembles a second power-law component. Since the anomaly varies only slightly with depth, it is unlikely to be directly linked to the muon flux. Our highest-quality data, acquired at 583 m.w.e., suggest a possible structure similar to the emission of approximately 74, 106, 143, and 214 neutrons from the target. We propose new underground measurements using low-cost, large-area, position-sensitive neutron arrays surrounding multi-ton Pb targets to verify and investigate these suspected anomalies and to determine their origin.

We investigate the scattering of two distinguishable particles with unequal masses and a mutual short-range interaction with the aim of quantifying the impact of a tunneling ``projectile'' particle on the quantum mechanical state of the ``barrier'' particle. We find that the states of the barrier particle after the tunneling or reflection of the projectile are displaced by a finite distance that is given by the derivative of the phase of the transmission or reflection amplitudes multiplied by factors dependent on particles' masses, respectively. We demonstrate these results on a numerical example with a resonant interaction between a projectile and barrier. Our work demonstrates physical implication of the concept of phase time delay in the form of finite displacements of particles that are, in principle, experimentally measurable.

We show that the variety of residuated lattices does not have the amalgamation property, thereby settling a long-standing open problem. In addition, we show that the amalgamation property fails for several subvarieties, including idempotent residuated lattices, involutive residuated lattices, and (integral) distributive residuated lattices.

The Deep Material Network (DMN) has emerged as a powerful framework for multiscale materials modeling, enabling efficient and accurate prediction of material behavior across different length scales. Unlike conventional data-driven approaches, the trainable parameters in DMN possess clear physical interpretations-they encode the geometric characteristics of representative volume elements (RVEs) rather than serving as purely statistical fitting parameters . By employing a hierarchical tree structure, DMN learns the homogenization behavior associated with microstructural geometry. Consequently, it can be trained exclusively on linear elastic datasets while effectively extrapolating to nonlinear responses during online prediction, making it a highly efficient and scalable approach for multiscale simulations. From a broader perspective, DMN can be viewed as a physics-informed reduced-order model that captures the essential micromechanical features governing macroscopic behavior. Its hierarchical formulation provides a compact yet interpretable representation of the RVE response, significantly reducing computational costs compared to direct numerical simulations. This review elaborates on the theoretical foundation, training methodology, and recent extensions of DMN, emphasizing its role as a unifying framework that connects data-driven learning with physically interpretable multiscale modeling.

Optically addressable spin defects in wide-bandgap semiconductors are promising building blocks for quantum sensing and quantum networks. Establishing their atomic structure is essential for understanding functionality and enabling controlled engineering. In 4H-SiC, the PL5 and PL6 centers have long been recognized for their exceptional charge stability and room-temperature optically detected magnetic resonance (ODMR) performance, but their structural origin has remained elusive for over a decade. Here, we provide direct evidence for their oxygen-vacancy (${\rm O_C V_{Si}}$) origins through a combined chemical and isotopic control strategy. Under oxygen ion implantation, we observe over tenfold enhancement in the yield of PL5 and PL6 compared to nitrogen ion implantation. Furthermore, implantation with $^{17}{\rm O}$ ions produces PL5 and PL6 defects that exhibit a characteristic six-fold $^{17}{\rm O}$ hyperfine splitting in their ODMR spectra. These results affirm PL6 as the ${\rm O_C V_{Si}}$ defect in the $hh$ configuration. For PL5, the oxygen-related evidence, together with \textit{ab initio} calculations and additional measurements of the zero-field splitting and hyperfine structure, establishes it as the ${\rm O_C V_{Si}}$ defect in the $kh$ configuration. This unambiguous structural identification, achieved through materials-level chemical control, provides the microscopic foundation for deterministic engineering of these defects, paving the way for scalable photonic devices and high-sensitivity ensemble quantum sensors based on oxygen-vacancy centers.

This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first formula employs a block generalization of the residual polynomial, while the second leverages the block collinearity of the residuals. A posteriori error bounds based on the knowledge of spectral information of the argument are derived and tested on a set of examples. Notably, both error formulas and their corresponding upper bounds do not require the use of quadratures for their practical evaluation.

Feedback optimization algorithms compute inputs to a system using real-time output measurements, which helps mitigate the effects of disturbances. However, existing work often models both system dynamics and computations in either discrete or continuous time, which may not accurately model some applications. In this work, we model linear system dynamics in continuous time, and we model the computations of inputs in discrete time. Therefore, we present a novel hybrid systems model of feedback optimization. We first establish the well-posedness of this hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then we show the state of the system converges exponentially fast to a ball of known radius about a desired goal state. Next we analytically show that this system is robust to perturbations in (i) the values of measured outputs, (ii) the matrices that model the linear time-invariant system, and (iii) the times at which inputs are applied to the system. Simulation results confirm that this approach successfully mitigates the effects of disturbances.

We investigate the global numerical approximation of a class of extended mean field control problems (MFC), where the dynamics and costs depend on the joint distribution of the state and the control. We propose a framework to approximate the value function globally over the Wasserstein space, moving beyond the restriction of fixed initial conditions. Our approach exploits the propagation of chaos by approximating the infinite-dimensional MFC problem by an $N$-player cooperative game, together with the usage of finite-dimensional solvers. This method avoids the need to parametrise functions on an infinite-dimensional space, offering a balance between probabilistic rigor and computational efficiency.

The purpose of this second part of the series is to show a technical result on Chow groups of toric varieties. This is a crucial ingredient for the first part.

Understanding the interplay between nonstabilizerness and entanglement is crucial for uncovering the fundamental origins of quantum complexity. Recent studies have proposed entanglement spectral quantities, such as antiflatness of the entanglement spectrum and entanglement capacity, as effective complexity measures, establishing direct connections to stabilizer Rényi entropies. In this work, we systematically investigate quantum complexity across a diverse range of spin models, analyzing how entanglement structure and nonstabilizerness serve as distinctive signatures of quantum phases. By studying entanglement spectra and stabilizer entropy measures, we demonstrate that these quantities consistently differentiate between distinct phases of matter. Specifically, we provide a detailed analysis of spin chains including the XXZ model, the transverse-field XY model, its extension with Dzyaloshinskii-Moriya interactions, as well as the Cluster Ising and Cluster XY models. Our findings reveal that entanglement spectral properties and magic-based measures serve as intertwined, robust indicators of quantum phase transitions, highlighting their significance in characterizing quantum complexity in many-body systems.

Non-diffusive, fluid-like transport of charge and heat has been observed in several materials, raising the question of whether they can emerge simultaneously and how they are related to bi-component electron-phonon fluids. Here we introduce a first-principles theory and computational framework to quantitatively describe these phenomena from atomistic to continuum scales in complex device geometries. Starting from the microscopic coupled electron-phonon Boltzmann transport equation, we formalize the emergence of composite "relaxon" electron-phonon excitations, show that they determine the viscosity tensors of the two fluids, and quantify the impact of electron-phonon drag on thermoelectric transport coefficients. We then demonstrate that the coupled Boltzmann equation can be coarse-grained into a set of mesoscopic Viscous Thermoelectric Equations, formally unifying Gurzhi's hydrodynamic equation for electrons [Sov. Phys. Usp., 1968] and the recently developed Viscous Heat Equations for phonons [PRX 10, 011019, 2020], while extending them to cover the intermediate regime of mixed electron and phonon fluids. We leverage this framework to elucidate how electron and phonon fluids can coexist or mix, rationalizing pioneering experiments on electron-phonon drag in graphite, and predicting smoking-gun signatures of non-diffusive behavior such as non-harmonic temperature and electric potential fields, and compressible thermoelectric backflow.

The GDPR's Right of Access aims to empower users with control over their personal data via Data Download Packages (DDPs). However, their effectiveness is often compromised by inconsistent platform implementations, questionable data reliability, and poor user comprehensibility. This paper conducts a comprehensive audit of DDPs from three social media platforms (TikTok, Instagram, and YouTube) to systematically assess these critical drawbacks. Despite offering similar services, we find that these platforms demonstrate significant inconsistencies in implementing the Right of Access, evident in varying levels of shared data. Critically, the failure to disclose processing purposes, retention periods, and other third-party data recipients serves as a further indicator of non-compliance. Our reliability evaluations, using bots and user-donated data, reveal that while TikTok's DDPs offer more consistent and complete data, others exhibit notable shortcomings. Similarly, our assessment of comprehensibility, based on surveys with 400 participants, indicates that current DDPs substantially fall short of GDPR's standards. To improve the comprehensibility, we propose and demonstrate a two-layered approach by: (1)~enhancing the data representation itself using stakeholder interpretations; and (2)~incorporating a user-friendly extension (\textit{Know Your Data}) for intuitive data visualization where users can control the level of transparency they prefer. Our findings underscore the need for clearer and non-conflicting regulatory guidance, stricter enforcement, and platform commitment to realize the goal of GDPR's Right of Access.

The study of the celestial phenomena in the hard X-ray/soft gamma-ray band(20 keV--1 MeV) is very intriguing but also very difficult to be performed with the needed sensitivity. In this review I will discuss the astrophysical importance of the soft gamma-ray astronomy, its difficulties to solve its issues with the current instrumentation, and a possible solution achievable using focusing Laue lens. Concerning these instruments, I will discuss their functioning principle, how to achieve a high reflection efficiency, their imaging properties, the current feasibility studies, the technological developments and observation prospects.

This paper introduces a method for pricing insurance policies using market data. The approach is designed for scenarios in which the insurance company seeks to enter a new market, in our case: pet insurance, lacking historical data. The methodology involves an iterative two-step process. First, a suitable parameter is proposed to characterize the underlying risk. Second, the resulting pure premium is linked to the observed commercial premium using an isotonic regression model. To validate the method, comprehensive testing is conducted on synthetic data, followed by its application to a dataset of actual pet insurance rates. To facilitate practical implementation, we have developed an R package called IsoPriceR. By addressing the challenge of pricing insurance policies in the absence of historical data, this method helps enhance pricing strategies in emerging markets.

Tensor networks provide a powerful framework for compressing multi-dimensional data. The optimal tensor network structure for a given data tensor depends on both data characteristics and specific optimality criteria, making tensor network structure search a difficult problem. Existing solutions typically rely on sampling and compressing numerous candidate structures; these procedures are computationally expensive and therefore limiting for practical applications. We address this challenge by viewing tensor network structure search as a program synthesis problem and introducing an efficient constraint-based assessment method that avoids costly tensor decomposition. Specifically, we establish a correspondence between transformation programs and network structures. We also design a novel operation named output-directed splits to reduce the search space without hindering expressiveness. We then propose a synthesis algorithm to identify promising network candidates through constraint solving, and avoid tensor decomposition for all but the most promising candidates. Experimental results show that our approach improves search speed by up to $10\times$ and achieves compression ratios $1.5\times$ to $3\times$ better than state-of-the-art. Notably, our approach scales to larger tensors that are unattainable by prior work. Furthermore, the discovered topologies generalize well to similar data, yielding compression ratios up to $ 2.4\times$ better than a generic structure while the runtime remains around $110$ seconds.

In the case of any bounded open set $Ω$ $\subset$ R d with boundary $\partial$$Ω$, we first construct a directional trace in any direction $θ$ of the unit sphere, for any u $\in$ L 2 ($Ω$) whose the directional derivative $\partial$ $θ$ u in the direction $θ$ belongs to L 2 ($Ω$). This directional trace is shown to belong to L 2 ($\partial$$Ω$, $μ$ $θ$ ), where $μ$ $θ$ is a measure supported by the closure of all points of $\partial$$Ω$ which are the extremity of an open segment directed by $θ$, included in $Ω$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($Ω$) containing the elements of H 1 ($Ω$) whose the directional trace does not depend on $θ$ is closed. It therefore contains the closure of H 1 ($Ω$) $\cap$ C 0 ($Ω$) in H 1 ($Ω$). Examples where H 1 tr ($Ω$) = H 1 ($Ω$) and H 1 tr ($Ω$) __ = H 1 ($Ω$) are provided.

Altermagnets are collinear antiferromagnets with spin-split electronic states. We introduce Ruddlesden-Popper chromates Sr$_{n+1}$Cr$_n$O$_{3n+1}$ (including SrCrO$_3$) as candidate materials in which altermagnetism can emerge from spontaneous orbital ordering rather than crystal symmetry. First-principles calculations reveal a layer-dependent spin splitting: if the spin and orbital orders align in adjacent layers, the system exhibits non-relativistic spin splitting, and thus altermagnetism. In contrast, if either the spin or the orbital order is reversed in adjacent layers, we observe a layerwise uncompensated spin splitting, that is compensated in the adjacent layer, giving rise to the concept of anti-altermagnetism. In the RP series, odd $n$ members support coexistence of altermagnetism and anti-altermagnetism, whereas even $n$ and the perovskite limit are strictly anti-altermagnetic. In both cases, larger $n$ favors metallicity, and in odd $n$ compounds strain can further stabilize altermagnetism.

We consider renewal-type processes whose positive inter-renewal times may be dependent, non-identically distributed, and may have mixed distributions. We introduce a generalised intensity measure extending the classical hazard-rate representation to this setting. Under a two-sided comparison scheme for the inter-renewal laws and an additional renewal-measure domination condition \textnormal{(RD)}, we prove a Lorden-type bound for the forward recurrence time. This bound provides an explicit first-moment input for coupling constructions and, once the remaining coupling parameters are controlled, yields a total-variation estimate. We illustrate the result on exponential, mixed, Markov-modulated, and Pareto benchmarks. In the i.i.d.\ benchmarks, the bound has the correct renewal scale up to a universal factor; in the Markov-modulated benchmark, the explicit Lorden constant is verified while the final convergence consequence remains conditional on \textnormal{(RD)}; and in the Pareto case the construction identifies the natural second-moment threshold for finiteness of the Lorden input.

Time-reversal (TR) symmetry is crucial for understanding a wide range of physical phenomena, and plays a key role in constraining fundamental particle interactions and in classifying phases of quantum matter. In this work, we introduce an ensemble of random quantum circuits that are representative of the dynamics of generic TR-invariant many-body quantum systems. We derive a general statistical mechanics model describing entanglement, many-body quantum chaos and quantum information dynamics in such TR-invariant circuits. As an example of application of our formalism, we study the universal properties of measurement-induced phase transitions (MIPT) in monitored TR-invariant systems, with measurements performed in a TR-invariant basis. We find that TR-invariance of the unitary part of the dynamics does not affect the universality class, unless measurement outcomes are post-selected to satisfy the global TR-invariance of each quantum trajectory. We confirm these predictions numerically, and find, for both generic and Clifford-based evolutions, novel critical exponents in the case of ``strong'', i.e. global TR-invariance where each quantum trajectory is TR-invariant.