liangzid/robench2024b_all_setphysicsSCP-s · Datasets at Hugging Face

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A: The second is highly non-trivial, since now m1subscript𝑚1m_{1}italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is externalB: of the kite in the integrand of (5)C: The relation F1,2,3=F1,4,6subscript𝐹123subscript𝐹146F_{1,2,3}=F_{1,4,6}italic_F start_POSTSUBSCRIPT 1 , 2 , 3 end_POSTSUBSCRIPT = italic_F start_POSTSUBSCRIPT 1 , 4 , 6 end_POSTSUBSCRIPT makes m3subscript𝑚3m_{3}italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT external.	BAC	CAB	ABC	CAB	Selection 1
A: we used the triangle inequalityB: The third inequality follows again from EqsC: (23). The above arguments prove that ρ𝜌\rhoitalic_ρ can be converted into σ𝜎\sigmaitalic_σ via DIO with approximate catalysis whenever an asymptotic conversion via DIO is possible with rate at least one.	CAB	ABC	CAB	BAC	Selection 2
A: that every PMC solution is a collection of superalgebra irrepsB: This evidence is still missing. C: In order to elevate this simple and elegant argument to the level of a proof of chiral symmetry breaking, one should demonstrate that it works for a generic spectrum with exotics and that it gives the most general solution of PMC, i.e	CBA	ABC	BCA	CAB	Selection 3
A: We used predominantly general-purpose software packages to solve the SDPs involved. We used YALMIP [86] for modeling the problems and solved primarily using MOSEK [87]B: The unfavorable memory scaling of interior-point methods (see e.g. [88]) prevented us from going to bond dimensions higher than 7 in the MPS-based variant and 5 in the tree-tensor-network-based variant. To overcome this, we tried to use the splitting conic solver (SCS) [89] which is a first-order method and requires less memory, but observed slow convergence and were not able to improve the results in any of the models, with the exception of the S=1𝑆1S=1italic_S = 1 Heisenberg chain. In this model we used our own matrix-free implementation of SCS (where the constraint matrix is not explicitly constructed as a sparse matrix, but only implicitly applied, which provides significant savings due to the tensor-network structure of the constraints) to produce the results for bond dimensions 6 through 9. In addition, SCS was used to solve the larger instances of the LTI problem, Eq. 5, where it performed very well. C: We conclude this section with some details regarding our numerical implementations. Sample code is available at [85]	ACB	CBA	ABC	BCA	Selection 4
A: This model showed the capability to already perform better across all the relevant metrics, achieving a precision of 29%percent2929\%29 %, 1%percent11\%1 % above the benchmark, with just 90909090 learners (against 1200120012001200) and runtimes of around 20202020 minutes compared to more than 3333 hours for the benchmarked Random ForestB: C: We also report a classification model based on the proposed heterogeneous structure and leveraging the boosting procedure that, although is not amenable to be trained on current hardware, was trained using a quantum-inspired optimizer based on Tensor Networks	BAC	CBA	BCA	CBA	Selection 3
A: Figure 11: Amplification of the energy (top) and angular momentum (bottom) against the frequency of the Q-ball ωQsubscript𝜔𝑄\omega_{Q}italic_ω start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT with mQ=1subscript𝑚𝑄1m_{Q}=1italic_m start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 1B: The coupling is g=1/3𝑔13g=1/3italic_g = 1 / 3. C: η+subscript𝜂\eta_{+}italic_η start_POSTSUBSCRIPT + end_POSTSUBSCRIPT is the only ingoing mode with m+=−2subscript𝑚2m_{+}=-2italic_m start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = - 2 and various frequencies ω+subscript𝜔\omega_{+}italic_ω start_POSTSUBSCRIPT + end_POSTSUBSCRIPT	BCA	CAB	ACB	ABC	Selection 3
A: The construction is characterized by a process that finds a multi-linear polynomial, which in this case is the Walsh-Hadamard transformB: The subsequent constructions differ precisely in the process of finding an equivalent multi-linear polynomial for the function. Thus, it is this process alone that distinguishes between them, as the conversion from the multi-linear polynomial to the measurement assignment will be always the same in every case. Therefore, this process will not be repeated for the next constructions.C: This first construction, taken from [21] was presented with a detailed translation from a valid multi-linear polynomial, for a specific Boolean function, to a correct and deterministic measurement assignment	BCA	ACB	BAC	ACB	Selection 1
A: Alternatively, in Fig. 1 the high-level strategy of a repeater protocol following a layered approach for entanglement swapping is illustrated.B: Naturally, the chosen repeater strategy decides which events exactly are used, e.g., a repeater protocol without entanglement purification may decide to schedule an entanglement swapping event at a station, immediately after entangled pairs in both directions have been establishedC: The other moving part of the system is the repeater protocol, i.e. the high-level strategy, the simulation should follow, because this defines which events should be scheduled and when	ACB	BAC	CBA	BCA	Selection 3
A: As for applications to enhance couplings, the BO is readily applicable to systems with weak couplings to photons and strong intrinsic dephasing so that the squeezing-induced dephasing is marginal. Common examples are quantum dots [9] and spin ensembles [10]B: Yet, the future of BOs is not prescribed to the injection of squeezed vacuum. Regarding the BO alone, the new regime of amplification that evades the gain-bandwidth product constraint [31] is immediately applicable for broadband quantum limited amplification with no hardware overhead [54]C: This enhanced coupling could then be leveraged by the BO gain for improved qubit readout [43]. Finally, the ability to dynamically tune qubit-photon interactions comes as a great resource for the study of squeezing-induced phase transitions [19, 20, 21], quantum transduction [36], and the exploration of the ultra-strong coupling regime [16, 17, 18].	CBA	BAC	CBA	CAB	Selection 2
A: We begin with the simplest modified conformally flat spacetimes. B: The current benchmark model of cosmology is the flat FLRW solution; therefore, we focus on the way the FLRW model is modified in our approachC: The rest of this paper is devoted to finding the simplest exact cosmological models of modified TEGR	ACB	CAB	CBA	ABC	Selection 3
A: The fellowship code is LCF/BQ/PI20/11760016B: JCB is also supported by the research grant PID2020-118635GB-I00 from the Spain-Ministerio de Ciencia e Innovación.C: JCB is supported by a fellowship from “la Caixa” Foundation (ID 100010434) and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847648	ABC	BCA	ACB	ACB	Selection 2
A: We allow only configurations that differ by a finite number of sites to produce offspringB: To address the concerns highlighted above, we preserve the framework they used for finite-volume Gibbs measures introducing infertility into the reproduction dynamicsC: This approach is similar to that employed in a model of sex-linked inheritance presented in Wörz-Busekros [16, Sec. 8.B] and it reflects the nature of Gibbs measures, which are commonly described by their microscopic components.	CAB	CAB	BAC	CAB	Selection 3
A: Figure 3 shows that the amount of contextuality decreases monotonically as coherence increases (at least according to these measures of contextuality and coherence)B: However, this intuition only goes so far, as we will give an example below where one achieves twice the robustness to dephasing by including effects that have maximal coherence in the dephased basis.C: While this might at first seem counterintuitive, one can understand it by noting that even under large dephasing noise, states and effects with little coherence to begin with are barely affected; that is, the dephasing channel is close to identity on such states and effects	CAB	CBA	ACB	BAC	Selection 3
A: If such an experiment is accomplished, the principles used in the present model could be translated to other systems with the same logical capabilities and even for qudits with d>3𝑑3d>3italic_d > 3B: The latter condition could be also realised using atomic qudits based on an extension of our simple qutrit modelC: It is always important to have thought experiments illustrating how future realistic implementations should work, and it really matters when the illustrative models show possible drawbacks and obstacles one should face when implementing theoretical ideas in the laboratory. We strongly recommend a serious analysis by experimentalists about how to implement a version of such an experiment, at least as a proof-of-principle, to assess possible unforeseen new questions that might further improve the technological aspects of quantum-information processing and quantum computing.	ACB	BAC	CBA	ABC	Selection 4
A: In this work, we develop a unified treatment of nonlinear quantum optical effects based on phase space fillingB: Our theory is applicable to a wide range of exciton-polariton lattices, also in the presence of both Pauli and Coulomb blockadeC: We describe three distinct NPSF regimes being the planar, fractured, and ultralocalized regimes. For each case, we present an analysis and show that in the fractured case a sharp decrease of Rabi frequency in the low-density regime can be facilitated by the Coulomb blockade. Our theory can shed light onto recent experiments in moiré heterobilayers, and open a way for enhancing the nonlinear response.	BAC	ABC	BCA	CAB	Selection 2
A: There are plenty of works, devoted to theoretical analysis of the energy propagation, induced by the supratransmission. In general, theoretical studies are focused on determining of conditions of the energy propagation by ILMs, with which emergence of the supratransmission is associated. The bibliography of such studies starts, probably, from the paper [18], wherein the supratransmission was shown to be accompanied by sharp increasing of the supplied energyB: This process was described by dint of the sine-Gordon equation, which became hereinafter a model equation for studying of the supratransmission (in particular, in the Josephson junctions) both in the discrete (see, e.g., [19, 20, 21]) and in the continuum (see, e.g., [22, 23, 24, 25]) formulations. Similar results were obtained for the models obeying the discrete nonlinear Schrödinger equation [26, 27], for FPUT chains [28, 32, 33], electrical lattices [30, 31] and others [34]. On the other hand, evolution of the energy, transmitted into the lattice by the loading at driving frequencies, lying both in the stop-band and also in the pass-band, is the rarely studied problemC: In the paper [11], evolution of the energy, transmitted into the infinite chain with on-site potential interactions and supplied by kinematic loading, was demonstrated. The non-stationary processes of the energy supply may be described analytically in the harmonic approximation, what was done for the infinite harmonic chain on the linear elastic substrate [35] for the cases of both kinematic and force loadings. For small driving amplitudes, the rate of supply of the total energy into the nonlinear chain is shown [11] to be indistinguishable from the one described in the harmonic approximation.	BCA	CBA	ABC	CAB	Selection 3
A: There arise 4 Higgs particles from the b⁢c𝑏𝑐bcitalic_b italic_c sector. B: Its detailed particle spectrum is shown in table 3C: Model 1 is a rare case without any filler branes and the hidden sector group SU(2) only comes from the d𝑑ditalic_d-stack of D6-branes	CBA	BAC	BCA	ABC	Selection 1
A: The first is the family of tangent X-states previously considered in Sec 5.1B: Now we apply Theorem 9 to two families of statesC: The second is a new family, which we call tangent spheroid states, which includes the family of the tangent sphere states, Sec. 5.3, as a special case. (Note that the canonical obese states, Sec. 5.2, are a special case of the tangent spheroid states.)	BAC	CBA	CAB	ABC	Selection 1
A: The type-II model is insteadB: The type-I model corresponds to ϵ𝒅=𝒀𝒖=𝟎superscriptbold-italic-ϵ𝒅superscript𝒀𝒖0\epsilon^{d}=Y^{u}=0bold_italic_ϵ start_POSTSUPERSCRIPT bold_italic_d end_POSTSUPERSCRIPT bold_= bold_italic_Y start_POSTSUPERSCRIPT bold_italic_u end_POSTSUPERSCRIPT bold_= bold_0C: of the four Yukawa matrices in Eqs. (1) and (21) are absent	ABC	ACB	ABC	CBA	Selection 4
A: Orbits evolve to become aligned and prograde with respect to the disc and circularize as they alignB: The final semimajor axis can be estimated analytically. This estimate is exact for circular orbits (see Appendix B, Rauch 1995, and Šubr & Karas 1999).C: Low inclination orbits remain close to their initial semimajor axis, while high inclination orbits inspiral significantly	CBA	BCA	BCA	ACB	Selection 4
A: We compare our results with other crystal detectors in literature as shown in Fig. 14. Experiments using doped crystals featured light yields smaller than 20 p.e./keVeeB: Recent results in Ref. Ding et al. (2022) used a 0.6×\times×0.6×\times×1.0 cm3 cryogenic pure CsI crystal readout by SiPM chips and achieved about 43 p.e./keVeeC: It’s not trivial to perform an apple-to-apple comparison between our results with it because the CT effect was not corrected in Ref. Ding et al. (2022).	CAB	BAC	ABC	ACB	Selection 3
A: Thus we can conclude that small (3%) amount of free spins responsible for the observed absorption signal and for low-temperature magnetization offset (see Fig. 4) is decoupled from the main magnetic system of nabokoiteB: The main copper matrix of nabokoite does not contribute to the observed paramagnetic ESR absorptionC: Presumably, it is due to the strong relaxation processes in the copper matrix of nabokoite which broadens ESR line beyond limits of observation. Note, that quite large ESR linewidth of 1—2 T was also reported for kagomé compound herbertsmithite [35].	CBA	ACB	ABC	BAC	Selection 3
A: This is the effective Majorana neutrino mass parameter of the neutrinoless double beta decay, which gives information on the Majorana nature of the neutrinosB: Furthermore, in our model, another observable can be obtainedC: This mass parameter has the form:	CAB	BAC	ACB	ACB	Selection 2
A: SMaity would like to thank Ms. Diana Vaclavkova, Ms. Anushree Dey, Mr. Sanjib Naskar, Mr. Rahul Paramanik and Mr. Soumik Das. MP and BD is grateful to IACS for the fellowship. SD acknowledges the financial support from DST-SERB grant No. ECR/2017/002037, SCP/2022/000411 and CRG/2021/004334. SD also acknowledges support from the Central Scientific Service (CSS) and the Technical Research Centre (TRC), IACS, Kolkata.B: SMaity and TK are grateful to DST-INSPIRE for their fellowships. D.D. and L.Y. gratefully acknowledge the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0021127 for financial assistance and Advanced Computing Group of the University of Maine System for providing computational resources for this work. SMasanta is grateful to Council of Scientific & Industrial Research (CSIR), New Delhi, for the financial support through the award of NET-SRF (File No: 09/015(0531)/2018-EMR-I). The authors are thankful to the facilities at UGC-DAE-CSR-Indore. Magneto-Raman scattering at low temperature were performed at LNCMI, European Magnetic Field Laboratory at Grenoble under the project GSC08-119C: Acknowledgments The authors would like to thank Dr. Vasant Sathe, Prof. K Sengupta, Dr. Marek Potemski, Dr. Clément Faugeras, Prof. Achintya Singha, Mr. Somsubhra Ghosh, Dr. Mintu Mondal, and Dr. Devajyoti Mukherjee for fruitful discussion	BAC	BCA	BAC	CBA	Selection 4
A: Compared with the classical counterpart, this quantum process has exponential acceleration in terms of data scale and quadratic speedup in data dimensionality. The other is a quantum secure multi-party computation (QSMC) protocol, which allows the aggregation of gradients to securely be done with quantum communication networks. That is, the server is able to calculate the federated gradients without the client sharing the local gradients. Furthermore, the application of the Chinese remainder theorem in QSMC makes it possible to avoid errors and overflow problems that may occur during the calculation of large numbers. The proposed quantum federated learning framework can improve the local computing efficiency and data privacy of FL. We also apply QFLGD to train the federated linear regression (FLR) and give its numerical experiment to verify the correctness. B: In this paper, we focus on the quantum algorithm running on ordinary quantum computers and present a quantum federated learning based on gradient descent (QFLGD). It aims to provide a unified, secure, and effective gradient distribution estimation scheme with distributed quantum networks. In QFLGD, we propose two data preparation methods by analyzing the different acquisition frequencies of static data (the local training data) and dynamic data (the parameters that need to be updated during iteration). That can reduce the requirement of QFLGD on the performance of quantum random memoryC: At the same time, two main processes of FL are implemented in QFLGD, which exploit quantum properties. The first one is a quantum gradient descent (QGD) algorithm. It facilitates the acceleration of the training gradient for the client. QGD provides the client with a classical gradient at each iteration, which can be directly used to learn classical model parameters	CAB	BCA	CBA	CBA	Selection 1
A: In Sec. III, the model of the likelihood and the Bayesian method are introduced. The constraints and relevant discussion are given in Sec. IV. Conclusions are given in Sec. V. B: In Sec. II, we briefly describe the galaxy catalog and FRB dataC: This paper is organized as follows	BAC	CAB	CAB	CBA	Selection 4
A: As we mentioned previously, the chip architecture will influence our parameterization. Depending on the architecture, the depth of the parameterization will be seriously influenced due to the need to apply several SWAP portsB: Thus, we are naturally led to ask whether for quantum machine learning models it is indeed necessary to make use of these non-parameterized quantum gates involving qubits that do not interact directly. To carry out this analysis, we begin by defining the cost function asC: However, although we previously discussed this issue only for CNOT gates, this same analysis applies to any gate that acts on more than one qubit. In fact, the more restricted the chip connectivity is, the deeper the parameterization will be	ABC	CBA	ACB	BAC	Selection 3
A: The basic idea is to study the Heisenberg evolution of an operator 𝒪𝒪\mathcal{O}caligraphic_O in a given quantum system governed by a Hamiltonian H𝐻Hitalic_H. The time evolution of 𝒪𝒪\mathcal{O}caligraphic_O is determined by the Heisenberg equationB: In this section we provide an outline of the recursion method, which is the basis for the Lanczos algorithmC: The reader can refer to RecursionBook ; Parker:2018yvk for further details	BAC	ACB	CAB	BAC	Selection 3
A: The importance of the bulk viscosity in the hydrodynamic evolution of quark gluon plasma has been emphasized in Refs. [51, 52, 53]. Our framework provides a direct control over this first-order transport coefficient through a choice of matching condition via b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. B: We identified a class of matching conditions for which the homogeneous part of the solution to the relativistic Boltzmann equation vanishes, and RTA turns out to be a special case of that. We examined the effect of choice of matching condition on dissipative coefficients and also studied scaling properties of the ratio of coefficients of bulk viscosity to shear viscosity on the conformality measureC: It is important to note that the BGK or MBGK collision kernels are affected by the matching conditions, which in turn affects the dissipative processes in the system. Moreover, at finite chemical potential, two descriptions become identical	CBA	BCA	ACB	ACB	Selection 1
A: The structure of the behavioral landscape is similar to the one found in wild type [19], with the dominating kinetics being the transitions between “runs” and “pirouettes”, see Fig. S7(a). However, mutants and wild type crucially differ at long timescales as mutants do not exhibit heavy tails, see Fig. 6(b). Instead, the tail of the first passage time is close to an exponential, and correlations decay to zero within a minute. This is due to the mutants’ inability to adapt their pirouette rates over time, Fig. 6(a), contrary to the modulation highlighted in Fig. 3(a) for the wild type.B: The NPR-1 neuropeptide receptor is known to impact several C. elegans behaviors, viz., aerotaxis and food response [45]C: We collected a public dataset where worms of the npr-1 loss-of-function strain npr-1(ad609) are allowed to freely explore an agar plate with a uniform food patch (see Appendix A) and used the same method as in Fig. 2 for wild type worms. The upshot is that the short-time behavior of npr-1 mutants on food is similar to wild type N2 worms off food	ACB	ABC	CAB	BCA	Selection 3
A: The signatures of Virgo West and the group around NGC4709, that are respectively beyond Virgo and Centaurus in the lines-of-sight, can also be identified as secondary wavesB: There exist a visual agreement between the observed lines-of-sight dynamical state of Virgo and Centaurus and those reproduced by CLONE. Comparisons with another constrained simulation, called SIBELIUS (McAlpine et al., 2022), shown in the appendix, reveal that an agreement at this level of detail between the full observed and constrained simulated lines-of-sight is not utterly expected. C: These smaller waves follow the highest ones representing the main clusters in both the observational and the simulated lines-of-sight	ABC	BCA	CAB	ACB	Selection 4
A: In this section, we demonstrate the superiority of the enhanced EPR over the HJB loss alone and over the normalizing flow (NF), which is a class of generative models used for density estimation that leverage invertible transformations to map between complex data distributions and simple latent distributions [39], through 2D benchmark examplesB: We then apply the enhanced EPR to the 3D Lorenz model and a 12D Gaussian mixture model to show its effectiveness in constructing energy landscapes in higher dimensionsC: We remark that alternative approaches have been investigated for potential construction in limit cycles [40] and the Lorenz system [41] distinct from our methodology.	ACB	CAB	ABC	BCA	Selection 3
A: The generalization can be done in multiple ways, differing in the number of required stochastic variables and the symmetry properties of the non-unitary perturbation. B: Notice the difference between 𝒩𝒩\mathcal{N}caligraphic_N (the size of the measurement apparatus) and N𝑁Nitalic_N (the number of pointer states with nonzero weight in the initial superposition)C: Having a model for DQSR based on SUV for the specific case of a two-state superposition of pointer states, we will now generalize the approach to initial superpositions over N𝑁Nitalic_N pointer states	BAC	CBA	BCA	BCA	Selection 2
A: We also showed that training and generalization are also possible using time series dataB: As expected, there exists a limit for how much generalization can be achieved, however, we note the presence of regularity that is observed in the gradual increase in test loss. Using the proposed d𝑑ditalic_d-statistic we can therefore evaluate the confidence in the inferred prediction, even in cases when we depart from standard SLT, UAT assumptions of i.i.d. sampling and a compact support. C: Overall, the neural networks can approximate various dynamical models and extrapolate predictions even when statistical properties of the input data, or the graph structure change, transcending the formal boundaries of SLT and UAT	BCA	CAB	BAC	CBA	Selection 1
A: The simulation geometry is described in more detail in Section A.2. Fig. 2 shows these simulated measurements on an α𝛼\alphaitalic_α-curve for the {6,4}64\{6,4\}{ 6 , 4 } MeV case, demonstrating that the theoretically predicted α𝛼\alphaitalic_α-line overlaps are present both at low area densities and at high-Z𝑍Zitalic_Z.B: To validate these results, transparency simulations were run in Geant4 Geant4 ; grasshopper C: 10101010, 6666, and 4444 MeV bremsstrahlung beams were directed through different materials of known thicknesses, and the beam transparencies were recorded	BAC	ACB	CAB	BCA	Selection 3
A: This method combines the linear decomposition method Cho & Lazarian (2003), singular value decomposition (SVD) method Santolík et alB: (2003), and multi-spacecraft timing analysis Pincon & Glassmeier (2008). We perform the calculations in each moving time window with a five-hour length and five-minute moving step. The window length selection (5 hours) provides low-frequency (large-scale) measurements while ensuring 𝐁0subscript𝐁0\mathbf{B}_{0}bold_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is approaching the local background magnetic field. C: We calculate wavenumber-frequency distributions of Alfvénic magnetic field and proton velocity power by an improved Alfvén mode decomposition method	ABC	CAB	BCA	ACB	Selection 3
A: These are the generalizations of closed string operators in the case of anyons. These operators are not necessarily unitary.B: The first step to generalizing this construction to more general excitation types (in general dimension, and including gapped boundaries) is the notion of flexible operator algebraC: This is an algebra of operators acting on a region ΩΩ\Omegaroman_Ω with the property that when acting on the reference state, their support can be deformed topologically within ΩΩ\Omegaroman_Ω	CBA	BCA	ACB	CAB	Selection 4
A: In this work we have connected several problems of majorization to the theory of preordered semiringsB: Using this framework, we have derived new sufficient and generically necessary conditions for large-sample and catalytic matrix majorization, i.e., Blackwell dominance in the finite-outcome settingC: In this latter context, we have in particular identified the relevant monotone quantities as matrix α𝛼\textstyle\alphaitalic_α-divergences which can be viewed as a generalization of the one-parameter family of the standard Rényi divergences.	CBA	ABC	CAB	ACB	Selection 2
A: Also a large non-zero 𝐄⋅𝐁⋅𝐄𝐁{\bf E}\cdot{\bf B}bold_E ⋅ bold_B can enhance production of axions. The gravitational waves, in the early Universe also have the potential to generate large-scale magnetic fields, corresponding to resonant modes. B: Our numerical simulations suggest that gravitational waves also will lead to parametric resonance in gauge fields. The gravitational waves will be dampened more efficiently due to the color factorC: Further, the gravitational waves can induce large fluctuations in C⁢P𝐶𝑃CPitalic_C italic_P violating physical observables. These observables can subsequently lead to a large-scale imbalance in local chiral charge distributions	CBA	CBA	CAB	BAC	Selection 3
A: We receive support from Ontario Research Fund—research Excellence Program (ORF-RE), Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Institute for Advanced Research (CIFAR), Canadian Foundation for Innovation (CFI), the National Science Foundation of China (Grants NoB: Computations were performed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile and/or Large Memory System] computing platform(s). SOSCIP is funded by the Federal Economic Development Agency of Southern Ontario, the Province of Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario academic member institutions.C: 11929301), Thoth Technology Inc, Alexander von Humboldt Foundation, and the Ministry of Science and Technology(MOST) of Taiwan(110-2112-M-001-071-MY3)	CBA	CAB	ACB	CAB	Selection 3
A: If ΩΩ\Omegaroman_Ω is semisimple, its topological part can be evaluated explicitly in the idempotent basis, see e.gB: Generating functions of a CohFT ΩΩ\Omegaroman_Ω can be defined by integrating CohFT classesC: [18, Section 2.5.1]. A consequence of the Givental-Teleman classification is that the generating functions of ΩΩ\Omegaroman_Ω can be explicitly written as sums of graphs. A reference for this can be found in [18, Section 2.5.2].	CBA	BAC	CBA	CBA	Selection 2
A: This pattern has been demonstrated to be independent of the input data which exhibits white noiseB: Since the input data consists of real world natural language sentences that our brain processes, our results demonstrate that artificial neural networks with close to human-level performance exhibit very similar 1/f1𝑓1/f1 / italic_f patterns as their biological counterpartsNovikov et al. (1997); He (2014, 2011).C: In this study, we have found convincing evidence of 1/f1𝑓1/f1 / italic_f noise in deep neural networks like the Long-Short-Term Memory (LSTM) network	ABC	CAB	CBA	BCA	Selection 4
A: To investigate the properties of non-equilibrium QGP medium, we take into account the viscous effect and study the responses of viscous QGP medium to the magnetic fieldB: Subsequently, we explore how the magnetized viscous effect impacts both the HQ potential and momentum diffusion coefficientC: In the strong magnetic limit within the LLL approximation, Landau level quantization ensures that only the longitudinal component of bulk viscosities, arising from the contributions of LLL (anti)-quarks, exists Hattori:2017qih . The response of viscous quark matter to the magnetic field is manifested in the longitudinal bulk viscous modified distribution function of light (anti-)quarks, which can be obtained using the Boltzmann kinetic theory.	CBA	ACB	ABC	ACB	Selection 3
A: This allows us to provide a rigorous error and complexity analysis when uploading first a truncated and discretized approximation of the multivariate log-normal distribution and then uploading an approximation of the CPWA payoff function in rotated form, where the approximation consists of truncation as well as the rounding of the coefficients of the CPWA payoff function. This together with a rigorous error and complexity analysis when applying the modified iterative quantum amplitude estimation algorithm [fukuzawa2022modified] allows us to control the output error of our algorithm to be bounded by the pre-specified accuracy level ε∈(0,1)𝜀01\varepsilon\in(0,1)italic_ε ∈ ( 0 , 1 ), while bounding its computational complexity; we refer to Theorem 1 for the precise statement of our main resultB: In particular, we prove that the computational complexity of our algorithm only grows polynomially in the space dimension d𝑑ditalic_d of the Black-Scholes PDE and in the (reciprocal of the) accuracy level ε𝜀\varepsilonitalic_ε. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. To the best of our knowledge, this is the first work in the literature whichC: Our main contribution lies in a rigorous error analysis as well as complexity analysis of our algorithm. To that end, we first introduce quantum circuits that can perform arithmetic operations on two complement’s numbers representing signed dyadic rational numbers, together with its complexity analysis	BAC	BCA	BAC	BAC	Selection 2
A: We see that the traditional method accepts some regions of meta-stable EW-breaking vacuum, but checking the cosmological history proves that incorrectB: The exclusion of such regions is computationally fast, because one only needs to trace the minima of the potential with temperatureC: On the other hard, it is more time-consuming to exclude the yellow region, as it involves calculation of the transition probability at finite temperature.	BAC	ABC	CBA	BCA	Selection 2
A: We then add nodes and links following preferential attachment as described in [2] where each new node (labeled with a consecutive number) is attached to existing nodes via two linksB: BA: For this network, we start with a simple path graph with 4 numbered nodesC: However, and in contrast to the reference, to ensure that the graph has a chromatic number of 2222, for an even (odd) number of already existing nodes, a newly added node can only connect to nodes with an odd (even) label.	BCA	BAC	CBA	ACB	Selection 2
A: Since the current can be polarization, time and space dependent, the effective speed can also have the same dependencyB: In this approach the effects of the current on the propagation of the EMWs has been encoded in the effective speedC: This is in agreement with the experimental evidence of the change of the speed of EMWs, depending on the the medium in which are propagating. In this approach the physical properties of the medium are modelled mathematically by the effective speed.	BCA	ABC	BAC	ACB	Selection 3
A: Figure 9 shows the Tully-Fisher relation from a single mock catalogue, for comparison with Figure 3B: There are no outliers as the generating model is purely Gaussian. Figure 10 shows the distributions of the actual and mock observables for the CF4 Tully-Fisher dataset and their pairwise correlations. It demonstrates that the mocks provide a reasonably realistic representation of the CF4 Tully-Fisher data, including the effective magnitude selection imposed by the flux selection. C: Note that the ‘observed’ sample includes galaxies below the subsequently-imposed lower limit to the velocity width	ACB	CBA	CBA	CAB	Selection 1
A: For MCMC, this is in some part due to higher mean values, and partly due to inflated errors. B: For GP, the alleviation is primarily because of the shifting of the mean to somewhat higher valuesC: With F1 (ΛΛ\Lambdaroman_ΛCDM), F2 (CPL) and F3 (PEDE), tensions decrease progressively from Fisher to MCMC to GP	BAC	BCA	CAB	CBA	Selection 4
A: In the finite system, there is a small energy gap even in the gapless system. Thus, we make use of the finite-size scaling to check the energy gap in the thermodynamic limitB: In the VBS phase (Fig.6(a)), the energy gaps at different sizes are larger than 12⁢t3/V212superscript𝑡3superscript𝑉212t^{3}/V^{2}12 italic_t start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / italic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. After linear fitting, the energy gap is still positively largeC: Here, we have to mention that the results of VBS present a strong deviation from the fitting results, and it may result from the finite size effects. In comparison, at DQCP (Fig.6(b)), the gap energies at different system sizes exhibit good linear behavior. The negative value at the thermodynamic limit indicates the close of the energy gap.	ABC	CAB	BAC	CAB	Selection 1
A: For example, in [9] the asymptotics of the fundamental solution was constructed depending on the properties of the kernel of the process describing the jumpsB: It was also noted that in the case of pure jumps (without diffusion), the fundamental solution always contains a singular component.C: The properties of fundamental solutions for integro-differential equations have been studied in previous works	BAC	BCA	CBA	ACB	Selection 2
A: The work demonstrates evidence that the feature of non-linearity from classical ML might be useful in quantum circuit models. B: This work put forth a preliminary investigation into the learning capabilities of the entire generative model, known as the Quantum Neuron Born Machine (QNBM), along with a demonstration of better performance over the more widely investigated Quantum Circuit Born Machine (QCBM) model [Benedetti_2019, Liu_2018, 2020Coyle, gili_qcbm], which does not contain these non-linear activations in the architectureC: Recent work proposed a quantum generative architecture that achieves non-linearity in the quantum state evolution using repeat-until-success (RUS) sub-routines containing mid-circuit measurements, similar to non-linear activations in a classical feed-forward network [gili_qnbm]	CAB	ABC	CAB	CBA	Selection 4
A: Despite the fact that the coded-aperture telescopes are unable to construct the images of extended objects, some information about their angular sizes can still be obtained from the IBIS data (see, e.g., Kuznetsova et al., 2019)B: In addition, special methods different from those that are used for point sources are required to be applied to properly extract the flux from an extended object. In particular, for the flux from an X-ray source to be correctly estimated, the convolution function, when reconstructing its image, must maximally coincide with the profile of this source for a given telescope. C: The X-ray emission from the Ophiuchus galaxy cluster is known to be an extended one with a characteristic size of about 15 arcmin (Nevalainen et al., 2009; Werner et al., 2016), which is comparable to the angular resolution of the IBIS telescope (12 arcmin)	BCA	CAB	ABC	CAB	Selection 1
A: We have that the bulk where the branes are embedded has a particular distribution of matter, however the homogeneity and isotropy is fulfilled in the internal space of the brane, having in turn an arbitrary sectional curvatureB: For this case, we also find equations for the evolution of perturbations in branes, and their respective scalar, vector and tensor modes, choosing also the Newtonian conformal and synchronous gauges, these in their version of extra dimensions.C: In that same line of thought and looking for a formulation of perturbations in extra dimensions, we present a development of perturbations for branes where we start from evolutionary equations and in where the geometry of space-time is determined by the extrinsic curvature tensor	CBA	BCA	BAC	CAB	Selection 2
A: Gamma-ray bursts are some of the most powerful astrophysical phenomena in the universeB: The collapsar model describes GRBs as the consequence of a failed supernova where all the energy of the gravitational collapse is funneled into a jet (Woosley, 1993)C: The jet is fed by the rotation of a central engine, a BH, through the BZ mechanism.	CAB	ABC	BCA	CBA	Selection 2
A: Linear combinations of spectral projections are simple functions on the spectrum — “simple” here is a technical term, meaning a finite linear combination of indicator functions. The statement that a positive operator can be approximated arbitrarily well with simple functions is then the standard measure theory fact that simple functions are dense in any L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT spaceB: See appendix A.2. by positive linear combinations of its spectral projections, so we should ask first what it means for a projection to be a density matrix, and then ask about whether the property “being a density matrix” is preserved under linear combinations and limits.C: Every positive operator can be approximated arbitrarily well171717Here “approximated” is meant in the sense of the norm topology. Using spectral theory, a bounded positive operator can be thought of as the identity function on a compact measure space, corresponding to the spectrum of the operator	ACB	ABC	CBA	BCA	Selection 4
A: The alignment of FTLE behaviour, particularly into the neutral subspace, has been previously observed in similar conceptual chaotic models when they undergo a transition between regions of the attractor, as well as in models of regime shifts in atmospheric flows [16, 17, 18]B: [20], [21]). C: In general, however, more useful measures for alignment strength already exist in regards to Lyapunov vectors, which not only help characterize the dynamics of a system [19], but strong levels of alignment can be used to predict chaotic transitions (e.g	CBA	CAB	ACB	BCA	Selection 3
A: The operator form of (1) first appeared in [15] in relation with conformal field theoryB: In [16] it was considered inside the context of three-dimensional integrable systems, while the set-theoretical version of (1) was studied in [17] in connection with Poisson maps, see also [18]C: Furthermore, the pentagon relation (1) itself, as it was shown in [19], serves as a manifestation of the 3↔2↔323\leftrightarrow 23 ↔ 2 Pachner move [20], where three tetrahedra with a common edge are replaced by two tetrahedra with a common face on a triangulation of a piecewise-linear 3-manifold.	CAB	ABC	ACB	CAB	Selection 2
A: NHSE can also occur in the synthetic field moments space of zero-dimensional bosonic quantum dimers [372]. An new, interacting form of the NHSE also occurs in the presence of interacting impurities, as manifested by so-called squeezed polarons which are impurity-localized dipole-like density profiles that are impervious to the lattice boundaries [373]. More interesting multi-polar NHSE signatures have been suggested in [374], and it remains to be seen if such geometric anisotropy may be generalized to give rise to quantum Hall-like states [375] with NHSE interplay.B: The NSHE can also emerge in the effective descriptions of various interacting models, even though the interactions themselves are not asymmetric hoppingsC: For instance, real-space dynamical mean-field theory reveals the NHSE in the pseudospectrum of some strongly-correlated systems [371]	CBA	ACB	BCA	CAB	Selection 4
A: and P.Y. were supported by NSFC Grant No. 12074438, Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100, and the Open Project of Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices under Grant No. 2022B1212010008. M.CB: acknowledges support from NSF under award number DMR-1846109. J.-Y.C. was supported by Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Project No. KF202207), Fundamental Research Funds for the Central Universities, Sun Yat-sen University (Project No. 23qnpy60), a startup fund from Sun Yat-sen University, the Innovation Program for Quantum Science and Technology 2021ZD0302100, Guangzhou Basic and Applied Basic Research Foundation (grant No. 2024A04J4264),C: We acknowledge conversations with Johannes Hauschild, Xiao-Yu Dong, Hui-Ke Jin and Hong-Hao Tu. The MPS calculations were performed using the TeNPy Library (version 0.9.0) [48]. L.M.C	ACB	ABC	BAC	BCA	Selection 4
A: The red cross marks the estimate in (37). The color encodes the expectation value (left) or the average (right) of the cut sizeB: On the left, we mimic an ideal quantum device by evaluation of (19). On the right, values are results from the quantum hardware. Here, every pixel represents the average taken over 1,02410241,0241 , 024 samples.C: Figure 4: Visualized results for QAOA parameter-estimate on instances a, b and c from Fig. 3. The x- and y-axis represent values of the parameters β𝛽\betaitalic_β and γ𝛾\gammaitalic_γ, respectively	CAB	ABC	BCA	BAC	Selection 3
A: 4(a)-(d), respectively. Around the cross points in the curves, the interference stripes are clearly visible,B: by shifting the phase will be discussed in the next subsectionC: The quantum Lissajous scars for ωx/ωy=2/1,3/1,3/2,4/3subscript𝜔𝑥subscript𝜔𝑦21313243\omega_{x}/\omega_{y}=2/1,3/1,3/2,4/3italic_ω start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT / italic_ω start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 2 / 1 , 3 / 1 , 3 / 2 , 4 / 3 are shown in Figs	CAB	ACB	ABC	BAC	Selection 1
A: [Eq. (14)]B: We have taken theC: This has the effect of replacing D𝐷Ditalic_D in A⁢(ℓ−)𝐴superscriptℓA(\ell^{-})italic_A ( roman_ℓ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) with Dnsubscript𝐷𝑛D_{n}italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT	BCA	ACB	CAB	BAC	Selection 2
A: Furthermore, this method could miss potential dark UDGs — ones with extremely faint stellar light that are of high astrophysical interest (see Li et al., 2022, for more detailed discussions). B: The two major challenges with this method are: (1) the high computational cost of image segmentation needed to separate UDGs from the night sky background, and (2) contamination of samples of UDGs by imaging artifacts (such as scattered light) and light-reflecting diffuse gas clouds in the interstellar medium (so-called ‘Galactic cirrus’)C: Currently, UDGs are found using a computational search of their associated faint stellar light in images	CBA	ACB	BCA	ACB	Selection 1
A: The completeness (or recall rate) of the RF classifier for different bins in planetary period-radius space is measured using the full set of synthetic planetary injections based on the Kepler planet sample (§4.1). The analysis is based on 7,751 of the 10,850 synthetic planetary injections that were processed by the RF classifier; the remaining objects were omitted due to missing features. B: It is important to understand how the classifier performs across the planetary radius-period distribution to evaluate the completeness of our intermediate DTARPS-S Analysis List with 7,377 objects and the smaller DTARPS-S Planet Candidate catalog produced in Paper IIC: The recall rate of the injected planetary signals across the range of the injected period and radii can quantitatively measure the ability of the RF classifier to recover planets in the DIAmante data set	ACB	CAB	CAB	CBA	Selection 4
A: The DTARPS-S Candidates catalog (Table 1) and DTARPS Galactic Plane list (Table 4) are ready for spectroscopic follow-up to remove remaining False Positives, measure other planetary properties, and further investigate the planetary systemsB: The results are informed by preliminary results of reconnaissance spectroscopy of a small portion of the DTARPS-S Candidate sample. Paper III also returns to the completeness heat maps introduced in Paper I showing that the DTARPS-S Candidate planet occurrence rates are compatible with those obtained with the Kepler mission.C: Paper III examines various astronomical properties of these samples, discussing exoplanet types such as the hot Neptune population, (extreme) Ultra Short Period planets, planets orbiting low-mass stars, and candidates for atmospheric characterization by transmission spectroscopy	CBA	CAB	ACB	CAB	Selection 3
A: In case of EqB: (8) with (12), the heavy-light diquarks satisfy the universal description irrespective of both mass of the light constituents and mass of the heavy constituentC: In case of Eq. (14) with (15), the heavy-light diquarks satisfy the universal description irrespective of mass of the heavy constituents.	BCA	ACB	BAC	ABC	Selection 4
A: Though we have provided some lower bounds on the size, we are also lacking a non-trivial upper bound. A second problem that we left open is the efficient construction of minimal generating sets of qudit Pauli subgroups in all cases. The missing step to get from a near-minimal generating set to a minimal one involves determining whether appropriate phases can be added to each generator as in Theorem 6.9.B: We have applied the theory of modules over commutative rings to study properties of the qudit Pauli groupC: One problem we have left open is the maximum size of a non-commuting set on n>1𝑛1n>1italic_n > 1 qudits	CAB	ACB	CBA	ABC	Selection 1
A: Here, polarized electron beams can play a crucial role in increasing the capability of analysisB: It allows a more selective probing of the process by enhancing the deviations from the SM via anomalous couplings and also the signal-background ratio. Eventually, it provides to make stringent tests and to get better constraints on anomalous couplings. C: On the other hand, CLIC experiment program provides an opportunity for ±%80\pm\%80± % 80 polarized electron beams and no positron polarization at the center-of-mass energy of 3 TeV with the integrated luminosity of 1 ab-1 and 4 ab-1, respectively	ACB	BCA	BAC	BAC	Selection 2
A: These algorithms are based on iterative optimization techniques, such as the Gerchberg-Saxton algorithm, the Hybrid Input-Output algorithm, and the Error Reduction algorithm. They are powerful and can recover images with high qualities. However most of the time, iterative phase retrieval algorithms suffers from twin images and stagnationsB: When the source becomes non-coherent. The CDI becomes hopeless. To overcome this problem, we report the new CDI method. C: Coherent diffraction imaging (CDI) is an imaging technique that uses coherent beam to reconstruct images of micro and nanoscale objects. It is a powerful tool for imaging small scale structures with high resolution, and it has been used to image a wide range of materials, including proteins, viruses, and nanomaterials. Phase retrieval algorithms are used in coherent diffraction imaging (CDI) to reconstruct the phase of a diffracted wavefront from its intensity measurements	ABC	BCA	BAC	ACB	Selection 2
A: 3B: Taking the indirect effects in the production network into account leads to 3 to 42 times higher estimates of job loss and 6 to 23 times higher estimates of economic output loss for the decarbonization strategies depicted in FigC: Note that EW-ESRI and OW-ESRI enable us to calculate both, the direct job and economic output loss incurred by the removed firms, and the indirect losses experienced by all subsequently affected firms in the production network, for every decarbonization strategy, respectively	ABC	BAC	CBA	CAB	Selection 3
A: It is tantalizing that the rich theoretical scenario of USC has an experimental counterpart limited so far to standard spectroscopyB: What has prevented a broader experimental investigation? To gain insight into this issue we address a fundamental problem posed since the birth of the field [1] namely the experimental detection of VPs, which still awaits demonstration despite several theoretical proposals [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 2, 3]. The specific question we ask is whether it is possible to overcome experimental challenges posed by available quantum hardwareC: This work shows that the answer is positive but not trivial. Indeed detecting VPs in an efficient and faithful way requires combining state-of-the-art technologies, such as a multilevel AA unconventional design [42, 43, 44, 45], and a multiphoton coherent control protocol with a tailored integrated measurement technique. The implementation of this setup is feasible with present-day superconducting technology [46].	ACB	ABC	ACB	CAB	Selection 2
A: In general, Q-ball potentials produce non-linearities in the equation of motion which require higher order modes to be nonzero. We have shown that this suggests the existence of a non-local, hence radiating, portion of the scalar fieldB: For small rotation, however, these radiation modes are only required to appear at subleading order in the expansion parameter. Therefore, though the localized perturbation is only an approximate solution to the equations of motion, the approximation can be quite close to a true solutionC: We estimate the life-time of the localized perturbation from the radiated power contained in the non-localized part of the field. We find that small angular momentum solitons are classically metastable in the small angular velocity limit with lifetimes that can be relevant to cosmological studies.	CAB	BAC	ABC	CBA	Selection 3
A: On the other hand, with rigid confinement, dilation is frustrated. When the dimension of a local high-density region is comparable to the gap between boundaries, a sudden rise in the stress response is expected [9, 5]B: The intense stress compels particles into profound interactions, revealing features that might otherwise remain hidden, such as the role of particle adhesion [46] and the occurrence of hysteresis [26]. Furthermore, macroscopic clusters of particles exist only briefly under intense stress, leaving behind smaller ones that subsequently promote the reformation of high-density clusters. From a heterogeneous perspective, the overall stress response of a suspension in the conventional shear-thickened state (with rigid boundaries) is primarily governed by the increasingly prominent formation and collapse of these high-stress regions [14, 15, 49]. The constitutive relation, calibrated with bulk rheology measurements, only captures these intermittent microscopic events on average.C: With soft boundaries, such as the free surface experiments here, dilation is allowed to a certain extent. In this scenario, the inhomogeneity develops into a persistent density-wave state, where particles do not make long-lived contact	CAB	ACB	CAB	BCA	Selection 4
A: The Pcal can give a modulation the mirror surface with photon pressureB: The summary of the calibration overview in O3GK is described in the summary paper. [8] The initial characterization of photon calibrator instruments are summarized elsewhere. [17, 18] Pcal can modulate the mirror displacement by injecting the power-stabilized laser with intensity modulation. The main applications of the photon calibrator are (i) calibration of an interferometer response, (ii) monitoring of time dependency of the interferometer response, and (iii) hardware injection to verify the analysis pipeline [19, 18]. The displacement of the mirror can be written as:C: In the joint observation run 3 (2020 April) with KAGRA and GEO600 (O3GK), [15, 6, 8, 16] KAGRA employs the photon calibrator as a primary calibrator	BCA	CBA	ABC	ACB	Selection 4
A: Set v⁢(x)=∫0x∫0t{q⁢(s)⁢u⁢(s)−g⁢(s)}⁢𝑑s⁢𝑑t𝑣𝑥superscriptsubscript0𝑥superscriptsubscript0𝑡𝑞𝑠𝑢𝑠𝑔𝑠differential-d𝑠differential-d𝑡v(x)=\int_{0}^{x}\int_{0}^{t}\{q(s)u(s)-g(s)\}dsdtitalic_v ( italic_x ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT { italic_q ( italic_s ) italic_u ( italic_s ) - italic_g ( italic_s ) } italic_d italic_s italic_d italic_tB: Hence v∈W2,1⁢(xj,xj+1)𝑣superscript𝑊21subscript𝑥𝑗subscript𝑥𝑗1v\in W^{2,1}(x_{j},x_{j+1})italic_v ∈ italic_W start_POSTSUPERSCRIPT 2 , 1 end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT ), v′′⁢(x)=q⁢(x)⁢u⁢(x)−g⁢(x)superscript𝑣′′𝑥𝑞𝑥𝑢𝑥𝑔𝑥v^{\prime\prime}(x)=q(x)u(x)-g(x)italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x ) = italic_q ( italic_x ) italic_u ( italic_x ) - italic_g ( italic_x ) a.eC: x∈(xj,xj+1)𝑥subscript𝑥𝑗subscript𝑥𝑗1x\in(x_{j},x_{j+1})italic_x ∈ ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT ), and we get the equality	ABC	BAC	CBA	BAC	Selection 1
A: We believe MIQS can simplify initialization of a qutrit, by coupling it to a qubit. B: However, controlling qutrits can be a difficult taskC: There are various factors that need to be calibrated, such as frequency of the drive, amplitidue of the drive, leakage, etc	CAB	CBA	ACB	BAC	Selection 1
A: The structure of this paper is as follows. In Sec. II, we provide the general equations required to construct a bouncing solutionB: To begin with, we demonstrate how two popular inflationary models can be tweaked (albeit in an ad-hoc manner) to achieve this goal. Thereafter, we show how a simple augmentation of a very well-motivated model also leads to very similar results. As these models suffer from a ghost instability, we propose, in Sec. V, a way to mitigate the problem by invoking a coupling between the two fields. Finally, in Sec. VI, we summarise and present a future outlook.C: In Sec. III, we focus on constructing a bouncing phase followed by the inflationary dynamics. We do this in two steps	CAB	ACB	CBA	ABC	Selection 2
A: Another domain feature that we have not considered is the inclusion of more than one target, one or more of which may be of non-circular shapeB: The multiple-target problem has been considered at length for Brownian motion on flat 2- and 3-dimensional geometries using hybrid asymptotic-numerical methods see (e.g., [32, 35, 30, 51, 55, 56, 57, 26] and the references therein)C: Several numerical optimization studies have been done to find optimal target arrangements that minimize the spatial average of the stopping time (e.g., [58, 59, 60, 61]). The inclusion of more than one target also gives rise to the question of splitting probabilities (see e.g., [31, 62, 63]) and shielding effects [31], and how they compare to their Brownian counterparts.	BCA	BCA	BAC	ABC	Selection 4
A: In the convention, θ~−,csubscript~𝜃𝑐\tilde{\theta}_{-,c}over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT - , italic_c end_POSTSUBSCRIPT is locked at 0 and π/2𝜋2\pi/2italic_π / 2 for s𝑠sitalic_s- and d𝑑ditalic_d- pairing states respectively. Thus if C0=Cπsubscript𝐶0subscript𝐶𝜋C_{0}=C_{\pi}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, only quasi-long-range order of ΔSsubscriptΔS\Delta_{\mathrm{S}}roman_Δ start_POSTSUBSCRIPT roman_S end_POSTSUBSCRIPT exists for s𝑠sitalic_s-wave states; the same applies to d𝑑ditalic_d-waveB: Given that C0≠Cπsubscript𝐶0subscript𝐶𝜋C_{0}\neq C_{\pi}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ italic_C start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT in general, we have both quasi-long-range orders in either s𝑠sitalic_s- and d𝑑ditalic_d-wave states; in other words, ⟨ΔS⁢j⁢ΔS,j+d†⟩delimited-⟨⟩subscriptΔS𝑗subscriptsuperscriptΔ†S𝑗𝑑\langle\Delta_{\mathrm{S}j}\Delta^{\dagger}_{\mathrm{S},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_S italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_S , italic_j + italic_d end_POSTSUBSCRIPT ⟩ and ⟨ΔD⁢j⁢ΔD,j+d†⟩delimited-⟨⟩subscriptΔD𝑗subscriptsuperscriptΔ†D𝑗𝑑\langle\Delta_{\mathrm{D}j}\Delta^{\dagger}_{\mathrm{D},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_D italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_D , italic_j + italic_d end_POSTSUBSCRIPT ⟩ decay algebraically in \|d\|𝑑\|d\|\| italic_d \| with the same exponent. However, we can still have a microscopic definition of s𝑠sitalic_s-wave and d𝑑ditalic_d-wave statesC: If the leading algebraic decay prefactor of ⟨ΔS⁢j⁢ΔS,j+d†⟩delimited-⟨⟩subscriptΔS𝑗subscriptsuperscriptΔ†S𝑗𝑑\langle\Delta_{\mathrm{S}j}\Delta^{\dagger}_{\mathrm{S},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_S italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_S , italic_j + italic_d end_POSTSUBSCRIPT ⟩ is larger, we call the state s𝑠sitalic_s-wave, otherwise d𝑑ditalic_d-wave. This is equivalent to the definition from the relative sign of the coefficient of the leading algebraic decay components of ⟨Δ0,j⁢Δ0,j+d†⟩delimited-⟨⟩subscriptΔ0jsubscriptsuperscriptΔ†0jd\langle\Delta_{\mathrm{0,j}}\Delta^{\dagger}_{\mathrm{0,j+d}}\rangle⟨ roman_Δ start_POSTSUBSCRIPT 0 , roman_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , roman_j + roman_d end_POSTSUBSCRIPT ⟩ and ⟨Δπ,j⁢Δπ,j+d†⟩delimited-⟨⟩subscriptΔ𝜋jsubscriptsuperscriptΔ†𝜋jd\langle\Delta_{\mathrm{\pi,j}}\Delta^{\dagger}_{\mathrm{\pi,j+d}}\rangle⟨ roman_Δ start_POSTSUBSCRIPT italic_π , roman_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_π , roman_j + roman_d end_POSTSUBSCRIPT ⟩, where Δky,j=cj,ky,↑⁢cj,ky,↓subscriptΔsubscriptkyjsubscript𝑐𝑗subscript𝑘𝑦↑subscript𝑐𝑗subscript𝑘𝑦↓\Delta_{\mathrm{k_{y},j}}=c_{j,k_{y},\uparrow}c_{j,k_{y},\downarrow}roman_Δ start_POSTSUBSCRIPT roman_k start_POSTSUBSCRIPT roman_y end_POSTSUBSCRIPT , roman_j end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_j , italic_k start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ↑ end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j , italic_k start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ↓ end_POSTSUBSCRIPT A positive relative sign is defined as s𝑠sitalic_s-wave states and a negative relative sign is defined as d𝑑ditalic_d-wave states.	ACB	BCA	ABC	BAC	Selection 3
A: The entanglement spectrum can show different behaviors from the energy spectrum of the subsystem as an independent system under open boundary condition. Tuning the coupling between the subsystem and environment can change the ES gap, which indicates that the coupling accounts for the inconsistency of the Li-Haldane conjectureB: In summary, we have systematically studied the relation between the energy and entanglement spectrum on the boundary in the AKLT phase of the S=1/2𝑆12S=1/2italic_S = 1 / 2 Heisenberg model on the square-octagon lattice with a symmetry-breaking term on the boundary and a tunable coupling between the subsystem and its environment. The correspondence does not always apply if the perturbation is added to the environmentC: This inconsistency can be naturally explained in the wormhole picture for the partition function in the replica manifold. In the path-integral picture, the wormhole effect induced by the coupling plays an important role in transporting the interaction of the environment and provides a pathway to control the ES. The wormhole picture can explain the correspondence between the entanglement spectrum and energy spectrum for the gapless and gapped cases which is indeed the fundamental mechanism of the ES.	BAC	CAB	ABC	ACB	Selection 1
A: In particular, it would be interesting to compare a policy that incorporates entanglement distillation to one without entanglement distillation that simply does the memory-cutoff policy with a single memory [72]. It is not obvious that entanglement distillation will always give a better overall fidelity, and understanding in what parameter regimes this is the case will help guide us towards a concept of a noise and loss threshold for quantum communication, similar to the fault-tolerance thresholds for quantum computation that are defined by the point at which the logical error rate of an error-corrected quantum computation is less than the raw, physical error rate. B: Our work opens up several interesting avenues for future work. One direction for future work is to add the possibility of doing entanglement distillation, which is a form of quantum error correction, to the MDP formalism presented hereC: As entanglement distillation is now within the reach of experiments [121, 122, 123], this question is especially pertinent, and much remains to be explored about what policies are optimal in this scenario, building on prior works that have considered specific policies [38, 111, 93, 94, 124, 125, 126, 96]; see also Refs. [71, 72], which numerically investigate several of these scenarios. We therefore expect our insights on the features of improved policies to be a valuable starting point when designing better protocols for quantum repeater chains with multiple memories and entanglement distillation	CBA	ACB	CAB	BCA	Selection 3
A: Sequences of symbols that disagree in every place could find application in authentication protocols [14]B: The use of checking sequences for elements that were not sent, that is the sequence of states sent and the sequence of measurement results disagree, has been used in digital signature schemes [15].C: Alice can authenticate herself to Bob by providing elements of her sequence to him, and he can check that what Alice sent disagrees with his sequence in every place	CAB	BAC	CAB	ACB	Selection 4
A: When the HNLS condition is satisfied and the HL is achievable, for qubit probes where the signal and noise acts individually on each probe, an optimal ancilla-free repetition code can be constructed Dür et al. (2014); Arrad et alB: (2019). In the case where the HNLS condition is violated, weakly spin-squeezed states were known to achieve the optimal SQL for a Pauli-Z signal under local dephasing noise Ulam-Orgikh and Kitagawa (2001); Escher et al. (2011); while in general ancilla-assisted QEC strategies are necessary. In this paper, we will address the open question whether there exist ancilla-free QEC codes for Hamiltonian estimation under Markovian noise that achieve the optimal HL (or SQL). We will exploit quantum entanglement among multiple probes to design two classes of optimal ancilla-free QEC codes.C: (2014); Peng and Fan (2020). Additionally, when the signal and the noise commute, e.g., a Pauli-Z signal acting on multiple probes under correlated dephasing noise, optimal ancilla-free codes were also proven to exist Layden et al	ACB	CBA	BAC	CBA	Selection 1
A: N000142212764. B.P and A.M.A. acknowledge support from ONR Grant No. N000141712793. B: Office of Naval Research (ONR) through Grants No. N000142112450 and MURI NoC: This work is supported in part by DARPA Topological Excitations in Electronics (TEE) program under grant No. DP18AP900007, the U.S	CBA	CAB	BCA	BAC	Selection 1
A: Figure 3: (a) Energy errors for TC calculations of the H2 bond dissociation w.r.tB: Å). TC-VarQITE calculations are based on HF orbitals in a STO-6G (4 qubits), 6-31G (8 qubits), and cc-pVDZ (20 qubits) basis sets. Also shown are no-TC FCI/cc-pVDZ calculations.C: CBS limit results (mH vs	ACB	CBA	BAC	BCA	Selection 1
A: It started with Sutherland’s discovery of a flat band in the dice lattice [28]B: Historically, developing flat band models has been a long and arduous processC: It continued with Lieb’s work on the Hubbard model, demonstrating that certain bipartite lattices with chiral flat bands exhibit ferromagnetism [29]. However, in recent years, there has been a growing interest in the development of new topological flat band models [2] that will allow for a better understanding of the properties of these materials and their potential applications and which can support quantum Hall-like states, including integer quantum Hall (QH) effect [30, 31], fractional quantum Hall (FQH) effect [32, 33, 34], and the existence of electronic fractional Chern states [35, 32, 11, 12].	ABC	BAC	CBA	ACB	Selection 2
A: It is worth mentioning that the studies in RefsB: predictions regarding the mixing angles with θ23=π4subscript𝜃23𝜋4\theta_{23}=\frac{\pi}{4}italic_θ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = divide start_ARG italic_π end_ARG start_ARG 4 end_ARG and θ13=0subscript𝜃130\theta_{13}=0italic_θ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT = 0C: [49, 50] have concisely explored small deviations from θ13=0subscript𝜃130\theta_{13}=0italic_θ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT = 0 and	ACB	BAC	ABC	CBA	Selection 2
A: The circuit is characterized by less depth and only one layer gives high accuracy of classification resultsB: The variant (B) of the circuit was created to directly build states consistent with the general structure of the data encoded in the undertaken classification taskC: This allowed us to built less complicated circuit with smaller number of rotation gates needed to implement the VQE circuit.	CBA	ACB	BAC	CAB	Selection 3
A: The authors are grateful to Oleg Teryaev for useful discussions and to Andrey Kotov for the help at the initial stage of the projectB: This work has been carried out using computing resources of the Federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/ and the Supercomputer “Govorun” of Joint Institute for Nuclear ResearchC: This work was supported by the Russian Science Foundation (project no. 23-12-00072).	CAB	ABC	CBA	BAC	Selection 2
A: The paramagnet is a channel spin liquid, since local moments are screened, and no local order parameter existsB: The transition between Phase Ia and Ib is reminiscent of the phase diagram of the NLσ𝜎\sigmaitalic_σM describing a channel ferromagnet and a quantum paramagnet with order destroyed by topological defectsC: This agreement with Fig. 1(b) is not surprising, as in the presence of the gap, fermions can be safely integrated out, and hence the effective interaction of Eq. (2) is expected to be valid. These phases are depicted in the phase diagram of Fig. 5.	CAB	ACB	BAC	ACB	Selection 3
A: In Section 2, we discuss the spectral decomposition of multimode thermal states after providing the necessary mathematical backgroundB: The structure of this article is as followsC: In Section 3, we include the main results of this article where Subsection 3.1 discusses the case of thermal states, and Subsection 3.2 provides results for displaced faithful thermal states.	BAC	BCA	BCA	CAB	Selection 1
A: We start from Eq. (21), up until which every computation was exact for finite L𝐿Litalic_LB: In this section, we numerically analyse the behaviour of the S-matrix for a finite number of interaction sitesC: Note that throughout this section we do not use the circulant approximation used previously, in Sec. 3.3—the purpose of that approximation was only to recover known results in the L→∞→𝐿L\rightarrow\inftyitalic_L → ∞ limit.	BCA	CAB	BAC	ABC	Selection 3
A: The first lemma contains the Sobolev embedding and the trace theoremB: These results are well-known so we omit the proof.C: We remind the reader that their constants are uniform for all 𝖢∈𝒞(1)𝖢superscript𝒞1\mathsf{C}\in\mathcal{C}^{(1)}sansserif_C ∈ caligraphic_C start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT	CAB	ACB	BCA	CBA	Selection 2
A: The AMPT model includes four main processes: initial conditions, partial scattering, hadronisation, and hadronic interactionsB: The initial conditions are generated from the heavy ion jet interaction generator (HIJING) model Wang:1991hta ; Gyulassy:1994ew , where minijet partons and soft-excited strings are produced and then converted to primordial hadrons based on Lund fragmentation. Under the string-melting mechanism, primordial hadrons are converted into partons, a process determined by their flavor and spin structures. Elastic scattering between the partons was simulated using Zhang’s parton cascade (ZPC) model Zhang:1997ej , which includes two-body scattering with a cross-section described by the following simplified equation: C: The string-melting version of the AMPT model (v2.26t9b, available online) Lin:2004en ; Lin:2021mdn was employed in this study to calculate v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of the final-state particles in high-multiplicity p–Pb at 5.02 TeV	CBA	BAC	BCA	CAB	Selection 3
A: The relevant information necessary to formulate the problem is discussed in Sec. IIB: This includes identifying the system Hamiltonian, the initial state, and providing a formal definition of local temperature of the individual qubits. In Section III, we analyseC: The remainder of the paper is arranged as follows	ACB	CBA	BCA	CBA	Selection 3
A: In 2014, the transmission phase limit theory for multilayer TA designs was introduced [5]B: 1(a) depicts a schematic illustration of the multilayer TA. Using microwave network models and S-parameters, researchers discovered the constraints on the transmission amplitude and phase of TA elements, demonstrating that the phase tuning range within a 1-dB transmission amplitude loss is 54∘, 170∘, 308∘, and 360∘ for single-, double-, triple-, and quad-layers. C: Fig	CAB	ACB	CAB	CAB	Selection 2
A: In this way, when passing from the limit cycle to the stationary solution the time-averaged magnetizations change continuously, i.e., the system undergoes a second-order phase transition. On the other hand, above the tricritical point, when passing from one phase to the other, the system experiences sudden jumps between two already existing solutions, which live in different regions of the “phase space”. This fact gives rise to a first-order phase transition with the associated jump of the order parameters. B: The presence of different types of bifurcations (infinite-period bifurcation below the tricritical point [115], saddle-node and homoclinic ones above) explains the appearance of different phase-transition behavior [cf. Fig. 2(a-b)]C: Approaching the critical line below the tricritical point, the limit cycle acquires an infinite period and spends an infinite amount of time close to where the stable solution emerges	ACB	CAB	ABC	ACB	Selection 2
A: In this section, we will apply the results of the simulations describing the formation of PBHs to the calculation of the PBH mass function and abundance. It is desirable to compute the PBH abundance and mass function directly from ζ𝜁\zetaitalic_ζ, which appears in the FLRW metric, equation (11). To perform the calculation, we will follow the method outlined in Young et alB: (2021), which can have a large impact on the PBH abundance and mass function (see e.g. Young (2022); Ferrante et al. (2023); Gow et al. (2023) for recent discussions of the effect of non-Gaussianities on the PBH abundance). C: (2019), applying peaks theory and accounting for the non-linearity between the curvature perturbation ζ𝜁\zetaitalic_ζ and the density - and will only briefly summarise the method here. We will also make the standard assumption that ζ𝜁\zetaitalic_ζ follows a Gaussian distribution, although it has been argued that inflationary models which predict a large PBH abundance typically also predict a non-Gaussian distribution Figueroa et al. (2021); Biagetti et al	BAC	ACB	CAB	BCA	Selection 2
A: Even though this qubit is not measured for adding the next qubit, it still influences the corresponding constraint measurement via a CNOT gate. This influence must be taken into account in the fully parallel implementation, where all measurements are performed before the corrections (see Fig. 5). B: Similarly, when removing the two qubits from the code, a sequential implementation would require qubit (0,2)02{(0,2)}( 0 , 2 ) to be added before qubit (0,3)03{(0,3)}( 0 , 3 )C: Full parallelization would be blocked by the corrective bit-flip on qubit (0,2)02{(0,2)}( 0 , 2 )	ACB	CBA	CAB	ACB	Selection 3

**A**: The second is highly non-trivial, since now m1subscript𝑚1m_{1}italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is external**B**: of the kite in the integrand of (5)**C**: The relation F1,2,3=F1,4,6subscript𝐹123subscript𝐹146F_{1,2,3}=F_{1,4,6}italic_F start_POSTSUBSCRIPT 1 , 2 , 3 end_POSTSUBSCRIPT = italic_F start_POSTSUBSCRIPT 1 , 4 , 6 end_POSTSUBSCRIPT makes m3subscript𝑚3m_{3}italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT external.

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**A**: we used the triangle inequality**B**: The third inequality follows again from Eqs**C**: (23). The above arguments prove that ρ𝜌\rhoitalic_ρ can be converted into σ𝜎\sigmaitalic_σ via DIO with approximate catalysis whenever an asymptotic conversion via DIO is possible with rate at least one.

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**A**: that every PMC solution is a collection of superalgebra irreps**B**: This evidence is still missing. **C**: In order to elevate this simple and elegant argument to the level of a proof of chiral symmetry breaking, one should demonstrate that it works for a generic spectrum with exotics and that it gives the most general solution of PMC, i.e

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**A**: We used predominantly general-purpose software packages to solve the SDPs involved. We used YALMIP [86] for modeling the problems and solved primarily using MOSEK [87]**B**: The unfavorable memory scaling of interior-point methods (see e.g. [88]) prevented us from going to bond dimensions higher than 7 in the MPS-based variant and 5 in the tree-tensor-network-based variant. To overcome this, we tried to use the splitting conic solver (SCS) [89] which is a first-order method and requires less memory, but observed slow convergence and were not able to improve the results in any of the models, with the exception of the S=1𝑆1S=1italic_S = 1 Heisenberg chain. In this model we used our own matrix-free implementation of SCS (where the constraint matrix is not explicitly constructed as a sparse matrix, but only implicitly applied, which provides significant savings due to the tensor-network structure of the constraints) to produce the results for bond dimensions 6 through 9. In addition, SCS was used to solve the larger instances of the LTI problem, Eq. 5, where it performed very well. **C**: We conclude this section with some details regarding our numerical implementations. Sample code is available at [85]

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**A**: This model showed the capability to already perform better across all the relevant metrics, achieving a precision of 29%percent2929\%29 %, 1%percent11\%1 % above the benchmark, with just 90909090 learners (against 1200120012001200) and runtimes of around 20202020 minutes compared to more than 3333 hours for the benchmarked Random Forest**B**: **C**: We also report a classification model based on the proposed heterogeneous structure and leveraging the boosting procedure that, although is not amenable to be trained on current hardware, was trained using a quantum-inspired optimizer based on Tensor Networks

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**A**: Figure 11: Amplification of the energy (top) and angular momentum (bottom) against the frequency of the Q-ball ωQsubscript𝜔𝑄\omega_{Q}italic_ω start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT with mQ=1subscript𝑚𝑄1m_{Q}=1italic_m start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 1**B**: The coupling is g=1/3𝑔13g=1/3italic_g = 1 / 3. **C**: η+subscript𝜂\eta_{+}italic_η start_POSTSUBSCRIPT + end_POSTSUBSCRIPT is the only ingoing mode with m+=−2subscript𝑚2m_{+}=-2italic_m start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = - 2 and various frequencies ω+subscript𝜔\omega_{+}italic_ω start_POSTSUBSCRIPT + end_POSTSUBSCRIPT

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**A**: The construction is characterized by a process that finds a multi-linear polynomial, which in this case is the Walsh-Hadamard transform**B**: The subsequent constructions differ precisely in the process of finding an equivalent multi-linear polynomial for the function. Thus, it is this process alone that distinguishes between them, as the conversion from the multi-linear polynomial to the measurement assignment will be always the same in every case. Therefore, this process will not be repeated for the next constructions.**C**: This first construction, taken from [21] was presented with a detailed translation from a valid multi-linear polynomial, for a specific Boolean function, to a correct and deterministic measurement assignment

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**A**: Alternatively, in Fig. 1 the high-level strategy of a repeater protocol following a layered approach for entanglement swapping is illustrated.**B**: Naturally, the chosen repeater strategy decides which events exactly are used, e.g., a repeater protocol without entanglement purification may decide to schedule an entanglement swapping event at a station, immediately after entangled pairs in both directions have been established**C**: The other moving part of the system is the repeater protocol, i.e. the high-level strategy, the simulation should follow, because this defines which events should be scheduled and when

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**A**: As for applications to enhance couplings, the BO is readily applicable to systems with weak couplings to photons and strong intrinsic dephasing so that the squeezing-induced dephasing is marginal. Common examples are quantum dots [9] and spin ensembles [10]**B**: Yet, the future of BOs is not prescribed to the injection of squeezed vacuum. Regarding the BO alone, the new regime of amplification that evades the gain-bandwidth product constraint [31] is immediately applicable for broadband quantum limited amplification with no hardware overhead [54]**C**: This enhanced coupling could then be leveraged by the BO gain for improved qubit readout [43]. Finally, the ability to dynamically tune qubit-photon interactions comes as a great resource for the study of squeezing-induced phase transitions [19, 20, 21], quantum transduction [36], and the exploration of the ultra-strong coupling regime [16, 17, 18].

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**A**: We begin with the simplest modified conformally flat spacetimes. **B**: The current benchmark model of cosmology is the flat FLRW solution; therefore, we focus on the way the FLRW model is modified in our approach**C**: The rest of this paper is devoted to finding the simplest exact cosmological models of modified TEGR

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**A**: The fellowship code is LCF/BQ/PI20/11760016**B**: JCB is also supported by the research grant PID2020-118635GB-I00 from the Spain-Ministerio de Ciencia e Innovación.**C**: JCB is supported by a fellowship from “la Caixa” Foundation (ID 100010434) and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847648

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**A**: We allow only configurations that differ by a finite number of sites to produce offspring**B**: To address the concerns highlighted above, we preserve the framework they used for finite-volume Gibbs measures introducing infertility into the reproduction dynamics**C**: This approach is similar to that employed in a model of sex-linked inheritance presented in Wörz-Busekros [16, Sec. 8.B] and it reflects the nature of Gibbs measures, which are commonly described by their microscopic components.

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**A**: Figure 3 shows that the amount of contextuality decreases monotonically as coherence increases (at least according to these measures of contextuality and coherence)**B**: However, this intuition only goes so far, as we will give an example below where one achieves twice the robustness to dephasing by including effects that have maximal coherence in the dephased basis.**C**: While this might at first seem counterintuitive, one can understand it by noting that even under large dephasing noise, states and effects with little coherence to begin with are barely affected; that is, the dephasing channel is close to identity on such states and effects

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**A**: If such an experiment is accomplished, the principles used in the present model could be translated to other systems with the same logical capabilities and even for qudits with d>3𝑑3d>3italic_d > 3**B**: The latter condition could be also realised using atomic qudits based on an extension of our simple qutrit model**C**: It is always important to have thought experiments illustrating how future realistic implementations should work, and it really matters when the illustrative models show possible drawbacks and obstacles one should face when implementing theoretical ideas in the laboratory. We strongly recommend a serious analysis by experimentalists about how to implement a version of such an experiment, at least as a proof-of-principle, to assess possible unforeseen new questions that might further improve the technological aspects of quantum-information processing and quantum computing.

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**A**: In this work, we develop a unified treatment of nonlinear quantum optical effects based on phase space filling**B**: Our theory is applicable to a wide range of exciton-polariton lattices, also in the presence of both Pauli and Coulomb blockade**C**: We describe three distinct NPSF regimes being the planar, fractured, and ultralocalized regimes. For each case, we present an analysis and show that in the fractured case a sharp decrease of Rabi frequency in the low-density regime can be facilitated by the Coulomb blockade. Our theory can shed light onto recent experiments in moiré heterobilayers, and open a way for enhancing the nonlinear response.

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**A**: There are plenty of works, devoted to theoretical analysis of the energy propagation, induced by the supratransmission. In general, theoretical studies are focused on determining of conditions of the energy propagation by ILMs, with which emergence of the supratransmission is associated. The bibliography of such studies starts, probably, from the paper [18], wherein the supratransmission was shown to be accompanied by sharp increasing of the supplied energy**B**: This process was described by dint of the sine-Gordon equation, which became hereinafter a model equation for studying of the supratransmission (in particular, in the Josephson junctions) both in the discrete (see, e.g., [19, 20, 21]) and in the continuum (see, e.g., [22, 23, 24, 25]) formulations. Similar results were obtained for the models obeying the discrete nonlinear Schrödinger equation [26, 27], for FPUT chains [28, 32, 33], electrical lattices [30, 31] and others [34]. On the other hand, evolution of the energy, transmitted into the lattice by the loading at driving frequencies, lying both in the stop-band and also in the pass-band, is the rarely studied problem**C**: In the paper [11], evolution of the energy, transmitted into the infinite chain with on-site potential interactions and supplied by kinematic loading, was demonstrated. The non-stationary processes of the energy supply may be described analytically in the harmonic approximation, what was done for the infinite harmonic chain on the linear elastic substrate [35] for the cases of both kinematic and force loadings. For small driving amplitudes, the rate of supply of the total energy into the nonlinear chain is shown [11] to be indistinguishable from the one described in the harmonic approximation.

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**A**: There arise 4 Higgs particles from the b⁢c𝑏𝑐bcitalic_b italic_c sector. **B**: Its detailed particle spectrum is shown in table 3**C**: Model 1 is a rare case without any filler branes and the hidden sector group SU(2) only comes from the d𝑑ditalic_d-stack of D6-branes

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**A**: The first is the family of tangent X-states previously considered in Sec 5.1**B**: Now we apply Theorem 9 to two families of states**C**: The second is a new family, which we call tangent spheroid states, which includes the family of the tangent sphere states, Sec. 5.3, as a special case. (Note that the canonical obese states, Sec. 5.2, are a special case of the tangent spheroid states.)

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**A**: The type-II model is instead**B**: The type-I model corresponds to ϵ𝒅=𝒀𝒖=𝟎superscriptbold-italic-ϵ𝒅superscript𝒀𝒖0\epsilon^{d}=Y^{u}=0bold_italic_ϵ start_POSTSUPERSCRIPT bold_italic_d end_POSTSUPERSCRIPT bold_= bold_italic_Y start_POSTSUPERSCRIPT bold_italic_u end_POSTSUPERSCRIPT bold_= bold_0**C**: of the four Yukawa matrices in Eqs. (1) and (21) are absent

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**A**: Orbits evolve to become aligned and prograde with respect to the disc and circularize as they align**B**: The final semimajor axis can be estimated analytically. This estimate is exact for circular orbits (see Appendix B, Rauch 1995, and Šubr & Karas 1999).**C**: Low inclination orbits remain close to their initial semimajor axis, while high inclination orbits inspiral significantly

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**A**: We compare our results with other crystal detectors in literature as shown in Fig. 14. Experiments using doped crystals featured light yields smaller than 20 p.e./keVee**B**: Recent results in Ref. Ding et al. (2022) used a 0.6×\times×0.6×\times×1.0 cm3 cryogenic pure CsI crystal readout by SiPM chips and achieved about 43 p.e./keVee**C**: It’s not trivial to perform an apple-to-apple comparison between our results with it because the CT effect was not corrected in Ref. Ding et al. (2022).

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**A**: Thus we can conclude that small (3%) amount of free spins responsible for the observed absorption signal and for low-temperature magnetization offset (see Fig. 4) is decoupled from the main magnetic system of nabokoite**B**: The main copper matrix of nabokoite does not contribute to the observed paramagnetic ESR absorption**C**: Presumably, it is due to the strong relaxation processes in the copper matrix of nabokoite which broadens ESR line beyond limits of observation. Note, that quite large ESR linewidth of 1—2 T was also reported for kagomé compound herbertsmithite [35].

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**A**: This is the effective Majorana neutrino mass parameter of the neutrinoless double beta decay, which gives information on the Majorana nature of the neutrinos**B**: Furthermore, in our model, another observable can be obtained**C**: This mass parameter has the form:

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**A**: SMaity would like to thank Ms. Diana Vaclavkova, Ms. Anushree Dey, Mr. Sanjib Naskar, Mr. Rahul Paramanik and Mr. Soumik Das. MP and BD is grateful to IACS for the fellowship. SD acknowledges the financial support from DST-SERB grant No. ECR/2017/002037, SCP/2022/000411 and CRG/2021/004334. SD also acknowledges support from the Central Scientific Service (CSS) and the Technical Research Centre (TRC), IACS, Kolkata.**B**: SMaity and TK are grateful to DST-INSPIRE for their fellowships. D.D. and L.Y. gratefully acknowledge the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0021127 for financial assistance and Advanced Computing Group of the University of Maine System for providing computational resources for this work. SMasanta is grateful to Council of Scientific & Industrial Research (CSIR), New Delhi, for the financial support through the award of NET-SRF (File No: 09/015(0531)/2018-EMR-I). The authors are thankful to the facilities at UGC-DAE-CSR-Indore. Magneto-Raman scattering at low temperature were performed at LNCMI, European Magnetic Field Laboratory at Grenoble under the project GSC08-119**C**: Acknowledgments The authors would like to thank Dr. Vasant Sathe, Prof. K Sengupta, Dr. Marek Potemski, Dr. Clément Faugeras, Prof. Achintya Singha, Mr. Somsubhra Ghosh, Dr. Mintu Mondal, and Dr. Devajyoti Mukherjee for fruitful discussion

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**A**: Compared with the classical counterpart, this quantum process has exponential acceleration in terms of data scale and quadratic speedup in data dimensionality. The other is a quantum secure multi-party computation (QSMC) protocol, which allows the aggregation of gradients to securely be done with quantum communication networks. That is, the server is able to calculate the federated gradients without the client sharing the local gradients. Furthermore, the application of the Chinese remainder theorem in QSMC makes it possible to avoid errors and overflow problems that may occur during the calculation of large numbers. The proposed quantum federated learning framework can improve the local computing efficiency and data privacy of FL. We also apply QFLGD to train the federated linear regression (FLR) and give its numerical experiment to verify the correctness. **B**: In this paper, we focus on the quantum algorithm running on ordinary quantum computers and present a quantum federated learning based on gradient descent (QFLGD). It aims to provide a unified, secure, and effective gradient distribution estimation scheme with distributed quantum networks. In QFLGD, we propose two data preparation methods by analyzing the different acquisition frequencies of static data (the local training data) and dynamic data (the parameters that need to be updated during iteration). That can reduce the requirement of QFLGD on the performance of quantum random memory**C**: At the same time, two main processes of FL are implemented in QFLGD, which exploit quantum properties. The first one is a quantum gradient descent (QGD) algorithm. It facilitates the acceleration of the training gradient for the client. QGD provides the client with a classical gradient at each iteration, which can be directly used to learn classical model parameters

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**A**: In Sec. III, the model of the likelihood and the Bayesian method are introduced. The constraints and relevant discussion are given in Sec. IV. Conclusions are given in Sec. V. **B**: In Sec. II, we briefly describe the galaxy catalog and FRB data**C**: This paper is organized as follows

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**A**: As we mentioned previously, the chip architecture will influence our parameterization. Depending on the architecture, the depth of the parameterization will be seriously influenced due to the need to apply several SWAP ports**B**: Thus, we are naturally led to ask whether for quantum machine learning models it is indeed necessary to make use of these non-parameterized quantum gates involving qubits that do not interact directly. To carry out this analysis, we begin by defining the cost function as**C**: However, although we previously discussed this issue only for CNOT gates, this same analysis applies to any gate that acts on more than one qubit. In fact, the more restricted the chip connectivity is, the deeper the parameterization will be

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**A**: The basic idea is to study the Heisenberg evolution of an operator 𝒪𝒪\mathcal{O}caligraphic_O in a given quantum system governed by a Hamiltonian H𝐻Hitalic_H. The time evolution of 𝒪𝒪\mathcal{O}caligraphic_O is determined by the Heisenberg equation**B**: In this section we provide an outline of the recursion method, which is the basis for the Lanczos algorithm**C**: The reader can refer to RecursionBook ; Parker:2018yvk for further details

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**A**: The importance of the bulk viscosity in the hydrodynamic evolution of quark gluon plasma has been emphasized in Refs. [51, 52, 53]. Our framework provides a direct control over this first-order transport coefficient through a choice of matching condition via b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. **B**: We identified a class of matching conditions for which the homogeneous part of the solution to the relativistic Boltzmann equation vanishes, and RTA turns out to be a special case of that. We examined the effect of choice of matching condition on dissipative coefficients and also studied scaling properties of the ratio of coefficients of bulk viscosity to shear viscosity on the conformality measure**C**: It is important to note that the BGK or MBGK collision kernels are affected by the matching conditions, which in turn affects the dissipative processes in the system. Moreover, at finite chemical potential, two descriptions become identical

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**A**: The structure of the behavioral landscape is similar to the one found in wild type [19], with the dominating kinetics being the transitions between “runs” and “pirouettes”, see Fig. S7(a). However, mutants and wild type crucially differ at long timescales as mutants do not exhibit heavy tails, see Fig. 6(b). Instead, the tail of the first passage time is close to an exponential, and correlations decay to zero within a minute. This is due to the mutants’ inability to adapt their pirouette rates over time, Fig. 6(a), contrary to the modulation highlighted in Fig. 3(a) for the wild type.**B**: The NPR-1 neuropeptide receptor is known to impact several C. elegans behaviors, viz., aerotaxis and food response [45]**C**: We collected a public dataset where worms of the npr-1 loss-of-function strain npr-1(ad609) are allowed to freely explore an agar plate with a uniform food patch (see Appendix A) and used the same method as in Fig. 2 for wild type worms. The upshot is that the short-time behavior of npr-1 mutants on food is similar to wild type N2 worms off food

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**A**: The signatures of Virgo West and the group around NGC4709, that are respectively beyond Virgo and Centaurus in the lines-of-sight, can also be identified as secondary waves**B**: There exist a visual agreement between the observed lines-of-sight dynamical state of Virgo and Centaurus and those reproduced by CLONE. Comparisons with another constrained simulation, called SIBELIUS (McAlpine et al., 2022), shown in the appendix, reveal that an agreement at this level of detail between the full observed and constrained simulated lines-of-sight is not utterly expected. **C**: These smaller waves follow the highest ones representing the main clusters in both the observational and the simulated lines-of-sight

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**A**: In this section, we demonstrate the superiority of the enhanced EPR over the HJB loss alone and over the normalizing flow (NF), which is a class of generative models used for density estimation that leverage invertible transformations to map between complex data distributions and simple latent distributions [39], through 2D benchmark examples**B**: We then apply the enhanced EPR to the 3D Lorenz model and a 12D Gaussian mixture model to show its effectiveness in constructing energy landscapes in higher dimensions**C**: We remark that alternative approaches have been investigated for potential construction in limit cycles [40] and the Lorenz system [41] distinct from our methodology.

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**A**: The generalization can be done in multiple ways, differing in the number of required stochastic variables and the symmetry properties of the non-unitary perturbation. **B**: Notice the difference between 𝒩𝒩\mathcal{N}caligraphic_N (the size of the measurement apparatus) and N𝑁Nitalic_N (the number of pointer states with nonzero weight in the initial superposition)**C**: Having a model for DQSR based on SUV for the specific case of a two-state superposition of pointer states, we will now generalize the approach to initial superpositions over N𝑁Nitalic_N pointer states

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**A**: We also showed that training and generalization are also possible using time series data**B**: As expected, there exists a limit for how much generalization can be achieved, however, we note the presence of regularity that is observed in the gradual increase in test loss. Using the proposed d𝑑ditalic_d-statistic we can therefore evaluate the confidence in the inferred prediction, even in cases when we depart from standard SLT, UAT assumptions of i.i.d. sampling and a compact support. **C**: Overall, the neural networks can approximate various dynamical models and extrapolate predictions even when statistical properties of the input data, or the graph structure change, transcending the formal boundaries of SLT and UAT

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**A**: The simulation geometry is described in more detail in Section A.2. Fig. 2 shows these simulated measurements on an α𝛼\alphaitalic_α-curve for the {6,4}64\{6,4\}{ 6 , 4 } MeV case, demonstrating that the theoretically predicted α𝛼\alphaitalic_α-line overlaps are present both at low area densities and at high-Z𝑍Zitalic_Z.**B**: To validate these results, transparency simulations were run in Geant4 Geant4 ; grasshopper **C**: 10101010, 6666, and 4444 MeV bremsstrahlung beams were directed through different materials of known thicknesses, and the beam transparencies were recorded

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**A**: This method combines the linear decomposition method Cho & Lazarian (2003), singular value decomposition (SVD) method Santolík et al**B**: (2003), and multi-spacecraft timing analysis Pincon & Glassmeier (2008). We perform the calculations in each moving time window with a five-hour length and five-minute moving step. The window length selection (5 hours) provides low-frequency (large-scale) measurements while ensuring 𝐁0subscript𝐁0\mathbf{B}_{0}bold_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is approaching the local background magnetic field. **C**: We calculate wavenumber-frequency distributions of Alfvénic magnetic field and proton velocity power by an improved Alfvén mode decomposition method

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**A**: These are the generalizations of closed string operators in the case of anyons. These operators are not necessarily unitary.**B**: The first step to generalizing this construction to more general excitation types (in general dimension, and including gapped boundaries) is the notion of flexible operator algebra**C**: This is an algebra of operators acting on a region ΩΩ\Omegaroman_Ω with the property that when acting on the reference state, their support can be deformed topologically within ΩΩ\Omegaroman_Ω

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**A**: In this work we have connected several problems of majorization to the theory of preordered semirings**B**: Using this framework, we have derived new sufficient and generically necessary conditions for large-sample and catalytic matrix majorization, i.e., Blackwell dominance in the finite-outcome setting**C**: In this latter context, we have in particular identified the relevant monotone quantities as matrix α𝛼\textstyle\alphaitalic_α-divergences which can be viewed as a generalization of the one-parameter family of the standard Rényi divergences.

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**A**: Also a large non-zero 𝐄⋅𝐁⋅𝐄𝐁{\bf E}\cdot{\bf B}bold_E ⋅ bold_B can enhance production of axions. The gravitational waves, in the early Universe also have the potential to generate large-scale magnetic fields, corresponding to resonant modes. **B**: Our numerical simulations suggest that gravitational waves also will lead to parametric resonance in gauge fields. The gravitational waves will be dampened more efficiently due to the color factor**C**: Further, the gravitational waves can induce large fluctuations in C⁢P𝐶𝑃CPitalic_C italic_P violating physical observables. These observables can subsequently lead to a large-scale imbalance in local chiral charge distributions

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**A**: We receive support from Ontario Research Fund—research Excellence Program (ORF-RE), Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Institute for Advanced Research (CIFAR), Canadian Foundation for Innovation (CFI), the National Science Foundation of China (Grants No**B**: Computations were performed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile and/or Large Memory System] computing platform(s). SOSCIP is funded by the Federal Economic Development Agency of Southern Ontario, the Province of Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario academic member institutions.**C**: 11929301), Thoth Technology Inc, Alexander von Humboldt Foundation, and the Ministry of Science and Technology(MOST) of Taiwan(110-2112-M-001-071-MY3)

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**A**: If ΩΩ\Omegaroman_Ω is semisimple, its topological part can be evaluated explicitly in the idempotent basis, see e.g**B**: Generating functions of a CohFT ΩΩ\Omegaroman_Ω can be defined by integrating CohFT classes**C**: [18, Section 2.5.1]. A consequence of the Givental-Teleman classification is that the generating functions of ΩΩ\Omegaroman_Ω can be explicitly written as sums of graphs. A reference for this can be found in [18, Section 2.5.2].

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**A**: This pattern has been demonstrated to be independent of the input data which exhibits white noise**B**: Since the input data consists of real world natural language sentences that our brain processes, our results demonstrate that artificial neural networks with close to human-level performance exhibit very similar 1/f1𝑓1/f1 / italic_f patterns as their biological counterpartsNovikov et al. (1997); He (2014, 2011).**C**: In this study, we have found convincing evidence of 1/f1𝑓1/f1 / italic_f noise in deep neural networks like the Long-Short-Term Memory (LSTM) network

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**A**: To investigate the properties of non-equilibrium QGP medium, we take into account the viscous effect and study the responses of viscous QGP medium to the magnetic field**B**: Subsequently, we explore how the magnetized viscous effect impacts both the HQ potential and momentum diffusion coefficient**C**: In the strong magnetic limit within the LLL approximation, Landau level quantization ensures that only the longitudinal component of bulk viscosities, arising from the contributions of LLL (anti)-quarks, exists Hattori:2017qih . The response of viscous quark matter to the magnetic field is manifested in the longitudinal bulk viscous modified distribution function of light (anti-)quarks, which can be obtained using the Boltzmann kinetic theory.

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**A**: This allows us to provide a rigorous error and complexity analysis when uploading first a truncated and discretized approximation of the multivariate log-normal distribution and then uploading an approximation of the CPWA payoff function in rotated form, where the approximation consists of truncation as well as the rounding of the coefficients of the CPWA payoff function. This together with a rigorous error and complexity analysis when applying the modified iterative quantum amplitude estimation algorithm [fukuzawa2022modified] allows us to control the output error of our algorithm to be bounded by the pre-specified accuracy level ε∈(0,1)𝜀01\varepsilon\in(0,1)italic_ε ∈ ( 0 , 1 ), while bounding its computational complexity; we refer to Theorem 1 for the precise statement of our main result**B**: In particular, we prove that the computational complexity of our algorithm only grows polynomially in the space dimension d𝑑ditalic_d of the Black-Scholes PDE and in the (reciprocal of the) accuracy level ε𝜀\varepsilonitalic_ε. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. To the best of our knowledge, this is the first work in the literature which**C**: Our main contribution lies in a rigorous error analysis as well as complexity analysis of our algorithm. To that end, we first introduce quantum circuits that can perform arithmetic operations on two complement’s numbers representing signed dyadic rational numbers, together with its complexity analysis

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**A**: We see that the traditional method accepts some regions of meta-stable EW-breaking vacuum, but checking the cosmological history proves that incorrect**B**: The exclusion of such regions is computationally fast, because one only needs to trace the minima of the potential with temperature**C**: On the other hard, it is more time-consuming to exclude the yellow region, as it involves calculation of the transition probability at finite temperature.

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**A**: We then add nodes and links following preferential attachment as described in [2] where each new node (labeled with a consecutive number) is attached to existing nodes via two links**B**: BA: For this network, we start with a simple path graph with 4 numbered nodes**C**: However, and in contrast to the reference, to ensure that the graph has a chromatic number of 2222, for an even (odd) number of already existing nodes, a newly added node can only connect to nodes with an odd (even) label.

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**A**: Since the current can be polarization, time and space dependent, the effective speed can also have the same dependency**B**: In this approach the effects of the current on the propagation of the EMWs has been encoded in the effective speed**C**: This is in agreement with the experimental evidence of the change of the speed of EMWs, depending on the the medium in which are propagating. In this approach the physical properties of the medium are modelled mathematically by the effective speed.

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**A**: Figure 9 shows the Tully-Fisher relation from a single mock catalogue, for comparison with Figure 3**B**: There are no outliers as the generating model is purely Gaussian. Figure 10 shows the distributions of the actual and mock observables for the CF4 Tully-Fisher dataset and their pairwise correlations. It demonstrates that the mocks provide a reasonably realistic representation of the CF4 Tully-Fisher data, including the effective magnitude selection imposed by the flux selection. **C**: Note that the ‘observed’ sample includes galaxies below the subsequently-imposed lower limit to the velocity width

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**A**: For MCMC, this is in some part due to higher mean values, and partly due to inflated errors. **B**: For GP, the alleviation is primarily because of the shifting of the mean to somewhat higher values**C**: With F1 (ΛΛ\Lambdaroman_ΛCDM), F2 (CPL) and F3 (PEDE), tensions decrease progressively from Fisher to MCMC to GP

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**A**: In the finite system, there is a small energy gap even in the gapless system. Thus, we make use of the finite-size scaling to check the energy gap in the thermodynamic limit**B**: In the VBS phase (Fig.6(a)), the energy gaps at different sizes are larger than 12⁢t3/V212superscript𝑡3superscript𝑉212t^{3}/V^{2}12 italic_t start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / italic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. After linear fitting, the energy gap is still positively large**C**: Here, we have to mention that the results of VBS present a strong deviation from the fitting results, and it may result from the finite size effects. In comparison, at DQCP (Fig.6(b)), the gap energies at different system sizes exhibit good linear behavior. The negative value at the thermodynamic limit indicates the close of the energy gap.

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**A**: For example, in [9] the asymptotics of the fundamental solution was constructed depending on the properties of the kernel of the process describing the jumps**B**: It was also noted that in the case of pure jumps (without diffusion), the fundamental solution always contains a singular component.**C**: The properties of fundamental solutions for integro-differential equations have been studied in previous works

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**A**: The work demonstrates evidence that the feature of non-linearity from classical ML might be useful in quantum circuit models. **B**: This work put forth a preliminary investigation into the learning capabilities of the entire generative model, known as the Quantum Neuron Born Machine (QNBM), along with a demonstration of better performance over the more widely investigated Quantum Circuit Born Machine (QCBM) model [Benedetti_2019, Liu_2018, 2020Coyle, gili_qcbm], which does not contain these non-linear activations in the architecture**C**: Recent work proposed a quantum generative architecture that achieves non-linearity in the quantum state evolution using repeat-until-success (RUS) sub-routines containing mid-circuit measurements, similar to non-linear activations in a classical feed-forward network [gili_qnbm]

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**A**: Despite the fact that the coded-aperture telescopes are unable to construct the images of extended objects, some information about their angular sizes can still be obtained from the IBIS data (see, e.g., Kuznetsova et al., 2019)**B**: In addition, special methods different from those that are used for point sources are required to be applied to properly extract the flux from an extended object. In particular, for the flux from an X-ray source to be correctly estimated, the convolution function, when reconstructing its image, must maximally coincide with the profile of this source for a given telescope. **C**: The X-ray emission from the Ophiuchus galaxy cluster is known to be an extended one with a characteristic size of about 15 arcmin (Nevalainen et al., 2009; Werner et al., 2016), which is comparable to the angular resolution of the IBIS telescope (12 arcmin)

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**A**: We have that the bulk where the branes are embedded has a particular distribution of matter, however the homogeneity and isotropy is fulfilled in the internal space of the brane, having in turn an arbitrary sectional curvature**B**: For this case, we also find equations for the evolution of perturbations in branes, and their respective scalar, vector and tensor modes, choosing also the Newtonian conformal and synchronous gauges, these in their version of extra dimensions.**C**: In that same line of thought and looking for a formulation of perturbations in extra dimensions, we present a development of perturbations for branes where we start from evolutionary equations and in where the geometry of space-time is determined by the extrinsic curvature tensor

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**A**: Gamma-ray bursts are some of the most powerful astrophysical phenomena in the universe**B**: The collapsar model describes GRBs as the consequence of a failed supernova where all the energy of the gravitational collapse is funneled into a jet (Woosley, 1993)**C**: The jet is fed by the rotation of a central engine, a BH, through the BZ mechanism.

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**A**: Linear combinations of spectral projections are simple functions on the spectrum — “simple” here is a technical term, meaning a finite linear combination of indicator functions. The statement that a positive operator can be approximated arbitrarily well with simple functions is then the standard measure theory fact that simple functions are dense in any L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT space**B**: See appendix A.2. by positive linear combinations of its spectral projections, so we should ask first what it means for a projection to be a density matrix, and then ask about whether the property “being a density matrix” is preserved under linear combinations and limits.**C**: Every positive operator can be approximated arbitrarily well171717Here “approximated” is meant in the sense of the norm topology. Using spectral theory, a bounded positive operator can be thought of as the identity function on a compact measure space, corresponding to the spectrum of the operator

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**A**: The alignment of FTLE behaviour, particularly into the neutral subspace, has been previously observed in similar conceptual chaotic models when they undergo a transition between regions of the attractor, as well as in models of regime shifts in atmospheric flows [16, 17, 18]**B**: [20], [21]). **C**: In general, however, more useful measures for alignment strength already exist in regards to Lyapunov vectors, which not only help characterize the dynamics of a system [19], but strong levels of alignment can be used to predict chaotic transitions (e.g

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**A**: The operator form of (1) first appeared in [15] in relation with conformal field theory**B**: In [16] it was considered inside the context of three-dimensional integrable systems, while the set-theoretical version of (1) was studied in [17] in connection with Poisson maps, see also [18]**C**: Furthermore, the pentagon relation (1) itself, as it was shown in [19], serves as a manifestation of the 3↔2↔323\leftrightarrow 23 ↔ 2 Pachner move [20], where three tetrahedra with a common edge are replaced by two tetrahedra with a common face on a triangulation of a piecewise-linear 3-manifold.

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**A**: NHSE can also occur in the synthetic field moments space of zero-dimensional bosonic quantum dimers [372]. An new, interacting form of the NHSE also occurs in the presence of interacting impurities, as manifested by so-called squeezed polarons which are impurity-localized dipole-like density profiles that are impervious to the lattice boundaries [373]. More interesting multi-polar NHSE signatures have been suggested in [374], and it remains to be seen if such geometric anisotropy may be generalized to give rise to quantum Hall-like states [375] with NHSE interplay.**B**: The NSHE can also emerge in the effective descriptions of various interacting models, even though the interactions themselves are not asymmetric hoppings**C**: For instance, real-space dynamical mean-field theory reveals the NHSE in the pseudospectrum of some strongly-correlated systems [371]

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**A**: and P.Y. were supported by NSFC Grant No. 12074438, Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100, and the Open Project of Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices under Grant No. 2022B1212010008. M.C**B**: acknowledges support from NSF under award number DMR-1846109. J.-Y.C. was supported by Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (Project No. KF202207), Fundamental Research Funds for the Central Universities, Sun Yat-sen University (Project No. 23qnpy60), a startup fund from Sun Yat-sen University, the Innovation Program for Quantum Science and Technology 2021ZD0302100, Guangzhou Basic and Applied Basic Research Foundation (grant No. 2024A04J4264),**C**: We acknowledge conversations with Johannes Hauschild, Xiao-Yu Dong, Hui-Ke Jin and Hong-Hao Tu. The MPS calculations were performed using the TeNPy Library (version 0.9.0) [48]. L.M.C

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**A**: The red cross marks the estimate in (37). The color encodes the expectation value (left) or the average (right) of the cut size**B**: On the left, we mimic an ideal quantum device by evaluation of (19). On the right, values are results from the quantum hardware. Here, every pixel represents the average taken over 1,02410241,0241 , 024 samples.**C**: Figure 4: Visualized results for QAOA parameter-estimate on instances a, b and c from Fig. 3. The x- and y-axis represent values of the parameters β𝛽\betaitalic_β and γ𝛾\gammaitalic_γ, respectively

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**A**: 4(a)-(d), respectively. Around the cross points in the curves, the interference stripes are clearly visible,**B**: by shifting the phase will be discussed in the next subsection**C**: The quantum Lissajous scars for ωx/ωy=2/1,3/1,3/2,4/3subscript𝜔𝑥subscript𝜔𝑦21313243\omega_{x}/\omega_{y}=2/1,3/1,3/2,4/3italic_ω start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT / italic_ω start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 2 / 1 , 3 / 1 , 3 / 2 , 4 / 3 are shown in Figs

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**A**: [Eq. (14)]**B**: We have taken the**C**: This has the effect of replacing D𝐷Ditalic_D in A⁢(ℓ−)𝐴superscriptℓA(\ell^{-})italic_A ( roman_ℓ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) with Dnsubscript𝐷𝑛D_{n}italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT

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**A**: Furthermore, this method could miss potential dark UDGs — ones with extremely faint stellar light that are of high astrophysical interest (see Li et al., 2022, for more detailed discussions). **B**: The two major challenges with this method are: (1) the high computational cost of image segmentation needed to separate UDGs from the night sky background, and (2) contamination of samples of UDGs by imaging artifacts (such as scattered light) and light-reflecting diffuse gas clouds in the interstellar medium (so-called ‘Galactic cirrus’)**C**: Currently, UDGs are found using a computational search of their associated faint stellar light in images

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**A**: The completeness (or recall rate) of the RF classifier for different bins in planetary period-radius space is measured using the full set of synthetic planetary injections based on the Kepler planet sample (§4.1). The analysis is based on 7,751 of the 10,850 synthetic planetary injections that were processed by the RF classifier; the remaining objects were omitted due to missing features. **B**: It is important to understand how the classifier performs across the planetary radius-period distribution to evaluate the completeness of our intermediate DTARPS-S Analysis List with 7,377 objects and the smaller DTARPS-S Planet Candidate catalog produced in Paper II**C**: The recall rate of the injected planetary signals across the range of the injected period and radii can quantitatively measure the ability of the RF classifier to recover planets in the DIAmante data set

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**A**: The DTARPS-S Candidates catalog (Table 1) and DTARPS Galactic Plane list (Table 4) are ready for spectroscopic follow-up to remove remaining False Positives, measure other planetary properties, and further investigate the planetary systems**B**: The results are informed by preliminary results of reconnaissance spectroscopy of a small portion of the DTARPS-S Candidate sample. Paper III also returns to the completeness heat maps introduced in Paper I showing that the DTARPS-S Candidate planet occurrence rates are compatible with those obtained with the Kepler mission.**C**: Paper III examines various astronomical properties of these samples, discussing exoplanet types such as the hot Neptune population, (extreme) Ultra Short Period planets, planets orbiting low-mass stars, and candidates for atmospheric characterization by transmission spectroscopy

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Selection 3

**A**: In case of Eq**B**: (8) with (12), the heavy-light diquarks satisfy the universal description irrespective of both mass of the light constituents and mass of the heavy constituent**C**: In case of Eq. (14) with (15), the heavy-light diquarks satisfy the universal description irrespective of mass of the heavy constituents.

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Selection 4

**A**: Though we have provided some lower bounds on the size, we are also lacking a non-trivial upper bound. A second problem that we left open is the efficient construction of minimal generating sets of qudit Pauli subgroups in all cases. The missing step to get from a near-minimal generating set to a minimal one involves determining whether appropriate phases can be added to each generator as in Theorem 6.9.**B**: We have applied the theory of modules over commutative rings to study properties of the qudit Pauli group**C**: One problem we have left open is the maximum size of a non-commuting set on n>1𝑛1n>1italic_n > 1 qudits

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Selection 1

**A**: Here, polarized electron beams can play a crucial role in increasing the capability of analysis**B**: It allows a more selective probing of the process by enhancing the deviations from the SM via anomalous couplings and also the signal-background ratio. Eventually, it provides to make stringent tests and to get better constraints on anomalous couplings. **C**: On the other hand, CLIC experiment program provides an opportunity for ±%80\pm\%80± % 80 polarized electron beams and no positron polarization at the center-of-mass energy of 3 TeV with the integrated luminosity of 1 ab-1 and 4 ab-1, respectively

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**A**: These algorithms are based on iterative optimization techniques, such as the Gerchberg-Saxton algorithm, the Hybrid Input-Output algorithm, and the Error Reduction algorithm. They are powerful and can recover images with high qualities. However most of the time, iterative phase retrieval algorithms suffers from twin images and stagnations**B**: When the source becomes non-coherent. The CDI becomes hopeless. To overcome this problem, we report the new CDI method. **C**: Coherent diffraction imaging (CDI) is an imaging technique that uses coherent beam to reconstruct images of micro and nanoscale objects. It is a powerful tool for imaging small scale structures with high resolution, and it has been used to image a wide range of materials, including proteins, viruses, and nanomaterials. Phase retrieval algorithms are used in coherent diffraction imaging (CDI) to reconstruct the phase of a diffracted wavefront from its intensity measurements

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**A**: 3**B**: Taking the indirect effects in the production network into account leads to 3 to 42 times higher estimates of job loss and 6 to 23 times higher estimates of economic output loss for the decarbonization strategies depicted in Fig**C**: Note that EW-ESRI and OW-ESRI enable us to calculate both, the direct job and economic output loss incurred by the removed firms, and the indirect losses experienced by all subsequently affected firms in the production network, for every decarbonization strategy, respectively

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**A**: It is tantalizing that the rich theoretical scenario of USC has an experimental counterpart limited so far to standard spectroscopy**B**: What has prevented a broader experimental investigation? To gain insight into this issue we address a fundamental problem posed since the birth of the field [1] namely the experimental detection of VPs, which still awaits demonstration despite several theoretical proposals [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 2, 3]. The specific question we ask is whether it is possible to overcome experimental challenges posed by available quantum hardware**C**: This work shows that the answer is positive but not trivial. Indeed detecting VPs in an efficient and faithful way requires combining state-of-the-art technologies, such as a multilevel AA unconventional design [42, 43, 44, 45], and a multiphoton coherent control protocol with a tailored integrated measurement technique. The implementation of this setup is feasible with present-day superconducting technology [46].

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Selection 2

**A**: In general, Q-ball potentials produce non-linearities in the equation of motion which require higher order modes to be nonzero. We have shown that this suggests the existence of a non-local, hence radiating, portion of the scalar field**B**: For small rotation, however, these radiation modes are only required to appear at subleading order in the expansion parameter. Therefore, though the localized perturbation is only an approximate solution to the equations of motion, the approximation can be quite close to a true solution**C**: We estimate the life-time of the localized perturbation from the radiated power contained in the non-localized part of the field. We find that small angular momentum solitons are classically metastable in the small angular velocity limit with lifetimes that can be relevant to cosmological studies.

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**A**: On the other hand, with rigid confinement, dilation is frustrated. When the dimension of a local high-density region is comparable to the gap between boundaries, a sudden rise in the stress response is expected [9, 5]**B**: The intense stress compels particles into profound interactions, revealing features that might otherwise remain hidden, such as the role of particle adhesion [46] and the occurrence of hysteresis [26]. Furthermore, macroscopic clusters of particles exist only briefly under intense stress, leaving behind smaller ones that subsequently promote the reformation of high-density clusters. From a heterogeneous perspective, the overall stress response of a suspension in the conventional shear-thickened state (with rigid boundaries) is primarily governed by the increasingly prominent formation and collapse of these high-stress regions [14, 15, 49]. The constitutive relation, calibrated with bulk rheology measurements, only captures these intermittent microscopic events on average.**C**: With soft boundaries, such as the free surface experiments here, dilation is allowed to a certain extent. In this scenario, the inhomogeneity develops into a persistent density-wave state, where particles do not make long-lived contact

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**A**: The Pcal can give a modulation the mirror surface with photon pressure**B**: The summary of the calibration overview in O3GK is described in the summary paper. [8] The initial characterization of photon calibrator instruments are summarized elsewhere. [17, 18] Pcal can modulate the mirror displacement by injecting the power-stabilized laser with intensity modulation. The main applications of the photon calibrator are (i) calibration of an interferometer response, (ii) monitoring of time dependency of the interferometer response, and (iii) hardware injection to verify the analysis pipeline [19, 18]. The displacement of the mirror can be written as:**C**: In the joint observation run 3 (2020 April) with KAGRA and GEO600 (O3GK), [15, 6, 8, 16] KAGRA employs the photon calibrator as a primary calibrator

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**A**: Set v⁢(x)=∫0x∫0t{q⁢(s)⁢u⁢(s)−g⁢(s)}⁢𝑑s⁢𝑑t𝑣𝑥superscriptsubscript0𝑥superscriptsubscript0𝑡𝑞𝑠𝑢𝑠𝑔𝑠differential-d𝑠differential-d𝑡v(x)=\int_{0}^{x}\int_{0}^{t}\{q(s)u(s)-g(s)\}dsdtitalic_v ( italic_x ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT { italic_q ( italic_s ) italic_u ( italic_s ) - italic_g ( italic_s ) } italic_d italic_s italic_d italic_t**B**: Hence v∈W2,1⁢(xj,xj+1)𝑣superscript𝑊21subscript𝑥𝑗subscript𝑥𝑗1v\in W^{2,1}(x_{j},x_{j+1})italic_v ∈ italic_W start_POSTSUPERSCRIPT 2 , 1 end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT ), v′′⁢(x)=q⁢(x)⁢u⁢(x)−g⁢(x)superscript𝑣′′𝑥𝑞𝑥𝑢𝑥𝑔𝑥v^{\prime\prime}(x)=q(x)u(x)-g(x)italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x ) = italic_q ( italic_x ) italic_u ( italic_x ) - italic_g ( italic_x ) a.e**C**: x∈(xj,xj+1)𝑥subscript𝑥𝑗subscript𝑥𝑗1x\in(x_{j},x_{j+1})italic_x ∈ ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT ), and we get the equality

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Selection 1

**A**: We believe MIQS can simplify initialization of a qutrit, by coupling it to a qubit. **B**: However, controlling qutrits can be a difficult task**C**: There are various factors that need to be calibrated, such as frequency of the drive, amplitidue of the drive, leakage, etc

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Selection 1

**A**: The structure of this paper is as follows. In Sec. II, we provide the general equations required to construct a bouncing solution**B**: To begin with, we demonstrate how two popular inflationary models can be tweaked (albeit in an ad-hoc manner) to achieve this goal. Thereafter, we show how a simple augmentation of a very well-motivated model also leads to very similar results. As these models suffer from a ghost instability, we propose, in Sec. V, a way to mitigate the problem by invoking a coupling between the two fields. Finally, in Sec. VI, we summarise and present a future outlook.**C**: In Sec. III, we focus on constructing a bouncing phase followed by the inflationary dynamics. We do this in two steps

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Selection 2

**A**: Another domain feature that we have not considered is the inclusion of more than one target, one or more of which may be of non-circular shape**B**: The multiple-target problem has been considered at length for Brownian motion on flat 2- and 3-dimensional geometries using hybrid asymptotic-numerical methods see (e.g., [32, 35, 30, 51, 55, 56, 57, 26] and the references therein)**C**: Several numerical optimization studies have been done to find optimal target arrangements that minimize the spatial average of the stopping time (e.g., [58, 59, 60, 61]). The inclusion of more than one target also gives rise to the question of splitting probabilities (see e.g., [31, 62, 63]) and shielding effects [31], and how they compare to their Brownian counterparts.

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Selection 4

**A**: In the convention, θ~−,csubscript~𝜃𝑐\tilde{\theta}_{-,c}over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT - , italic_c end_POSTSUBSCRIPT is locked at 0 and π/2𝜋2\pi/2italic_π / 2 for s𝑠sitalic_s- and d𝑑ditalic_d- pairing states respectively. Thus if C0=Cπsubscript𝐶0subscript𝐶𝜋C_{0}=C_{\pi}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, only quasi-long-range order of ΔSsubscriptΔS\Delta_{\mathrm{S}}roman_Δ start_POSTSUBSCRIPT roman_S end_POSTSUBSCRIPT exists for s𝑠sitalic_s-wave states; the same applies to d𝑑ditalic_d-wave**B**: Given that C0≠Cπsubscript𝐶0subscript𝐶𝜋C_{0}\neq C_{\pi}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ italic_C start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT in general, we have both quasi-long-range orders in either s𝑠sitalic_s- and d𝑑ditalic_d-wave states; in other words, ⟨ΔS⁢j⁢ΔS,j+d†⟩delimited-⟨⟩subscriptΔS𝑗subscriptsuperscriptΔ†S𝑗𝑑\langle\Delta_{\mathrm{S}j}\Delta^{\dagger}_{\mathrm{S},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_S italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_S , italic_j + italic_d end_POSTSUBSCRIPT ⟩ and ⟨ΔD⁢j⁢ΔD,j+d†⟩delimited-⟨⟩subscriptΔD𝑗subscriptsuperscriptΔ†D𝑗𝑑\langle\Delta_{\mathrm{D}j}\Delta^{\dagger}_{\mathrm{D},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_D italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_D , italic_j + italic_d end_POSTSUBSCRIPT ⟩ decay algebraically in |d|𝑑|d|| italic_d | with the same exponent. However, we can still have a microscopic definition of s𝑠sitalic_s-wave and d𝑑ditalic_d-wave states**C**: If the leading algebraic decay prefactor of ⟨ΔS⁢j⁢ΔS,j+d†⟩delimited-⟨⟩subscriptΔS𝑗subscriptsuperscriptΔ†S𝑗𝑑\langle\Delta_{\mathrm{S}j}\Delta^{\dagger}_{\mathrm{S},j+d}\rangle⟨ roman_Δ start_POSTSUBSCRIPT roman_S italic_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_S , italic_j + italic_d end_POSTSUBSCRIPT ⟩ is larger, we call the state s𝑠sitalic_s-wave, otherwise d𝑑ditalic_d-wave. This is equivalent to the definition from the relative sign of the coefficient of the leading algebraic decay components of ⟨Δ0,j⁢Δ0,j+d†⟩delimited-⟨⟩subscriptΔ0jsubscriptsuperscriptΔ†0jd\langle\Delta_{\mathrm{0,j}}\Delta^{\dagger}_{\mathrm{0,j+d}}\rangle⟨ roman_Δ start_POSTSUBSCRIPT 0 , roman_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , roman_j + roman_d end_POSTSUBSCRIPT ⟩ and ⟨Δπ,j⁢Δπ,j+d†⟩delimited-⟨⟩subscriptΔ𝜋jsubscriptsuperscriptΔ†𝜋jd\langle\Delta_{\mathrm{\pi,j}}\Delta^{\dagger}_{\mathrm{\pi,j+d}}\rangle⟨ roman_Δ start_POSTSUBSCRIPT italic_π , roman_j end_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_π , roman_j + roman_d end_POSTSUBSCRIPT ⟩, where Δky,j=cj,ky,↑⁢cj,ky,↓subscriptΔsubscriptkyjsubscript𝑐𝑗subscript𝑘𝑦↑subscript𝑐𝑗subscript𝑘𝑦↓\Delta_{\mathrm{k_{y},j}}=c_{j,k_{y},\uparrow}c_{j,k_{y},\downarrow}roman_Δ start_POSTSUBSCRIPT roman_k start_POSTSUBSCRIPT roman_y end_POSTSUBSCRIPT , roman_j end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_j , italic_k start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ↑ end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j , italic_k start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ↓ end_POSTSUBSCRIPT A positive relative sign is defined as s𝑠sitalic_s-wave states and a negative relative sign is defined as d𝑑ditalic_d-wave states.

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Selection 3

**A**: The entanglement spectrum can show different behaviors from the energy spectrum of the subsystem as an independent system under open boundary condition. Tuning the coupling between the subsystem and environment can change the ES gap, which indicates that the coupling accounts for the inconsistency of the Li-Haldane conjecture**B**: In summary, we have systematically studied the relation between the energy and entanglement spectrum on the boundary in the AKLT phase of the S=1/2𝑆12S=1/2italic_S = 1 / 2 Heisenberg model on the square-octagon lattice with a symmetry-breaking term on the boundary and a tunable coupling between the subsystem and its environment. The correspondence does not always apply if the perturbation is added to the environment**C**: This inconsistency can be naturally explained in the wormhole picture for the partition function in the replica manifold. In the path-integral picture, the wormhole effect induced by the coupling plays an important role in transporting the interaction of the environment and provides a pathway to control the ES. The wormhole picture can explain the correspondence between the entanglement spectrum and energy spectrum for the gapless and gapped cases which is indeed the fundamental mechanism of the ES.

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Selection 1

**A**: In particular, it would be interesting to compare a policy that incorporates entanglement distillation to one without entanglement distillation that simply does the memory-cutoff policy with a single memory [72]. It is not obvious that entanglement distillation will always give a better overall fidelity, and understanding in what parameter regimes this is the case will help guide us towards a concept of a noise and loss threshold for quantum communication, similar to the fault-tolerance thresholds for quantum computation that are defined by the point at which the logical error rate of an error-corrected quantum computation is less than the raw, physical error rate. **B**: Our work opens up several interesting avenues for future work. One direction for future work is to add the possibility of doing entanglement distillation, which is a form of quantum error correction, to the MDP formalism presented here**C**: As entanglement distillation is now within the reach of experiments [121, 122, 123], this question is especially pertinent, and much remains to be explored about what policies are optimal in this scenario, building on prior works that have considered specific policies [38, 111, 93, 94, 124, 125, 126, 96]; see also Refs. [71, 72], which numerically investigate several of these scenarios. We therefore expect our insights on the features of improved policies to be a valuable starting point when designing better protocols for quantum repeater chains with multiple memories and entanglement distillation

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Selection 3

**A**: Sequences of symbols that disagree in every place could find application in authentication protocols [14]**B**: The use of checking sequences for elements that were not sent, that is the sequence of states sent and the sequence of measurement results disagree, has been used in digital signature schemes [15].**C**: Alice can authenticate herself to Bob by providing elements of her sequence to him, and he can check that what Alice sent disagrees with his sequence in every place

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Selection 4

**A**: When the HNLS condition is satisfied and the HL is achievable, for qubit probes where the signal and noise acts individually on each probe, an optimal ancilla-free repetition code can be constructed Dür et al. (2014); Arrad et al**B**: (2019). In the case where the HNLS condition is violated, weakly spin-squeezed states were known to achieve the optimal SQL for a Pauli-Z signal under local dephasing noise Ulam-Orgikh and Kitagawa (2001); Escher et al. (2011); while in general ancilla-assisted QEC strategies are necessary. In this paper, we will address the open question whether there exist ancilla-free QEC codes for Hamiltonian estimation under Markovian noise that achieve the optimal HL (or SQL). We will exploit quantum entanglement among multiple probes to design two classes of optimal ancilla-free QEC codes.**C**: (2014); Peng and Fan (2020). Additionally, when the signal and the noise commute, e.g., a Pauli-Z signal acting on multiple probes under correlated dephasing noise, optimal ancilla-free codes were also proven to exist Layden et al

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Selection 1

**A**: N000142212764. B.P and A.M.A. acknowledge support from ONR Grant No. N000141712793. **B**: Office of Naval Research (ONR) through Grants No. N000142112450 and MURI No**C**: This work is supported in part by DARPA Topological Excitations in Electronics (TEE) program under grant No. DP18AP900007, the U.S

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Selection 1

**A**: Figure 3: (a) Energy errors for TC calculations of the H2 bond dissociation w.r.t**B**: Å). TC-VarQITE calculations are based on HF orbitals in a STO-6G (4 qubits), 6-31G (8 qubits), and cc-pVDZ (20 qubits) basis sets. Also shown are no-TC FCI/cc-pVDZ calculations.**C**: CBS limit results (mH vs

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Selection 1

**A**: It started with Sutherland’s discovery of a flat band in the dice lattice [28]**B**: Historically, developing flat band models has been a long and arduous process**C**: It continued with Lieb’s work on the Hubbard model, demonstrating that certain bipartite lattices with chiral flat bands exhibit ferromagnetism [29]. However, in recent years, there has been a growing interest in the development of new topological flat band models [2] that will allow for a better understanding of the properties of these materials and their potential applications and which can support quantum Hall-like states, including integer quantum Hall (QH) effect [30, 31], fractional quantum Hall (FQH) effect [32, 33, 34], and the existence of electronic fractional Chern states [35, 32, 11, 12].

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Selection 2

**A**: It is worth mentioning that the studies in Refs**B**: predictions regarding the mixing angles with θ23=π4subscript𝜃23𝜋4\theta_{23}=\frac{\pi}{4}italic_θ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = divide start_ARG italic_π end_ARG start_ARG 4 end_ARG and θ13=0subscript𝜃130\theta_{13}=0italic_θ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT = 0**C**: [49, 50] have concisely explored small deviations from θ13=0subscript𝜃130\theta_{13}=0italic_θ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT = 0 and

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Selection 2

**A**: The circuit is characterized by less depth and only one layer gives high accuracy of classification results**B**: The variant (B) of the circuit was created to directly build states consistent with the general structure of the data encoded in the undertaken classification task**C**: This allowed us to built less complicated circuit with smaller number of rotation gates needed to implement the VQE circuit.

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Selection 3

**A**: The authors are grateful to Oleg Teryaev for useful discussions and to Andrey Kotov for the help at the initial stage of the project**B**: This work has been carried out using computing resources of the Federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/ and the Supercomputer “Govorun” of Joint Institute for Nuclear Research**C**: This work was supported by the Russian Science Foundation (project no. 23-12-00072).

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Selection 2

**A**: The paramagnet is a channel spin liquid, since local moments are screened, and no local order parameter exists**B**: The transition between Phase Ia and Ib is reminiscent of the phase diagram of the NLσ𝜎\sigmaitalic_σM describing a channel ferromagnet and a quantum paramagnet with order destroyed by topological defects**C**: This agreement with Fig. 1(b) is not surprising, as in the presence of the gap, fermions can be safely integrated out, and hence the effective interaction of Eq. (2) is expected to be valid. These phases are depicted in the phase diagram of Fig. 5.

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Selection 3

**A**: In Section 2, we discuss the spectral decomposition of multimode thermal states after providing the necessary mathematical background**B**: The structure of this article is as follows**C**: In Section 3, we include the main results of this article where Subsection 3.1 discusses the case of thermal states, and Subsection 3.2 provides results for displaced faithful thermal states.

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Selection 1

**A**: We start from Eq. (21), up until which every computation was exact for finite L𝐿Litalic_L**B**: In this section, we numerically analyse the behaviour of the S-matrix for a finite number of interaction sites**C**: Note that throughout this section we do not use the circulant approximation used previously, in Sec. 3.3—the purpose of that approximation was only to recover known results in the L→∞→𝐿L\rightarrow\inftyitalic_L → ∞ limit.

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**A**: The first lemma contains the Sobolev embedding and the trace theorem**B**: These results are well-known so we omit the proof.**C**: We remind the reader that their constants are uniform for all 𝖢∈𝒞(1)𝖢superscript𝒞1\mathsf{C}\in\mathcal{C}^{(1)}sansserif_C ∈ caligraphic_C start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT

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Selection 2

**A**: The AMPT model includes four main processes: initial conditions, partial scattering, hadronisation, and hadronic interactions**B**: The initial conditions are generated from the heavy ion jet interaction generator (HIJING) model Wang:1991hta ; Gyulassy:1994ew , where minijet partons and soft-excited strings are produced and then converted to primordial hadrons based on Lund fragmentation. Under the string-melting mechanism, primordial hadrons are converted into partons, a process determined by their flavor and spin structures. Elastic scattering between the partons was simulated using Zhang’s parton cascade (ZPC) model Zhang:1997ej , which includes two-body scattering with a cross-section described by the following simplified equation: **C**: The string-melting version of the AMPT model (v2.26t9b, available online) Lin:2004en ; Lin:2021mdn was employed in this study to calculate v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of the final-state particles in high-multiplicity p–Pb at 5.02 TeV

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Selection 3

**A**: The relevant information necessary to formulate the problem is discussed in Sec. II**B**: This includes identifying the system Hamiltonian, the initial state, and providing a formal definition of local temperature of the individual qubits. In Section III, we analyse**C**: The remainder of the paper is arranged as follows

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Selection 3

**A**: In 2014, the transmission phase limit theory for multilayer TA designs was introduced [5]**B**: 1(a) depicts a schematic illustration of the multilayer TA. Using microwave network models and S-parameters, researchers discovered the constraints on the transmission amplitude and phase of TA elements, demonstrating that the phase tuning range within a 1-dB transmission amplitude loss is 54∘, 170∘, 308∘, and 360∘ for single-, double-, triple-, and quad-layers. **C**: Fig

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Selection 2

**A**: In this way, when passing from the limit cycle to the stationary solution the time-averaged magnetizations change continuously, i.e., the system undergoes a second-order phase transition. On the other hand, above the tricritical point, when passing from one phase to the other, the system experiences sudden jumps between two already existing solutions, which live in different regions of the “phase space”. This fact gives rise to a first-order phase transition with the associated jump of the order parameters. **B**: The presence of different types of bifurcations (infinite-period bifurcation below the tricritical point [115], saddle-node and homoclinic ones above) explains the appearance of different phase-transition behavior [cf. Fig. 2(a-b)]**C**: Approaching the critical line below the tricritical point, the limit cycle acquires an infinite period and spends an infinite amount of time close to where the stable solution emerges

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Selection 2

**A**: In this section, we will apply the results of the simulations describing the formation of PBHs to the calculation of the PBH mass function and abundance. It is desirable to compute the PBH abundance and mass function directly from ζ𝜁\zetaitalic_ζ, which appears in the FLRW metric, equation (11). To perform the calculation, we will follow the method outlined in Young et al**B**: (2021), which can have a large impact on the PBH abundance and mass function (see e.g. Young (2022); Ferrante et al. (2023); Gow et al. (2023) for recent discussions of the effect of non-Gaussianities on the PBH abundance). **C**: (2019), applying peaks theory and accounting for the non-linearity between the curvature perturbation ζ𝜁\zetaitalic_ζ and the density - and will only briefly summarise the method here. We will also make the standard assumption that ζ𝜁\zetaitalic_ζ follows a Gaussian distribution, although it has been argued that inflationary models which predict a large PBH abundance typically also predict a non-Gaussian distribution Figueroa et al. (2021); Biagetti et al

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BCA

Selection 2

**A**: Even though this qubit is not measured for adding the next qubit, it still influences the corresponding constraint measurement via a CNOT gate. This influence must be taken into account in the fully parallel implementation, where all measurements are performed before the corrections (see Fig. 5). **B**: Similarly, when removing the two qubits from the code, a sequential implementation would require qubit (0,2)02{(0,2)}( 0 , 2 ) to be added before qubit (0,3)03{(0,3)}( 0 , 3 )**C**: Full parallelization would be blocked by the corrective bit-flip on qubit (0,2)02{(0,2)}( 0 , 2 )

ACB

CBA

CAB

ACB

Selection 3