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Rapid and accurate initialization of qubits, reset, is a crucial building block for various tasks in quantum information processing, such as quantum error-correction and estimation of statistics of noisy quantum devices with many qubits. We demonstrate unconditional reset of a frequency-tunable transmon qubit that simultaneously resets multiple excited states by utilizing a metamaterial waveguide engineered to provide a cold bath over a wide spectral range, while providing strong protection against Purcell decay of the qubit. We report reset error below 0.13% (0.16%) when prepared in the first (second) excited state of the transmon within 88ns. Additionally, through the sharp roll-off in the density of states of the metamaterial waveguide, we implement a leakage reduction unit that selectively resets the transmon's second excited state to 0.285(3)% residual population within 44ns while acting trivially in the computational subspace as an identity operation that preserves encoded information with an infidelity of 0.72(1)%., Comment: 17 pages, 11 figures

In this study, we propose a novel approach to uncover subgroup-specific and subgroup-common latent factors addressing the challenges posed by the heterogeneity of neurological and mental disorders, which hinder disease understanding, treatment development, and outcome prediction. The proposed approach, sparse Group Factor Analysis (GFA) with regularised horseshoe priors, was implemented with probabilistic programming and can uncover associations (or latent factors) among multiple data modalities differentially expressed in sample subgroups. Synthetic data experiments showed the robustness of our sparse GFA by correctly inferring latent factors and model parameters. When applied to the Genetic Frontotemporal Dementia Initiative (GENFI) dataset, which comprises patients with frontotemporal dementia (FTD) with genetically defined subgroups, the sparse GFA identified latent disease factors differentially expressed across the subgroups, distinguishing between "subgroup-specific" latent factors within homogeneous groups and "subgroup common" latent factors shared across subgroups. The latent disease factors captured associations between brain structure and non-imaging variables (i.e., questionnaires assessing behaviour and disease severity) across the different genetic subgroups, offering insights into disease profiles. Importantly, two latent factors were more pronounced in the two more homogeneous FTD patient subgroups (progranulin (GRN) and microtubule-associated protein tau (MAPT) mutation), showcasing the method's ability to reveal subgroup-specific characteristics. These findings underscore the potential of sparse GFA for integrating multiple data modalities and identifying interpretable latent disease factors that can improve the characterization and stratification of patients with neurological and mental health disorders., Comment: 38 pages, 14 figures

One of the most challenging problems in the computational study of localization in quantum manybody systems is to capture the effects of rare events, which requires sampling over exponentially many disorder realizations. We implement an efficient procedure on a quantum processor, leveraging quantum parallelism, to efficiently sample over all disorder realizations. We observe localization without disorder in quantum many-body dynamics in one and two dimensions: perturbations do not diffuse even though both the generator of evolution and the initial states are fully translationally invariant. The disorder strength as well as its density can be readily tuned using the initial state. Furthermore, we demonstrate the versatility of our platform by measuring Renyi entropies. Our method could also be extended to higher moments of the physical observables and disorder learning.

Lattice gauge theories (LGTs) can be employed to understand a wide range of phenomena, from elementary particle scattering in high-energy physics to effective descriptions of many-body interactions in materials. Studying dynamical properties of emergent phases can be challenging as it requires solving many-body problems that are generally beyond perturbative limits. We investigate the dynamics of local excitations in a $\mathbb{Z}_2$ LGT using a two-dimensional lattice of superconducting qubits. We first construct a simple variational circuit which prepares low-energy states that have a large overlap with the ground state; then we create particles with local gates and simulate their quantum dynamics via a discretized time evolution. As the effective magnetic field is increased, our measurements show signatures of transitioning from deconfined to confined dynamics. For confined excitations, the magnetic field induces a tension in the string connecting them. Our method allows us to experimentally image string dynamics in a (2+1)D LGT from which we uncover two distinct regimes inside the confining phase: for weak confinement the string fluctuates strongly in the transverse direction, while for strong confinement transverse fluctuations are effectively frozen. In addition, we demonstrate a resonance condition at which dynamical string breaking is facilitated. Our LGT implementation on a quantum processor presents a novel set of techniques for investigating emergent particle and string dynamics.

Ionic liquids have proved to be excellent heat transfer fluids and alternatives to common HTFs used in industries for heat exchangers and other heat transfer equipment. However, its industrial utilization depends on the cost per kg of its production, to be competitive for industrial applications with biphenyl and diphenyl oxide, alkylated aromatics, and dimethyl polysiloxane oils, which degrade above 200 {\deg}C and possess some environmental problems. The efficiency of a heat transfer fluid depends on the fundamental thermophysical properties influencing convective heat transfer (density, heat capacity, thermal conductivity, and viscosity), as these properties are necessary to calculate the heat transfer coefficients for different heat exchanger geometries. In Part 1, the thermophysical properties of pure 1-ethyl-3-methylimidazolium methanesulfonate $[C_{2}mim][CH_{3}SO_{3}]$ (CAS no. 145022-45-3), (ECOENG 110), produced by BASF, under the trade name of Basionics ST35, with an assay $\geq$97% with $\leq$0.5% water and $\leq$2% chloride ($Cl^{-}$), were presented, for temperatures slightly below room temperature and up to 355 K. In this paper, we report the thermophysical properties of mixtures of [C2mim][CH3SO3] with water, in the whole concentration range, at $P$ = 0.1 MPa. The properties measured were density and speed of sound (293.15 < $T$/K < 343.15), viscosity, electrical and thermal conductivities, refractive index (293.15 < $T$/K < 353.15), and infinite dilution diffusion coefficient of the ionic liquid in water (298.15 K).

Motivated by the idea of using simple macroscopic examples to illustrate the physics of complex systems, we modify a historic experimental setup in which interacting floating magnets spontaneously self-assemble into ordered clusters. By making the cluster components mechanically stable against their natural tendency to flip and coalesce, we can monitor the torque experienced by individual magnets through the macroscopic tilt angles they acquire when exposed to non-collinear external fields. A mathematical model that reproduces the empirical observations is introduced, enabling us to go beyond the experimental cases considered. The model confirms the existence of alternative orderings as the number of objects increases. Furthermore, a simpler and more mathematically transparent version of our model enables us to establish the conditions under which the cluster structure is stable and when it will collapse. We argue that our simple experimental setup combined with the accompanying models may be useful to describe general features seen in systems composed of mutually repulsive particles in the presence of modulating confining fields., Comment: 14 pages, 8 figures, submitted to PRE

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $\Lambda$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $\mu$s at distance-5 up to a million cycles, with a cycle time of 1.1 $\mu$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms., Comment: 10 pages, 4 figures, Supplementary Information

We revisit two evolutionary game theory models, namely the Prisoner and the Snowdrift dilemmas, on top of small-world networks. These dynamics on networked populations (individuals occupying nodes of a graph) are mainly concerning on the competition between to cooperate or to defect, by allowing some process of revision of strategies. Cooperators avoid defectors by forming clusters in a process known as network reciprocity. This defense strategy is based on the fact that any individual interact only with its nearest neighbors. The minimum cluster, in turn, is formed by a set of three completely connected nodes and the bulk of these triplets is associated with the transitivity property of a network. Particularly, we show that the transitivity increases eventually assuming a constant behavior when observed as a function of the number of contacts of an individual. We investigate the influence of the network reciprocity on that transitivity increasing regime on the promotion of a cooperative behavior. The dynamics on small-world networks are compared with those random regular, and annealed networks, the later typically studied as the well-mixed approach. We observe that the Snowdrift Game converge to an annealed scenario as randonness and coordination number increase, whereas the Prisoner's Dilemma becomes more severe against the cooperative behavior under the regime of an increasing network reciprocity., Comment: 10 pages, 7 figures

We introduce the generalized diffusive epidemic process, which is a metapopulation model for an epidemic outbreak where a non-sedentary population of walkers can jump along lattice edges with diffusion rates $D_S$ or $D_I$ if they are susceptible or infected, respectively, and recovered individuals possess permanent immunity. Individuals can be contaminated with rate $\mu_c$ if they share the same lattice node with an infected individual and recover with rate $\mu_r$, being removed from the dynamics. Therefore, the model does not have the conservation of the active particles composed of susceptible and infected individuals. The reaction-diffusion dynamics are separated into two stages: (i) Brownian diffusion, where the particles can jump to neighboring nodes, and (ii) contamination and recovery reactions. The dynamics are mapped into a growing process by activating lattice nodes with successful contaminations where activated nodes are interpreted as infection sources. In all simulations, the epidemic starts with one infected individual in a lattice filled with susceptibles. Our results indicate a phase transition in the dynamic percolation universality class controlled by the population size, irrespective of diffusion rates $D_S$ and $D_I$ and a subexponential growth of the epidemics in the percolation threshold., Comment: 8 pages, 4 figures

Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators. Unlocking the full potential of such systems toward this goal requires flexible initial state preparation, precise time evolution, and extensive probes for final state characterization. We present a quantum simulator comprising 69 superconducting qubits which supports both universal quantum gates and high-fidelity analog evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. Emulating a two-dimensional (2D) XY quantum magnet, we leverage a wide range of measurement techniques to study quantum states after ramps from an antiferromagnetic initial state. We observe signatures of the classical Kosterlitz-Thouless phase transition, as well as strong deviations from Kibble-Zurek scaling predictions attributed to the interplay between quantum and classical coarsening of the correlated domains. This interpretation is corroborated by injecting variable energy density into the initial state, which enables studying the effects of the eigenstate thermalization hypothesis (ETH) in targeted parts of the eigenspectrum. Finally, we digitally prepare the system in pairwise-entangled dimer states and image the transport of energy and vorticity during thermalization. These results establish the efficacy of superconducting analog-digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

We discuss the short-time behavior of the majority vote dynamics on scale-free networks at the critical threshold. We introduce a heterogeneous mean-field theory on the critical short-time behavior of the majority-vote model on scale-free networks. In addition, we also compare the heterogeneous mean-field predictions with extensive Monte Carlo simulations of the short-time dependencies of the order parameter and the susceptibility. We obtained a closed expression for the dynamical exponent $z$ and the time correlation exponent $\nu_\parallel$. Short-time scaling is compatible with a non-universal critical behavior for $5/2 < \gamma < 7/2$, and for $\gamma \geq 7/2$, we have the mean-field Ising criticality with additional logarithmic corrections for $\gamma=7/2$, in the same way as the stationary scaling., Comment: 10 pages, 3 figures, 24 references

The Prelle-Singer method allows determining an elementary first integral admitted by a polynomial vector field in the plane. It is a semi-algorithm whose nonlinear step consists of determining the Darboux polynomials of the vector field. In this article we construct a linear procedure to determine the Darboux polynomials present in the integrating factor of a polynomial vector field in the plane. Next, we extend the procedure to deal with rational 2ODEs that admit an elementary first integral

We use the pair heterogeneous mean-field (PHMF) approximation for an asynchronous version of the susceptible-infected-removed (SIR) model to estimate the epidemic thresholds on complex quenched networks. Our results indicate an improvement compared to the heuristic heterogeneous mean-field theory developed for one vertex (HMF) when the dynamic evolves on top random regular and power-law networks. However, there is a slight overestimation of the transition point for the later network type. We also analyze scaling for random regular networks near the thresholds. For this region, collapses were shown at the subcritical and supercritical phases.

Time-lapse images carry out important information about dynamic changes in Earth's interior which can be inferred using different Full Waveform Inversion (FWI) schemes. The estimation process is performed by manipulating more than one seismic dataset, associated with the baseline and monitors surveys. The time-lapse variations can be so minute and localised that quantifying the uncertainties becomes fundamental to assessing the reliability of the results. The Bayesian formulation of the FWI problem naturally provides confidence levels in the solution, but evaluating the uncertainty of time-lapse seismic inversion remains a challenge due to the ill-posedness and high dimensionality of the problem. The Hamiltonian Monte Carlo (HMC) can be used to effectively sample over high dimensional distributions with affordable computational efforts. In this context, we propose a probabilistic Bayesian sequential approach for time-lapse FWI using the HMC method. Our approach relies on the integration of the baseline survey information as prior knowledge in the monitor estimation. We compare the proposed methodology with a parallel scheme in perfect and perturbed acquisition geometry scenarios. We also investigate the correlation effect between baseline and monitor samples in the propagated uncertainties. The results show that our strategy provides accurate times-lapse estimates with errors of similar magnitude to the parallel methodology., Comment: 12 pages, 6 figures

In this work, we constrain the Hubble constant parameter, $H_0$, using a combination of the Pantheon sample and galaxy clusters (GC) measurements from minimal cosmological assumptions. Assuming the validity of the cosmic distance duality relation, an estimator is created for $H_0$ that only depends on simple geometrical distances, which is evaluated from Pantheon and a GC angular diameter distance sample afterward. The statistical and systematic errors in GC measurements are summed in quadrature in our analysis. We find $H_0 = 67.22 \pm 6.07$ ${\rm km \ s^{-1} Mpc^{-1}}$ in $1\sigma$ confidence level (C.L.). This measurement presents an error of around 9\%, showing that future and better GC measurements can shed light on the current Hubble tension, Comment: 8 pages, two figures

In this work, we present a theoretical investigation of the elasticity properties of new exotic carbon allotropes studied recently called Irida-graphene and Sun-graphene. The Irida-graphene is a new 2D all-sp2 carbon allotrope composed of fused rings containing 3-6-8 carbon atoms. However, The Sun-graphene is a new carbon allotrope analogous to graphene, has a semi-metal nanostructure and presents two Dirac cones in its band structure. Despite being theoretically studied in single-layer (2D), no study of the elastic properties of Irida-graphene-based nanotubes (Irida-CNTs) and Sun-graphene-based nanotubes (Sun-CNTs) has not been studied yet. Thus, we seek to investigate the elastic properties of nanotubes of these new exotic allotropes of carbon in quasi-one-dimensional geometry (1D), nanotubes for different chiralities, diameters and lengths at room temperature. For the development of the theoretical investigation of the elasticity of the Irida-CNTs and Sun-CNTs, we developed a fully reactive (ReaxFF) atomistic classical molecular dynamics simulation method. Our theoretical results obtained in the computational simulations show that the Irida-CNTs nanotubes behave as flexible nanostructure and have a slightly higher value of strain rate than Sun-CNTs. The values of Young's modulus for Irida-CNTs are slightly larger range (640.34 GPa - 825.00 GPa), while the Young's Modulus range of Sun-CNTs is 200.44 GPa - 472.84 GPa, the lower values than those presented for conventional carbon nanotubes (CNT). These findings provide insights into the mechanical behavior of Irida-CNTs and Sun-CNTs and their potential applications in nanoscale devices.

The Baryon Acoustic Oscillations (BAO) are one of the most used probes to understand the accelerated expansion of the Universe. Traditional methods rely on fiducial model information within their statistical analysis, which may be a problem when constraining different families of models. This work aims to provide a method that constrains $\theta_{BAO}$ through a model-independent approach using the covariance matrix from the galaxy sample from thin redshift bins, later validated with a mock sample covariance matrix. We used widths of $\delta z = 0.002$ separation for all samples as the basis for a sample covariance matrix weighted by the statistical importance of the redshift bin. Each sample belongs to the Sloan Digital Sky Survey: BOSS1, BOSS2, and eBOSS, with effective redshift $z_{eff}$: 0.35, 0.51, 0.71, and different numbers of bins with 50, 100, and 200. To get $\theta_{BAO}$, we correct the angular separation from the polynomial fit ($\theta_{fit}$) by comparing each bin correlation function with the correlation function of the whole set, a parameter named $\tilde{\alpha}$. We also tested such correction by choosing the bin at $z_{eff}$ and found that for eBOSS $\theta_{BAO}$ is in $1 \sigma$ agreement with the Planck 18 model. Finally, we found that the sample covariances are noisy compared to the mocks for lower $z$ samples, something expected due to nonlinear effects. Such noise impact can be seen in the parameter constraints but does not affect the eBOSS covariance sample. It is shown that mocks' results do tend to its chosen fiducial cosmology $\theta_{BAO}$. BOSS1 and BOSS2 showed agreement with Planck 18 and an agreement with Pantheon + S$H_0$ES when $\tilde{\alpha}$ is based on the bin $z=z_{eff}$., Comment: 18 pages Changed figures and included comparison between old and new method with the mocks

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this paper, we obtain a nontrivial invariant subspace of $T^{*}_{\varphi}|_{M}$, where $T_{\varphi}$ is the Toeplitz operator on the Hardy space over the bidisk $H^{2}(\mathbb{D}^{2})$ induced by the symbol $\varphi\in H^{\infty}(\mathbb{D})$ and $M$ is a $T_{\varphi}^{*}$-invariant subspace. We use this fact to get sufficient conditions for the ISP., Comment: 7 pages

In this paper we find non-negative integer solutions for exponential Diophantine equations of the type $p \cdot 3^x+ p^y=z^2,$ where $p$ is a prime number. We prove that such equation has a unique solution $\displaystyle{(x,y,z)=\left(\log_3(p-2), 0, p-1\right)}$ if $2 \neq p \equiv 2 \pmod 3$ and $(x,y,z)=(0,1,2)$ if $p=2$. We also display the infinite solution set of that equation in the case $p=3$. Finally, a brief discussion of the case $p \equiv 1 \pmod 3$ is made, where we display an equation that does not have a non-negative integer solution and leave some open questions. The proofs are based on the use of the properties of the modular arithmetic.

In this work, we study and evaluate the impact of a periodic spin-lattice coupling in an Ising-like system on a 2D triangular lattice. Our proposed simple Hamiltonian considers this additional interaction as an effect of preferential phonon propagation direction augmented by the symmetry ofthe underline lattice. The simplified analytical description of this new model brought us consistent information about its ground state and thermal behavior, and allowed us to highlight a singularity where the model behaves as several decoupled one-dimensional Ising systems. A thorough analysis was obtained via entropic simulations based in the Wang-Landau method that estimates the density of states g(E) to explore the phase diagram and other thermodynamic properties of interest. Also, we used the finite size scaling technique to characterize the critical exponents and the nature of the phase transitions that, despite the strong influence of the spin-lattice coupling, turned out to be within the same universality class as the original 2D Ising model.

Engineered dissipative reservoirs have the potential to steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using up to 49 superconducting qubits, we prepared low-energy states of the transverse-field Ising model through coupling to dissipative auxiliary qubits. In one dimension, we observed long-range quantum correlations and a ground-state fidelity of 0.86 for 18 qubits at the critical point. In two dimensions, we found mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we explored transport in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors.

Undesired coupling to the surrounding environment destroys long-range correlations on quantum processors and hinders the coherent evolution in the nominally available computational space. This incoherent noise is an outstanding challenge to fully leverage the computation power of near-term quantum processors. It has been shown that benchmarking Random Circuit Sampling (RCS) with Cross-Entropy Benchmarking (XEB) can provide a reliable estimate of the effective size of the Hilbert space coherently available. The extent to which the presence of noise can trivialize the outputs of a given quantum algorithm, i.e. making it spoofable by a classical computation, is an unanswered question. Here, by implementing an RCS algorithm we demonstrate experimentally that there are two phase transitions observable with XEB, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak link model which allows varying the strength of noise versus coherent evolution. Furthermore, by presenting an RCS experiment with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers, even when accounting for the inevitable presence of noise. Our experimental and theoretical work establishes the existence of transitions to a stable computationally complex phase that is reachable with current quantum processors.

Measurement has a special role in quantum theory: by collapsing the wavefunction it can enable phenomena such as teleportation and thereby alter the "arrow of time" that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space-time that go beyond established paradigms for characterizing phases, either in or out of equilibrium. On present-day NISQ processors, the experimental realization of this physics is challenging due to noise, hardware limitations, and the stochastic nature of quantum measurement. Here we address each of these experimental challenges and investigate measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping, to avoid mid-circuit measurement and access different manifestations of the underlying phases -- from entanglement scaling to measurement-induced teleportation -- in a unified way. We obtain finite-size signatures of a phase transition with a decoding protocol that correlates the experimental measurement record with classical simulation data. The phases display sharply different sensitivity to noise, which we exploit to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realize measurement-induced physics at scales that are at the limits of current NISQ processors.

We consider the Majority Vote model coupled with scale-free networks. Recent works point to a non-universal behavior of the Majority Vote model, where the critical exponents depend on the connectivity while the network's effective dimension $D_\mathrm{eff}$ is unity for a degree distribution exponent $5/2<\gamma<7/2$. We present a finite-size theory of the Majority Vote Model for uncorrelated networks and present generalized scaling relations with good agreement with Monte-Carlo simulation results. The presented finite-size theory has two main sources of size dependence. The first source is an external field describing a mass media influence on the consensus formation and the second source is the scale-free network cutoff. The model indeed presents non-universal critical behavior where the critical exponents depend on the degree distribution exponent $5/2<\gamma<7/2$. For $\gamma \geq 7/2$, the model is on the same universality class of the Majority Vote model on Erd\"os-Renyi random graphs, while for $\gamma=7/2$, the critical behavior presents additional logarithmic corrections., Comment: arXiv admin note: text overlap with arXiv:1910.06046

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems., Comment: 11 pages, 6 figures

We present a finite-size scaling theory of a contact process with permanent immunity on uncorrelated scale-free networks. We model an epidemic outbreak by an analog of the susceptible-infected-removed model where an infected individual attacks only one susceptible in a time unit in a way we can expect a non-vanishing critical threshold at scale-free networks. As we already know, the susceptible-infected-removed model can be mapped in a bond percolation process, allowing us to compare the critical behavior of site and bond universality classes on networks. We used the external field finite-scale theory, where the dependence on the finite size enters the external field defined as the initial number of infected individuals. We can impose the scale of the external field as $N^{-1}$. The system presents an epidemic-endemic phase transition where the critical behavior obeys the mean-field universality class, as we show theoretically and by simulations.