For each poster contribution there will be one poster wall (width: 97 cm, height: 250 cm) available. Please do not feel obliged to fill the whole space. Posters can be put up for the full duration of the event.

Andreoni, Riccardo

Classifying phases of matter traditionally requires knowledge of the symmetries and properties of the system in question. However, inspired by experimental capabilities for stochastic sampling of wave functions via projective measurements, we explore whether classification can be achieved based solely on these measurements, without prior system knowledge. We introduce a novel classification method for states of matter based on stochastic sampling of the system's wave function. This method employs two distinct analyses. The first involves examining the intrinsic dimension of the dataset derived from the samples, which effectively discriminates between different phases. The second analysis constructs networks from the sample dataset, where the network properties are sensitive to phase transitions. Notably, we observe the emergence of scale-free networks near critical points and within critical phases. This method offers a practical way to classify phases of matter using only measurement data.

Biagetti, Leonardo

We consider quantum or classical many-body Hamiltonian systems, whose dynamics is given by integrable, contact interactions, and by a weak, possibly long-range, non-integrable two-body potential. We show how the dynamics of local observables is given in terms of a generalised version of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, built for the densities, and their correlations, of the quasiparticle of the underlying integrable model. Unlike the usual cases of perturbed free gases, the presence of local interactions lifts the so-called kinetic blockade, and the second layer of the hierarchy is enough to reproduce the dynamics at all time-scales. The latter consists in a fast pre-equilibration to a non-thermal steady state, and its subsequent thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded into a Boltzmann scattering integral which includes three or higher body-scatterings, and which, remarkably, it is completely determined by the diffusion constants of the underlying integrable model. We check our results with exact molecular dynamics simulations, finding perfect agreement. Our results show how BBGKY hierarchy can be successfully used in both quantum and classical systems to efficiently describe the dynamics near any integrable Hamiltonian.

Cook, Ashley Megan

A great variety of topological phases have been classified as a consequence of discovery of the quantum Hall effect, but this work has recently led to discovery of some topologically non-trivial phases of matter, which contradict key assumptions of established classification schemes. These phases, which are the topological skyrmion phases of matter, multiplicative topological phases of matter, and finite-size topological phases of matter, necessitate a paradigm shift from the quantum Hall effect framework to that of the quantum skyrmion Hall effect, in which the point charges of the quantum Hall effect are generalised to compactified p-branes. For compactification via fuzzification, these compactified p-branes carrying charge are necessarily expressed in terms of angular momentum as quantum skyrmions.

Eckstein, Finn

Teleportation is a facet where quantum measurements can act as a powerful resource in quantum physics, as local measurements allow to steer quantum information in a non-local way. While this has long been established for a single Bell pair, the teleportation of a fault-tolerant logical qubit presents a fundamentally different challenge as it requires the teleportation of a many-qubit state. Here we investigate a tangible protocol for teleporting a long-range entangled surface code state using elementary Bell measurements and its stability in the presence of tunable coherent errors. We relate the underlying threshold problem to the physics of anyon condensation under weak measurements and map it to a variant of the Ashkin-Teller model of statistical mechanics with Nishimori type disorder, which gives rise to a cascade of phase transitions. Tuning the angle of the local Bell measurements, we find a continuously varying threshold. Notably, the threshold moves to infinity for the X + Z angle along the self-dual line – indicating an optimal protocol that is fault-tolerant even in the presence of coherent noise. Our teleportation protocol, which can be readily implemented in dynamically configurable Rydberg atom arrays, thereby gives guidance for a practical demonstration of the power of quantum measurements.

Jacoby, Alexander

Thermalization in the Heisenberg picture is often conceptualized as the interchange of local, few-body operators with global many-body operators. Under unitary dynamics, starting from an initial area law state, initially local operators have fixed expectation values. These expectation values are held fixed as the operators spread. However, when they become global, the fixed expectation values no longer correspond to physical observables. Thus, the expectation values of later-time local operators are randomized (they are the expectation values of initially global operators), and the unitary dynamics of observable operators become, in a sense, dissipative. By introducing an artificial but arbitrarily weak dissipation in conjunction with a thermodynamic limit, some aspects of this dissipative loss of information (or relaxation) can be understood in a hydrodynamic picture for chaotic systems. From non-unitary operator spreading equations, we uncover a universal lower speed limit on the dissipation of observable operators and the relaxation of initially non-equilibrium states.

Li, Lauren

We demonstrate that non-integrable systems can exhibit Page curve-like behavior in their entanglement entropy dynamics. Specifically, we study the mixed-field Ising model using both full bath reconstruction and Lindbladian approaches. Both Markovian and non-Markovian models of the bath reveal an initial increase in entanglement entropy, followed by a subsequent decrease. Here, we characterize the evolution of the entanglement spectrum across the Page transition. Our findings can be explained by considering an entanglement Hamiltonian with spatially-dependent temperature.

Matrasulov, Jasur

Confined nanoscale systems appear in different problems of modern quantum science and technology. In most of the cases confinement is not static, but dynamic. In such cases, for optimization of functional materials of device optimization, one needs to control the quantum evolution of such system by tuning some parameters. In this work we propose a protocol for such control. In particular, using of the scheme of fast forward which realizes quasistatic or adiabatic dynamics in shortened timescale, we investigate a thermally isolated ideal quantum gas confined in a rapidly dilating one-dimensional (1D) cavity with the time-dependent size $????=????(????)$. In the fast-forward variants of equation of states, i.e., Bernoulli's formula and Poisson's adiabatic equation, the force or 1D analog of pressure can be expressed as a function of the velocity $(\dot{????})$ and acceleration $(\ddot{????})$ of $????$ besides rapidly changing state variables like effective temperature $(????)$ and $????$ itself. The force is now a sum of nonadiabatic (NAD) and adiabatic contributions with the former caused by particles moving synchronously with kinetics of $????$ and the latter by ideal bulk particles insensitive to such a kinetics. The ratio of NAD and adiabatic contributions does not depend on the particle number $(????)$ in the case of the soft-wall confinement, whereas such a ratio is controllable in the case of hard-wall confinement. We also reveal the condition when the NAD contribution overwhelms the adiabatic one and thoroughly changes the standard form of the equilibrium equation of states.

McRoberts, Adam

Motivated by recent experiments on Google’s sycamore NISQ platform on the spin transport resulting from a non-unitary periodic boundary drive of an XXZ chain, we study a classical variant thereof by a combination of analytical and numerical means. We find the classical model reproduces the quantum results in remarkable detail, and provides an analytical handle on the nature and shape of the spin transport’s three distinct regimes: ballistic (easy-plane), subdiffusive (isotropic) and insulating (easy-axis). Further, we show that this phenomenology is remarkably robust to the inclusion of bond disorder – albeit that the transient dynamics approaching the steady states differs qualitatively between the clean and disordered cases – providing an accessible instance of ballistic transport in a disordered setting.

Mondal, Suman

In the study of the thermalization of closed quantum systems, the role of kinetic constraints on the temporal dynamics and the eventual thermalization is attracting significant interest. Kinetic constraints typically lead to long-lived metastable states depending on initial conditions. We consider a model of interacting hardcore bosons with an additional kinetic constraint that was originally devised to capture glassy dynamics at high densities. As a main result, we demonstrate that the system is highly prone to localization in the presence of uncorrelated disorder. Adding disorder quickly triggers long-lived dynamics as evidenced in the time evolution of density autocorrelations. Moreover, the kinetic constraint favors localization also in the eigenstates, where a finite-size transition to a many-body localized phase occurs for much lower disorder strengths than for the same model without a kinetic constraint.

Mu, Sen

Originally associated with classical systems, Kardar-Parisi-Zhang (KPZ) universality class describes a broad range of processes, ranging from growth of rough interfaces to directed polymers. We have identified the key features of the KPZ universality class in the fluctuations of the wave density logarithm in two-dimensional Anderson localization. Our previous work [1] shows that these fluctuations exhibit algebraic scaling with distance, characterized by an exponent of 1/3, and follow a Tracy-Widom probability distribution. In this study, we explore the conditions necessary to realize genuine Anderson localization of a few-body wave function in Fock space, thereby uncovering the KPZ universality class in (1+1)D for the two-body localized wave function. To validate the universality of our results, we draw comparisons with the well-understood directed polymer problem in the presence of competing point and columnar disorders. Additionally, we discuss the generalization of these findings to the localized wave functions of multiple particles and the emergence of the KPZ universality class in higher dimensions. Our work paves the way for new investigations into the KPZ universality class in disordered quantum systems and establishes a bridge for applying the analytic framework of KPZ physics to the study of many-body localized quantum systems. [1] S. Mu, J. Gong, and G. Lemarié, Kardar-parisi-zhang physics in the density fluctuations of localized two-dimensional wave packets, Phys. Rev. Lett. 132, 046301 (2024).

Muzzi, Cristiano

We study how stronger noise can counter-intuitively enhance the entanglement in inhomogeneously monitored quantum systems. We consider a free fermions model composed of two coupled chains - a system chain and an ancilla chain, each subject to its own different noise - and explore the dynamics of entanglement within the system chain under different noise intensities. Our results demonstrate that, contrary to the detrimental effects typically associated with noise, certain regimes of noise on the ancilla can significantly enhance entanglement within the system. Numerical simulations demonstrate the robustness of such entanglement enhancement across various system sizes and noise parameters. This enhancement is found to be highly dependent on the hopping strength in the ancilla, suggesting that the interplay between unitary dynamics and noise can be tuned to optimize entanglement.

Nambiar, Gautam

Theories whose fluctuating degrees of freedom live on extended loops as opposed to points, are abundant in nature. One example is the action obtained upon eliminating the redundant gauge fields in a gauge theory. Formulating a Renormalization Group (RG) procedure for such a theory is an open problem. In this work, we outline a procedure that in principle computes the outcome of coarse-graining and rescaling of such a theory. We approximately solve the RG flow equations and we find qualitative agreement with known results of phase transitions in gauge theories and the XY-model. For concreteness, we focus on compact pure $U(1)$ gauge theory in 3+1 dimensions which is known to have a confinement-deconfinement phase transition. We start with a pedagogical review for a mapping from this theory to a dual sine Gordon-like theory of a Coulomb loop gas. We then illustrate our RG procedure on this theory by drawing parallels to known RG procedures for regular Sine-Gordon field theories, while at the same time emphasizing the loop nature of the theory. We show how our procedure readily generalizes to the $XY$ model in 2+1 dimensions and to odd compact $U(1)$ gauge theory in 3+1 dimensions. From a contemporary perspective, our work is an attempt at an RG procedure for a theory with one-form symmetries.

Pei, Yufei

I explore the dynamics of the random transverse field ferromagnetic Ising model (RTFIM), a one-dimensional chain with dilute, randomly placed transverse fields. At low temperatures, the classical Ising chain exhibits diffusive dynamics characterised by randomly walking domain walls (DWs), akin to magnetic monopoles in classical spin ice. Introducing transverse fields at select sites introduces nontrivial modifications to these dynamics, offering insights into recent experimental investigations involving similarly doped spin ice materials. Following an introductory chapter on the background and motivation of this research, I first examine the thermodynamics of the RTFIM. This study reveals a lowered chemical potential for the creation of DWs on spins with transverse fields, suggesting a transverse-field-induced trapping effect. Subsequently, I propose a Lindbladian approach that captures the dynamics in the classical Ising chain, enabling an investigation of the RTFIM dynamics using a combination of Monte Carlo and exact diagonalisation methods. In the next chapter, I present the results of my analysis, which reveal significant modifications to the power spectral density of the magnetic noise generated by the model's dynamics. These effects intensify with increasing doping concentrations and exhibit an intricate dependence on temperature and transverse field strength. I attribute this behaviour to the trapping effects of the transverse fields, which alter the random walking patterns of domain walls, a phenomenon I model in detail. My findings provide crucial insights into the magnetic noise of disordered non-Kramers oxides, with implications for experimental studies of oxygen-diluted spin ice.

Shnaiderov, David

Crosstalk between qubits induces unwanted entanglement and errors, affecting quantum computation regardless of hardware quality. Accurately measuring and mitigating this noise is crucial for improving quantum device performance. This poster explores effective methods to identify said crosstalk, aiming to enhance quantum processor reliability

Singh Roy, Monalisa

Recent theoretical studies have predicted the existence of caustics in many-body quantum dynamics, where they manifest as extended regions of enhanced probability density that obey temporal and spatial scaling relations. Using the transverse field Ising model (TFIM), we investigate the dynamics initiated by a local quench in a spin chain, resulting in outward-propagating domain walls that create a distinct caustic pattern. We calculate the scaling of the first two maxima of the interference fringes dressing the caustic, finding a universal exponent of 2/3, associated with an Airy function catastrophe. We demonstrate that this property is universal in the entire paramagnetic phase of the model, and starts varying at the quantum phase transition (QPT). This robust scaling persists even under integrability-breaking perturbations introduced through $S^z S^z$ interactions, until a critical threshold where the caustic pattern degrades. Additionally, we explore the effect of boundary conditions on the TFIM and find that while periodic boundaries preserve transnational invariance and uniform propagation of domain walls, open boundaries introduce significant edge effects, leading to complex interference patterns. Despite these edge-induced dynamics, the overall power-law scaling exponent remains robust. These findings highlight the potential of quantum caustics as a powerful diagnostic tool for QPTs, demonstrating resilience against integrability-breaking perturbations and boundary condition variations.

Sinha, Aritra

Phase Separation(PS) is today believed to be a common trait of high-temperature superconductors, with the widespread appearance of charge-inhomogenity in numerous cuprate materials serving as a key example [1]. The 2D Fermi Hubbard model on a square lattice (FHM) is the paradigmatic model for understanding cuprates. Recent numerical studies through multiple efforts predicted that the ground state of underdoped strongly coupled FHM is not phase separating but striped [2]. In our calculations, however, we find evidence of phase separation between Mott insulating and metallic hole-rich regions at finite temperatures (not ground state) using state-of-the-art tensor network techniques. Signatures of this phase separation are observed as a soft peak in compressibility as a function of chemical potential using infinite projected entangled pair states (iPEPS) data on an infinite lattice. Additionally, real-space snapshots obtained from minimally entangled typical thermal states (METTS) on a cylindrical geometry reveal clustering of holes at intermediate temperatures, transitioning to striped physics at lower temperatures (T< 0.05). Analysis of the histogram of hole cluster sizes across different temperatures shows that the mean cluster size reaches a maximum at intermediate temperatures (0.1 ≤ T ≤ 0.2). This peak coincides with the peak of spin structure factor at k = (π, π), indicating presence of strong antiferromagnetic correlations whenever there is strong hole clustering. These findings serve as hallmarks of phase separation in the system. [1]E. Sigmund and K. A. Muller, Phase separation in cuprate superconductors: Proceedings of the second international workshop, September 4-10, 1993, Cottbus, Germany (2012). [2]B.-X. Zheng et. al. Stripe order in the underdoped region of the two-dimensional hubbard model, Science 358, 1155 (2017).

Sonner, Michael

Computing local dynamics of a small quantum system coupled to large, complex quantum environments is a ubiquitous problem in quantum science. This problem can be approached by characterizing the quantum environments by their influence matrix~(IM) -- a multi-time tensor describing the repeated interactions between the system and environment. A priori, the number of parameters needed to describe such an IM grows exponentially with the evolution time. However, recent works argued that for many relevant classes of physical many-body environments, the IM can be parametrized by a low bond dimension Matrix Product State~(MPS). This is thanks to the slow growth of temporal entanglement, which quantifies the correlations between the past and the future states of the system. To perform practical computations using this approach the existence of such an efficient MPS representation is not sufficient. One also needs an algorithm to obtain it; currently such algorithms are only available for certain cases such as non-interacting and homogenous 1D environments. In this talk, we explore how a quantum computer can help us to obtain the IM even in cases where no classical algorithm is known. On a quantum processor auxiliary qubits are repeatedly coupled to the simulated many-body environment and measured. From these measurement samples we construct a MPS representation of the IM using a classical machine learning approach. We demonstrate that this hybrid algorithm allows for IM reconstruction even for long times by using a classically generated training dataset for 1D spin chains. We show that this reconstructed IM can be succesfully used to predict local dynamics of a qubit coupled to a spin chain. Furthermore we show that we can compute quantum transport even in cases with multiple leads by combining multiple independently reconstructed IM. These results indicate the feasibility of characterizing long-time dynamics of complex environments using a limited number of measurements, under the assumption of a moderate temporal entanglement. This talk would be mostly based on the article: Luchnikov, Ilia A., Michael Sonner, and Dmitry A. Abanin. "Scalable tomography of many-body quantum environments with low temporal entanglement." arXiv preprint arXiv:2406.18458 (2024).

Tarabunga, Poetri Sonya

Magic, also known as nonstabilizerness, quantifies the distance of a quantum state to the set of stabilizer states, and it serves as a necessary resource for potential quantum advantage over classical computing. In this work, we study magic in a measurement-only quantum circuit with competing types of Clifford and non-Clifford measurements, where magic is injected through the non-Clifford measurements. This circuit can be mapped to a classical model that can be simulated efficiently, and the magic can be characterized using any magic measure that is additive for tensor product of single-qubit states. Leveraging this observation, we study the magic transition in this circuit in both one- and two-dimensional lattices using large-scale numerical simulations. Our results demonstrate the presence of a magic transition between two different phases with extensive magic scaling, separated by a critical point in which the mutual magic exhibits scaling behavior analogous to entanglement. We further show that these two distinct phases can be distinguished by the topological magic. In a different regime, with a vanishing rate of non-Clifford measurements, we find that the magic saturates in both phases. Our work sheds light on the behavior of magic and its linear combinations in quantum circuits, employing genuine magic measures.

Tornow, Sabine

We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics is recorded using unitary dynamics interspersed stroboscopically by measurements, which is implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both, return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. Yet, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements. Our results point the way for the development of novel quantum walk algorithms that exploit measurement-induced effects on quantum computers.

Uria Valencia, Mariano

We present a protocol to deterministically prepare the electromagnetic field in a large photon number state. The field starts in a coherent state and, through resonant interaction with one or few two-level systems, it evolves into a coherently displaced Fock state without any postselection. We show the feasibility of the scheme under realistic parameters. The presented method opens a door to reach Fock states, with ????∼100 and optimal fidelities above 70%, blurring the line between macroscopic and quantum states of the field.

Ván , Peter

Quantum fluids are particular Korteweg fluids, simple fluids that are weakly nonlocal in density. The goal of the presentation is to show the physical and mathematical conditions that connect general Korteweg fluids, quantum fluids and the fluid analogy of quantum mechanics. The usual quantum to classical arguments are reversed, the starting point is classical fluid mechanics. It is shown, that the Second Law of Thermodynamics imposes strong constraints on the possible forms of the constitutive functions. One of the general consequences is that ideal Korteweg fluids are holographic, in the sense that the divergence of the pressure can be transformed to a force density: bulk force acting on a fluid body can be substituted with surface traction. The holographic property is the bridge between particle and field based interpretations of fluid-quantum systems. Another general consequence is the Euler-Lagrange form of the quantum potential, a property emerging without variational principles in the thermodynamical framework. I briefly treat the use of fluid mechanics based computation methods in quantum physics and also the inevitable questions regarding the foundations of physics. ---- Note: I do not know the difference between seminar and workshop talks, therefore I am a little confused with the choice below. My preference is to show my results to every potentially interested participants.

Verdel Aranda, Roberto

We introduce a simple yet powerful framework for inferring dynamical scaling exponents from wave function snapshots, which are routinely available in present-day quantum simulation experiments. This framework is grounded on a relatively simple unsupervised learning technique, namely principal component analysis, which is applied to data sets of wave function snapshots to examine how information propagates within these data sets. Our numerical investigations on nonequilibrium dynamics in several interacting quantum spin chains featuring distinct spin or energy transport reveal that the growth of data information spreading follows the same dynamical exponents as that of the underlying quantum transport of spin or energy. Specifically, our approach enables an easy, data-driven, and interpretable diagnostic to track energy transport with a limited number of samples — a task that is usually challenging without any assumption on the Hamiltonian form. These observations are obtained at a modest system size and evolution time, which aligns with present experimental and numerical constraints. Our framework directly applies to experimental quantum simulator data of dynamics in higher-dimensional systems, where classical simulation methods often encounter significant limitations. This contribution is based on the following preprint: D. S. Bhakuni, R. Verdel, C. Muzzi, R. Andreoni, M. Aidelsburger, M. Dalmonte, Diagnosing quantum transport from wave function snapshots, arXiv preprint, arXiv:2407.09092 (2024).

Wang, Yupeng

Most open quantum system models described by Lindblad master equation in previous literature exhibit diffusive transport. In this study, we explore the transport dynamics of the Lindblad master equation with free fermion Hamiltonian and quasi-particle projector Lindblad operator. We reveal that when the momentum distribution of quasi-particles has nodal points, superdiffusive transport could emerge. By studying the dynamics of the Wigner function, we rigorously elucidate how the dynamics of these enduring modes give rise to Levy walk processes, a renowned mechanism underlying superdiffusion phenomena. Our research not only demonstrates the controllability of dynamical scaling exponents through the selection of quasi-particles but also extends its applicability to higher dimensions, underlining the pervasive nature of superdiffusion in dephasing models.

Yang, Jing

Many-body interactions can introduce entanglement between particles and hence are valuable resources for quantum information processing. In quantum metrology, the precision can be further boosted by adding many-body interactions. In this talk, I will discuss a variational principle for controlling many-body quantum systems with restricted operations in the context of quantum sensing. We show that in a spin chain model containing three-body interactions, the Heisenberg scaling can be still achieved even if the control operations are restricted to one-body and two-body interactions, given an initial GHZ state can be prepared. When the GHZ state cannot be efficiently prepared in experiments, one may consider many-body sensing with separable initial states. We find that using separable initial states cannot beat the shot noise limit in locally interacting systems, unless long-range non-local interactions are utilized. These findings identify two important ingredients in many-body sensing: initial entanglement and long-range interactions. Finally, I will briefly discuss how local optimal measurements can be performed to extract the many-body precision limits. [1] Physical Review Letters 132, 100803(2024) [2] Physical Review Letters 128,160505(2022)

Yao, Jiangtian

We study temporal entanglement in dual-unitary Clifford circuits with probabilistic measurements preserving spatial unitarity. We exactly characterize the temporal entanglement barrier in the measurement-free regime, exhibiting ballistic growth and decay and a volume-law peak. In the presence of measurements, we relate the temporal entanglement to the scrambling properties of the circuit. For "good scramblers" measurements do not qualitatively change the temporal entanglement profile but only result in a reduced entanglement velocity, whereas for "poor scramblers" the initial ballistic growth of temporal entanglement with bath size is modified to diffusive. This difference is understood through a mapping of the underlying operator dynamics to a biased and an unbiased persistent random walk respectively. In the latter case measurements additionally modify the ballistic decay to the perfect dephaser limit, with vanishing temporal entanglement, to an exponential decay, which we describe through a spatial transfer matrix method. This spatial dynamics is shown to be described by a non-Hermitian hopping model, exhibiting a PT-breaking transition at a critical measurement rate p=1/2. In all cases the peak value of the temporal entanglement barrier exhibits volume-law scaling for all measurement rates.

Zhang, Frank

Preparing long-range entangled states poses significant challenges for near-term quantum devices. It is known that measurement and feedback (MF) can aid this task by allowing the preparation of certain paradigmatic long-range entangled states with only constant circuit depth. Here we systematically explore the structure of states that can be prepared using constant-depth local circuits and a single MF round. Using the framework of tensor networks, the preparability under MF translates to tensor symmetries. We detail the structure of matrix-product states (MPS) and projected entangled-pair states (PEPS) that can be prepared using MF, revealing the coexistence of Clifford-like properties and magic. Furthermore, we fully parameterize states exhibiting MF symmetries akin to the symmetry-protected topological order in one dimension and the topological order in two dimensions, and we discuss their characteristics. Finally, we discuss the analogous implementation of operators via MF, providing a structural theorem that connects to the well-known Clifford teleportation.