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Quantum spin liquids, exotic phases of matter with topological order, have been a major focus of explorations in physical science for the past several decades. Such phases feature long-range quantum entanglement that can potentially be exploited to realize robust quantum computation. We use a 219-atom programmable quantum simulator to probe quantum spin liquid states. In our approach, arrays of atoms are placed on the links of a kagome lattice and evolution under Rydberg blockade creates frustrated quantum states with no local order. The onset of a quantum spin liquid phase of the paradigmatic toric code type is detected by evaluating topological string operators that provide direct signatures of topological order and quantum correlations. Its properties are further revealed by using an atom array with nontrivial topology, representing a first step towards topological encoding. Our observations enable the controlled experimental exploration of topological quantum matter and protected quantum information processing.

We develop a novel approach aimed at solving the equations of motion of open quantum many-body systems. It is based on a combination of generalized wave function trajectories and matrix product states. We introduce an adaptive quantum stochastic propagator, which minimizes the expected entanglement in the many-body quantum state, thus minimizing the computational cost of the matrix product state representation of each trajectory. We illustrate this approach on the example of a one-dimensional open Brownian circuit. We show that this model displays an entanglement phase transition between area and volume law when changing between different propagators and that our method autonomously finds an efficiently representable area law unravelling.

Trapped neutral atoms have become a prominent platform for quantum science, where entanglement fidelity records have been set using highly-excited Rydberg states. However, controlled two-qubit entanglement generation has so far been limited to alkali species, leaving the exploitation of more complex electronic structures as an open frontier that could lead to improved fidelities and fundamentally different applications such as quantum-enhanced optical clocks. Here we demonstrate a novel approach utilizing the two-valence electron structure of individual alkaline-earth Rydberg atoms. We find fidelities for Rydberg state detection, single-atom Rabi operations, and two-atom entanglement surpassing previously published values. Our results pave the way for novel applications, including programmable quantum metrology and hybrid atom-ion systems, and set the stage for alkaline-earth based quantum computing architectures., ISM and JPC contributed equally to this work

Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a new strategy for accessing a family of fractionalized phases known as quantum spin liquids in two-dimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model, albeit from an entirely different framework. In particular, we argue that finite-range repulsive Rydberg interactions, which frustrate nearby symmetry-breaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full two-dimensional Rydberg array yields a gapless Z2 spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal power-law correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toric-code topological order, whereas introducing chirality generates non-Abelian Ising topological order. In the toric-code phase, we analytically construct microscopic incarnations of non-Abelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a cold-atoms counterpart of solid-state 'Kitaev materials' and, more generally, spotlights a new angle for pursuing experimental platforms for Abelian and non-Abelian fractionalization., 17 pages, 7 figures

Despite its scarcity, upward lightning initiated from tall structures causes more damage than common downward lightning. One particular subtype with a continuous current only is not detectable by conventional lightning location systems (LLS) causing a significantly reduced detection efficiency. Upward lightning has become a major concern due to the recent push in the field of renewable wind energy generation . The growing number of tall wind turbines increased lightning related damages. Upward lightning may be initiated by the tall structure triggering the flash itself (self-triggered) or by a flash striking close by (other-triggered).The major objective of this study is to find the driving atmospheric conditions influencing whether an upward flash is self-triggered or other-triggered and whether it is of the undetectable subtype. We explore upward flashes directly measured at the Gaisberg Tower in Salzburg (Austria) between 2000 and 2015. These upward flashes are combined with atmospheric reanalysis data stratified into five main meteorological groups: cloud physics, mass field, moisture field, surface exchange and wind field. We use classification methods based on tree-structured ensembles in form of conditional random forests. From these random forests we assess the meteorological influence and find the most important atmospheric drivers for one event or the other, respectively.Whether upward lightning is self-triggered or other-triggered can be reliably explained by meteorology. The closer the -10 °C isotherm is to the tall structure, the higher is the probability of self-triggered flashes. On the other hand, lower proportions of solid hydrometeors, supercooled liquid water and lower amounts of large scale precipitation increase the probability of an initial continuous current only flash type. However, the occurrence of nearby lightning discharges is about ten times more important for the type of upward flash. No nearby discharges (or them being further than 4 km away) considerably increases the probability of the initial continuous current only flash type.

Rydberg chains provide an appealing platform for probing conformal field theories (CFTs) that capture universal behavior in a myriad of physical settings. Focusing on a Rydberg chain at the Ising transition separating charge density wave and disordered phases, we establish a detailed link between microscopics and low-energy physics emerging at criticality. We first construct lattice incarnations of primary fields in the underlying Ising CFT including chiral fermions -- a nontrivial task given that the Rydberg chain Hamiltonian does not admit an exact fermionization. With this dictionary in hand, we compute correlations of microscopic Rydberg operators, paying special attention to finite, open chains of immediate experimental relevance. We further develop a method to quantify how second-neighbor Rydberg interactions tune the sign and strength of four-fermion couplings in the Ising CFT. Finally, we determine how the Ising fields evolve when four-fermion couplings drive an instability to Ising tricriticality. Our results pave the way to a thorough experimental characterization of Ising criticality in Rydberg arrays, and can inform the design of novel higher-dimensional phases based on coupled critical chains., 18+3 pages, 17+1 figures

Trimers are defined as two adjacent edges on a graph. We study the quantumstates obtained as equal-weight superpositions of all trimer coverings of alattice, with the constraint of having a trimer on each vertex: the so-calledtrimer resonating-valence-bond (tRVB) states. Exploiting their tensor networkrepresentation, we show that these states can host $\mathbb{Z}_3$ topologicalorder or can be gapless liquids with $\mathrm{U}(1) \times \mathrm{U}(1)$ localsymmetry. We prove that this continuous symmetry emerges whenever the latticecan be tripartite such that each trimer covers all the three sublattices. Inthe gapped case, we demonstrate the stability of topological order againstdilution of maximal trimer coverings, which is relevant for realistic modelswhere the density of trimers can fluctuate. Furthermore, we clarify theconnection between gapped tRVB states and $\mathbb{Z}_3$ lattice gauge theoriesby smoothly connecting the former to the $\mathbb{Z}_3$ toric code, and discussthe non-local excitations on top of tRVB states. Finally, we analyze via exactdiagonalization the zero-temperature phase diagram of a diluted trimer model onthe square lattice and demonstrate that the ground state exhibits topologicalproperties in a narrow region in parameter space. We show that a similar modelcan be implemented in Rydberg atom arrays exploiting the blockade effect. Weinvestigate dynamical preparation schemes in this setup and provide a viableroute for probing experimentally $\mathbb{Z}_3$ quantum spin liquids.

Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can in principle be solved on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hard-sphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states., Companion paper of arXiv:2005.14059

We examined the occurrence characteristics of the lightning observed at the Gaisberg Tower (GBT) in the years 2000 to 2018 and analyzed current waveforms (measured using a 0.25-mΩ shunt) of upward flashes initiated from the tower. During this period, 865 flashes were recorded at the GBT, of which 823 (95%) were upward and 4 (0.5%) were downward. For 18 flashes, the current waveforms were ambiguous and for 20 flashes they were unsuitable for analysis. Of the 823 upward flashes, 651 (79%) were negative, 35 (4.3%) were positive, and 137 (17%) were bipolar. The median initial stage (IS)-current durations in upward negative, positive, and bipolar flashes were 275, 96, and 282 ms, respectively. The median IS-current peaks in these flashes were 1.4, 3.2, and 1.8 kA, respectively. We expanded the traditional classification of bipolar flashes to include five categories. Of the 137 bipolar flashes, 45% were of Type 1S (single reversal of current polarity during IS), 47% of Type 1M (multiple reversals of current polarity during IS), 5.1% of Type 2 [different polarities of current during IS and return stroke (RS)], 1.5% of Type 3 (RSs of different polarities), and 0.73% (one flash) of Type 4 (different polarities of RS and the following continuing current).

Producing quantum states at random has become increasingly important in modern quantum science, with applications both theoretical and practical. In particular, ensembles of such randomly-distributed, but pure, quantum states underly our understanding of complexity in quantum circuits and black holes, and have been used for benchmarking quantum devices in tests of quantum advantage. However, creating random ensembles has necessitated a high degree of spatio-temporal control, placing such studies out of reach for a wide class of quantum systems. Here we solve this problem by predicting and experimentally observing the emergence of random state ensembles naturally under time-independent Hamiltonian dynamics, which we use to implement an efficient, widely applicable benchmarking protocol. The observed random ensembles emerge from projective measurements and are intimately linked to universal correlations built up between subsystems of a larger quantum system, offering new insights into quantum thermalization. Predicated on this discovery, we develop a fidelity estimation scheme, which we demonstrate for a Rydberg quantum simulator with up to 25 atoms using fewer than 10^4 experimental samples. This method has broad applicability, as we show for Hamiltonian parameter estimation, target-state generation benchmarking, and comparison of analog and digital quantum devices. Our work has implications for understanding randomness in quantum dynamics, and enables applications of this concept in a much wider context., JC and ALS contributed equally to this work

The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is central for building scalable quantum information systems. In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here, we demonstrate a quantum processor with dynamic, nonlocal connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, in between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation. We use this architecture to realize programmable generation of entangled graph states such as cluster states and a 7-qubit Steane code state. Furthermore, we shuttle entangled ancilla arrays to realize a surface code with 19 qubits and a toric code state on a torus with 24 qubits. Finally, we use this architecture to realize a hybrid analog-digital evolution and employ it for measuring entanglement entropy in quantum simulations, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars. Realizing a long-standing goal, these results pave the way toward scalable quantum processing and enable new applications ranging from simulation to metrology., Comment: 23 pages, 14 figures; movie attached as ancillary file

We develop an analytical approach for the description of quantum many-body scars in PXP models. We show that the scarred dynamics in the PXP model on a complete bipartite graph can be interpreted as a one-dimensional chiral scattering problem, and solve this problem analytically. The insights from this analysis allow us to predict that dynamical signatures of scars in PXP models can be enhanced by spin squeezing the initial states. We show numerically that this stabilization mechanism applies not only to the complete bipartite graph but also to one- and two-dimensional lattices, which are relevant for Rydberg atom array experiments. Moreover, our findings provide a physical motivation for Hamiltonian deformations reminiscent of those known to produce perfect scars., Comment: 12 pages, 6 figures

Motivated by far-reaching applications ranging from quantum simulations of complex processes in physics and chemistry to quantum information processing

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. Despite its promise for near-term quantum applications, not much is currently understood about QAOA's performance beyond its lowest-depth variant. An essential but missing ingredient for understanding and deploying QAOA is a constructive approach to carry out the outer-loop classical optimization. We provide an in-depth study of the performance of QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit non-adiabatic operations. Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasi-optimal $p$-level QAOA parameters in $O(\text{poly}(p))$ time, whereas the standard strategy of random initialization requires $2^{O(p)}$ optimization runs to achieve similar performance. We then benchmark QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps. The comparison reveals that QAOA can learn via optimization to utilize non-adiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps. Finally, we provide a realistic resource analysis on the experimental implementation of QAOA. When quantum fluctuations in measurements are accounted for, we illustrate that optimization will be important only for problem sizes beyond numerical simulations, but accessible on near-term devices. We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms., 13+10 pages, 15 figures

We study weak ergodicity breaking in a one-dimensional, nonintegrable spin-1 XY model. We construct for it an exact, highly excited eigenstate, which despite its large energy density, can be represented analytically by a finite bond-dimension matrix product state (MPS) with area-law entanglement. Upon a quench to a finite Zeeman field, the state undergoes periodic dynamics with perfect many-body revivals, in stark contrast to other generic initial states which instead rapidly thermalize. This dynamics can be completely understood in terms of the evolution of entangled virtual spin-1/2 degrees of freedom, which in turn underpin the presence of an extensive tower of strong-eigenstate thermalization hypothesis (ETH)-violating many-body eigenstates. The resulting quantum many-body scars are therefore of novel origin. Our results provide important analytical insights into the nature and entanglement structure of quantum many-body scars., Comment: 7 + 9 pages

Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on near-term quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.

Motivated by far-reaching applications ranging from quantum simulations of complex processes in physics and chemistry to quantum information processing, a broad effort is currently underway to build large-scale programmable quantum systems. Such systems provide unique insights into strongly correlated quantum matter, while at the same time enabling new methods for computation and metrology. Here, we demonstrate a programmable quantum simulator based on deterministically prepared two-dimensional arrays of neutral atoms, featuring strong interactions controlled via coherent atomic excitation into Rydberg states. Using this approach, we realize a quantum spin model with tunable interactions for system sizes ranging from 64 to 256 qubits. We benchmark the system by creating and characterizing high-fidelity antiferromagnetically ordered states, and demonstrate the universal properties of an Ising quantum phase transition in (2+1) dimensions. We then create and study several new quantum phases that arise from the interplay between interactions and coherent laser excitation, experimentally map the phase diagram, and investigate the role of quantum fluctuations. Offering a new lens into the study of complex quantum matter, these observations pave the way for investigations of exotic quantum phases, non-equilibrium entanglement dynamics, and hardware-efficient realization of quantum algorithms., Comment: 20 pages, 13 Figures

We analyze the zero-temperature phases of an array of neutral atoms on the kagome lattice, interacting via laser excitation to atomic Rydberg states. Density-matrix renormalization group calculations reveal the presence of a wide variety of complex solid phases with broken lattice symmetries. In addition, we identify a novel regime with dense Rydberg excitations that has a large entanglement entropy and no local order parameter associated with lattice symmetries. From a mapping to the triangular lattice quantum dimer model, and theories of quantum phase transitions out of the proximate solid phases, we argue that this regime could contain one or more phases with topological order. Our results provide the foundation for theoretical and experimental explorations of crystalline and liquid states using programmable quantum simulators based on Rydberg atom arrays., Comment: 10+11 pages, 7+8 figures; submitted to PNAS on July 28, 2020

We present simultaneous current and wideband electric field waveforms at 380 km associated with upward lightning flashes initiated from the Santis Tower, Switzerland. To the best of our knowledge, the dataset presented in this study includes the first simultaneous records of lightning currents and associated fields featuring ionospheric reflections for natural upward flashes, and the longest distance at which natural upward lightning fields have been measured simultaneously with their causative currents. The intervals between the groundwave and skywave arrival times are used to estimate ionospheric reflection heights during day and night times using the so-called zero-to-zero and peak-to-peak methods. During daytime, the mean ionospheric reflection heights, obtained using the two different delay estimation approaches, are about 78 and 76 km, corresponding to the D layer. The mean reflection height at nighttime is about 90 and 89 km, corresponding to the E layer. We present a full-wave, finite-difference time-domain (FDTD) analysis of the electric field propagation including the effect of the ionospheric reflections. The FDTD simulation results are compared with the measured fields associated with upward flashes initiated from the Santis Tower. It is found that the model reproduces reasonably well the measured waveforms and the times of arrival of the one-hop and two-hop skywaves relative to the groundwave.

Information about sport activity after short-stem total hip arthroplasty (THA) is scarce in the literature. We therefore aimed to evaluate the rate of return to sport after short-stem THA.We evaluated the sport pattern, rate of return to sport, activity level, extent of sport activity, and subjective rating and sense of well-being in 137 patients (137 hips) after short-stem THA. The minimum follow-up time was 18 months. All results were analyzed according to gender (male and female) and age (≤60,60-≤70, and70 years).Ninety-two percent of all patients practiced sport before surgery, and 91% of the patients returned to sport. Most patients returned to sport within the first 6 months after surgery. There was a decline in the number of sport disciplines from preoperatively to postoperatively, which was from 2.9 to 2.6 (P = 0.025). High-impact activities decreased postoperatively, but most low-impact activities did not change significantly. Eighty percent of all patients were involved in recreational sports.In this study, we observed an excellent rate of return to sport after short-stem THA. Most patients returned to the same level of sport activity that they had before the onset of restricting symptoms, with the majority of patients having a great sense of well-being during and after sports, and almost no pain in the affected hip.

We propose a quantum information based scheme to reduce the temperature of quantum many-body systems, and access regimes beyond the current capability of conventional cooling techniques. We show that collective measurements on multiple copies of a system at finite temperature can simulate measurements of the same system at a lower temperature. This idea is illustrated for the example of ultracold atoms in optical lattices, where controlled tunnel coupling and quantum gas microscopy can be naturally combined to realize the required collective measurements to access a lower, virtual temperature. Our protocol is experimentally implemented for a Bose-Hubbard model on up to 12 sites, and we successfully extract expectation values of observables at half the temperature of the physical system. Additionally, we present related techniques that enable the extraction of zero-temperature states directly., 8+4 pages, 4 figures; v2: New sections added, minor typos fixed

We examined current waveforms of upward flashes initiated from the Gaisberg Tower in 2000 to 2018. Current was measured at the tower-top using a $\pmb{0.25-\text{m}\Omega}$ shunt. During this period, 865 flashes were recorded at the Gaisberg Tower, of which 823 (95%) were upward and 4 (0.5%) were downward. For 18 flashes the current waveforms were ambiguous and for 20 flashes they were unsuitable for analysis. Of the 823 upward flashes, 651 (79%) were negative, 35 (4.3%) were positive, and 137 (17%) were bipolar. The median initial stage (IS)-current durations in upward negative, positive, and bipolar flashes were 275, 96, and 282 ms, respectively. The median IS-current peaks in these flashes were 1.4, 3.2, and 1.8 $\mathbf{kA}$ , respectively. We expanded the traditional classification of bipolar flashes to include five categories. Of the 137 bipolar flashes, 45% were of Type 1S (single current polarity reversal during IS), 47% of Type 1M (multiple reversals of current polarity during IS), 5.1% of Type 2 (different polarities of current during IS and return stroke), 1.5% of Type 3 (return strokes of different polarities), and 0.73% (one flash) of Type 4 (different polarities of return stroke and the following continuing current).

We analyze the coupling of atoms or atom-like emitters to nanophotonic waveguides in the presence of propagating acoustic waves. Specifically, we show that strong index modulations induced by such waves can drastically modify the effective photonic density of states and thereby influence the strength, the directionality, as well as the overall characteristics of photon emission and absorption processes. These effects enable a complete dynamical control of light-matter interactions in waveguide structures, which even in a two dimensional system can be used to efficiently exchange individual photons along selected directions and with a very high fidelity. Such a quantum acousto-optical control provides a versatile tool for various quantum networking applications ranging from the distribution of entanglement via directional emitter-emitter interactions to the routing of individual photonic quantum states via acoustic conveyor belts., 15 pages, 9 figures

We describe the zero-temperature phase diagram of a model of a two-dimensional square-lattice array of neutral atoms, excited into Rydberg states and interacting via strong van der Waals interactions. Using the density-matrix renormalization group algorithm, we map out the phase diagram and obtain a rich variety of phases featuring complex density wave orderings, upon varying lattice spacing and laser detuning. While some of these phases result from the classical optimization of the van der Waals energy, we also find intrinsically quantum-ordered phases stabilized by quantum fluctuations. These phases are surrounded by novel quantum phase transitions, which we analyze by finite-size scaling numerics and Landau theories. Our work highlights Rydberg quantum simulators in higher dimensions as promising platforms to realize exotic many-body phenomena., Comment: 7+4 pages, 4+1 figures

This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made., In this paper, a full-wave two-dimensional Finite-Difference-Time-Domain model is developed to evaluate the propagation effects of lightning electromagnetic fields over mountainous terrain in the Earth-ionosphere waveguide. In the model, we investigate the effect of the Earth-ionosphere waveguide structure and medium parameters, including the effect of the ionospheric cold plasma characteristics, the effect of the Earth curvature, and the propagation effects over mountainous terrain. For the first time, the obtained results are validated against simultaneous experimental data consisting of lightning currents measured at the Säntis Tower and electric fields measured in Neudorf, Austria, located at 380-km distance from the tower. It is shown that both the time delays and amplitudes of the lightning electromagnetic fields at 380-km distance can be strongly affected by the ionospheric electron density profile, the mountainous terrain, and the Earth curvature. After taking into account the effect of the irregular terrain between the Säntis Tower and the field measurement station, the vertical electric fields calculated by using our model are found to be in good agreement with the corresponding measured cases occurred in both daytime and nighttime. The ideal approximation used in either the classical solutions or the simplified models might lead to inaccuracies in the estimated reflection height. Furthermore, we discuss the sensitivity of our results by considering different return stroke models, as well as different typical values of the return stroke speed and of the ground conductivity. © 2019. The Authors., This work was supported by the European Research Council (ERC) under the European Union H2020 programme/ERC Grant Agreement 681257 and Swiss National Science Foundation under Project 200021_147058.

International audience; Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations 1 . These fluctuations play a dominant part in the quantum critical region surrounding the transition point, where the dynamics is governed by the universal properties associated with the QPT. Although time-dependent phenomena associated with classical, thermally driven phase transitions have been extensively studied in systems ranging from the early Universe to Bose–Einstein condensates 2$^{–}$5 , understanding critical real-time dynamics in isolated, non-equilibrium quantum systems remains a challenge 6 . Here we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations when crossing the QPT, we experimentally verify the quantum Kibble–Zurek mechanism (QKZM) 7$^{–}$9 for an Ising-type QPT, explore scaling universality and observe corrections beyond QKZM predictions. This approach is subsequently used to measure the critical exponents associated with chiral clock models 10$^{,}$11 , providing new insights into exotic systems that were not previously understood and opening the door to precision studies of critical phenomena, simulations of lattice gauge theories 12$^{,}$13 and applications to quantum optimization 14$^{,}$15 .