**Abstract:**

We explore a supervised machine learning approach to estimate the entanglement entropy of multi-qubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network’s estimate and benchmark against the best known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for non-unitary evolution by training our model on different noise strengths.

**M. Rieger, M. Reh, M. Gärttner, “Sample-efficient estimation of entanglement entropy through supervised learning”, Sept. 14, 2023, arXiv:2309.07556 (2023).
**

https://arxiv.org/abs/2309.07556

Related to Project A06

**Abstract:**

We investigate pattern formation in two-dimensional Bose-Einstein condensates (BECs) caused by temporal periodic modulation of the interatomic interaction. Temporal modulation of the interaction causes the so-called Faraday instability in the condensate, which we show generically leads to a stable square grid density pattern. We take the amplitudes in each of the two directions spanning the two-dimensional density pattern as order parameters in pattern formation and derive a set of simultaneous time evolution equations for those order parameters from the Gross–Pitaevskii (GP) equation with a time-periodic interaction. We identify the fixed points of the time evolution and show by stability analysis that the inhomogeneous density exhibits a square grid pattern as a stable fixed point.

**K. Fujii, S. L. Görlitz, N. Liebster, M. Sparn, E. Kath, H. Strobel, M. K. Oberthaler, T. Enss, “Square Pattern Formation as Stable Fixed Point in Driven Two-Dimensional Bose-Einstein Condensates”, Sept. 7, 2023, arXiv:2309.03829 (2023).
**

https://arxiv.org/abs/2309.03829

Related to Project A04, C02, C03

**Abstract:**

The emergence of patterns from simple physical laws belongs to the most striking topics in natural science. In particular, the spontaneous formation of structures from an initially homogeneous state can eventually lead to stable, non-homogeneous states of matter. Here we report on the spontaneous formation of square lattice patterns in a rotationally symmetric and driven Bose-Einstein condensate, confined in a two-dimensional box potential with absorptive boundaries. The drive is realized by globally modulating the two-particle interaction periodically in time. After a primary phase of randomly oriented stripes that emerge as a consequence of the Faraday instability, we observe the subsequent formation of persistent square lattice patterns in the highly occupied regime, where phonon-phonon interactions become relevant. We show theoretically that this state can be understood as an attractive fixed point of coupled nonlinear amplitude equations. Establishing the existence of this fixed point opens the perspective for engineering new, highly correlated states of matter in driven superfluids.

**N. Liebster, M. Sparn, E. Kath, K. Fujii, S. Görlitz, T. Enss, H. Strobel, M. K. Oberthaler, “Spontaneous formation of persistent square pattern in a driven superfluid”, Sept. 7, 2023, arXiv:2309.03792 (2023).
**

https://arxiv.org/abs/2309.03792

Related to Project A04, C02, C03

**Abstract:**

Hydrodynamics provides a successful framework to effectively describe the dynamics of complex many-body systems ranging from subnuclear to cosmological scales by introducing macroscopic quantities such as particle densities and fluid velocities. According to textbook knowledge, it requires coarse graining over microscopic constituents to define a macroscopic fluid cell, which is large compared to the interparticle spacing and the mean free path. In addition, the entire system must consist of many such fluid cells. The latter requirement on the system size has been challenged by experiments on high-energy heavy-ion collisions, where collective particle emission, typically associated with the formation of a hydrodynamic medium, has been observed with few tens of final-state particles. Here, we demonstrate emergence of hydrodynamics in a system with significantly less constituents. Our observation challenges the requirements for a hydrodynamic description, as in our system all relevant length scales, i.e. the system size, the inter-particle spacing, and the mean free path are comparable. The single particle resolution, deterministic control over particle number and interaction strength in our experiment allow us to explore the boundaries between a microscopic description and a hydrodynamic framework in unprecedented detail.

**S. Brandstetter, P. Lunt, C. Heintze, G. Giacalone, L. H. Heyen, M. Gaka, K. Subramanian, M. Holten, P. M. Preiss, S. Floerchinger, S. Jochim, “Emergent hydrodynamic behaviour of few strongly interacting fermions”, Aug. 18, 2023, arXiv:2308.09699 (2023).
**

https://arxiv.org/abs/2308.09699

Related to Project C01, C02, ABC

**Abstract:**

Vacuum birefringence produces a differential phase between orthogonally polarized components of a weak electromagnetic probe in the presence of a strong electromagnetic field. Despite representing a hallmark prediction of quantum electrodynamics, vacuum birefringence remains untested in pure light configurations due to the extremely large electromagnetic fields required for a detectable phase difference. Here, we exploit the programmable focal velocity and extended focal range of a flying focus laser pulse to substantially lower the laser power required for detection of vacuum birefringence. In the proposed scheme, a linearly polarized x-ray probe pulse counter-propagates with respect to a flying focus pulse, whose focus moves at the speed of light in the same direction as the x-ray probe. The peak intensity of the flying focus pulse overlaps the probe over millimeter-scale distances and induces a polarization ellipticity on the order of 10−10, which lies within the detection sensitivity of existing x-ray polarimeters.

**M. Formanek, J. P. Palastro, D. Ramsey, S. Weber, A. Di Piazza, “Signatures of vacuum birefringence in low-power flying focus pulses”, July 21, 2023, arXiv:2307.11734 (2023).
**

https://arxiv.org/abs/2307.11734

Related to Project B02

**Abstract:**

We study how isotropic and homogeneous far-from-equilibrium quantum systems relax to nonthermal attractors, which are of interest for cold atoms and nuclear collisions. We demonstrate that a first-order ordinary differential equation governs the self-similar approach to nonthermal attractors, i.e., the prescaling. We also show that certain natural scaling-breaking terms induce logarithmically slow corrections that prevent the scaling exponents from reaching the constant values during the system’s lifetime. We propose that, analogously to hydrodynamic attractors, the appropriate mathematical structure to describe such dynamics is the transseries. We verify our analytic predictions with state-of-the-art 2PI simulations of the large-N vector model and QCD kinetic theory.

**M. P. Heller, A. Mazeliauskas, T. Preis, “Prescaling relaxation to nonthermal attractors”, July 14, 2023, arXiv:2307.07545 (2023).
**

https://arxiv.org/abs/2307.07545

Related to Project A01

**Abstract:**

Inner-shell electrons naturally sense the electric field close to the nucleus, which can reach extreme values beyond 1015V/cm for the innermost electrons. Especially in few-electron highly charged ions, the interaction with the electromagnetic fields can be accurately calculated within quantum electrodynamics (QED), rendering these ions good candidates to test the validity of QED in strong fields. Consequently, their Lamb shifts were intensively studied in the last decades. Another approach is the measurement of g factors in highly charged ions. However, so far, either experimental accuracy or small field strength in low-Z ions limited the stringency of these QED tests. Here, we report on our high-precision, high-field test of QED in hydrogenlike 118Sn49+. The highly charged ions were produced with the Heidelberg-EBIT (electron beam ion trap) and injected into the ALPHATRAP Penning-trap setup, where the bound-electron g factor was measured with a precision of 0.5 parts-per-billion. For comparison, we present state-of-the-art theory calculations, which together test the underlying QED to about 0.012%, yielding a stringent test in the strong-field regime. With this measurement, we challenge the best tests via the Lamb shift and, with anticipated advances in the g-factor theory, surpass them by more than an order of magnitude.

**J. Morgner, B. Tu, C. M. König, T. Sailer, F. HeiSSe, H. Bekker, B. Sikora, C. Lyu, V. A. Yerokhin, Z. Harman, J. R. Crespo López-Urrutia, C. H. Keitel, S. Sturm, K. Blaum, “Stringent test of QED with hydrogenlike tin”, July 13, 2023, arXiv:2307.06613 (2023).
**

https://arxiv.org/abs/2307.06613

Related to Project B01

**Abstract:**

We investigate the excitation spectrum and compressibility of a dipolar Bose-Einstein condensate in an infinite tube potential in the parameter regime where the transition between superfluid and supersolid phases occurs. Our study focuses on the density range in which crystalline order develops continuously across the transition. Above the transition the superfluid shows a single gapless excitation band, phononic at small momenta and with a roton at a finite momentum. Below the transition, two gapless excitations branches (three at the transition point) emerge in the supersolid. We examine the two gapless excitation bands and their associated speeds of sound in the supersolid phase. Our results show that the speeds of sound and the compressibility are discontinuous at the transition, indicating a second-order phase transition. These results provide valuable insights into the identification of supersolid phenomena in dipolar quantum gases and the relationship to supersolidity in spin-orbit coupled gases.

**P. B. Blakie, L. Chomaz, D. Baillie, F. Ferlaino, “Compressibility and speeds of sound across the superfluid to supersolid phase transition of an elongated dipolar gas”, June 7, 2023, arXiv:2306.04794, (2023).
**

https://arxiv.org/abs/2306.04794

Related to Project A07

**Abstract:**

State-of-the-art hydrodynamic simulations of the quark-gluon plasma are unable to reproduce the elliptic flow of particles observed at the BNL Relativistic Heavy Ion Collider (RHIC) in relativistic 238U+238U collisions when they rely on information obtained from low-energy experiments for the implementation of deformation in the colliding 238U ions. We show that this is due to an inappropriate treatment of well-deformed nuclei in the modeling of the initial conditions of the quark-gluon plasma. Past studies have identified the deformation of the nuclear surface with that of the nuclear volume, though these are different concepts. In particular, a volume quadrupole moment can be generated by both a surface hexadecapole and a surface quadrupole moment. This feature was so far neglected in the modeling of heavy-ion collisions, and is particularly relevant for nuclei like 238U, which is both quadrupole deformed and hexadecapole deformed. With rigorous input from Skyrme density functional calculations, we show that correcting for such effects in the implementation of nuclear deformations in hydrodynamic simulations restores agreement with BNL RHIC data. This brings consistency to the results of nuclear experiments across energy scales, and demonstrates the impact of the hexadecapole deformation of 238U on high-energy collisions.

**W. Ryssens, G. Giacalone, B. Schenke, C. Shen, “Evidence of Hexadecapole Deformation in Uranium-238 at the Relativistic Heavy Ion Collider”, Feb. 27, 2023, Phys. Rev. Lett. 130, 212302**,** (2023).
**

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.212302

Related to Project C06

**Abstract:**

We investigate signal propagation in a quantum field simulator of the Klein–Gordon model realized by two strongly coupled parallel one-dimensional quasi-condensates. By measuring local phononic fields after a quench, we observe the propagation of correlations along sharp light-cone fronts. If the local atomic density is inhomogeneous, these propagation fronts are curved. For sharp edges, the propagation fronts are reflected at the system’s boundaries. By extracting the space-dependent variation of the front velocity from the data, we find agreement with theoretical predictions based on curved geodesics of an inhomogeneous metric. This work extends the range of quantum simulations of nonequilibrium field dynamics in general space–time metrics.

**M. Tajik, M. Gluza, N. Sebe, P. Schüttelkopf, F. Cataldini, J. Sabino, F. Møller, S.-C. Ji,
S. Erne, G. Guarnieri, S. Sotiriadis, J. Eisert, J. Schmiedmayer, “Experimental observation of curved light-cones in a quantum field simulator”, PNAS 120, (2023).**

https://www.pnas.org/doi/10.1073/pnas.2301287120

Related to Project A03

**Abstract:**

Quantum entanglement has been identified as a crucial concept underlying many intriguing phenomena in condensed matter systems such as topological phases or many-body localization. Recently, instead of considering mere quantifiers of entanglement like entanglement entropy, the study of entanglement structure in terms of the entanglement spectrum has shifted into the focus leading to new insights into fractional quantum Hall states and topological insulators, among others. What remains a challenge is the experimental detection of such fine-grained properties of quantum systems. The development of protocols for detecting features of the entanglement spectrum in cold atom systems, which are one of the leading platforms for quantum simulation, is thus highly desirable and will open up new avenues for experimentally exploring quantum many-body physics. Here we present a method to bound the width of the entanglement spectrum, or entanglement dimension, of cold atoms in lattice geometries, requiring only measurements in two experimentally accessible bases and utilizing ballistic time-of-flight (ToF) expansion. Building on previous proposals for entanglement certification for photon pairs, we first consider entanglement between two atoms of different atomic species and later generalize to higher numbers of atoms per species and multispecies configurations showing multipartite high-dimensional entanglement. Through numerical simulations we show that our method is robust against typical experimental noise effects and thus will enable high-dimensional entanglement certification in systems of up to 8 atoms using currently available experimental techniques.

**N. Euler, M. Gärttner, “Detecting high-dimensional entanglement in cold-atom quantum simulators”, arXiv:2305.07413 (2023).**

https://arxiv.org/abs/2305.07413

Related to Project A06

**Abstract:**

Physical systems can be used as an information processing substrate and with that extend traditional computing architectures. For such an application the experimental platform must guarantee pristine control of the initial state, the temporal evolution and readout. All these ingredients are provided by modern experimental realizations of atomic Bose Einstein condensates. By embedding the nonlinear evolution of a quantum gas in a Machine Learning pipeline, one can represent nonlinear functions while only linear operations on classical computing of the pipeline are necessary. We demonstrate successful regression and interpolation of a nonlinear function using a quasi one-dimensional cloud of potassium atoms and characterize the performance of our system.

**M. Hans, E. Kath, M. Sparn, N. Liebster, F. Draxler, C. Schnörr, H. Strobel, M. K. Oberthaler, “Bose Einstein condensate as nonlinear block of a Machine Learning pipeline”, Apr. 28, arXiv:2304.14905, (2023).
**

https://arxiv.org/abs/2304.14905

Related to Project A04

**Abstract:**

We present our experimental and theoretical framework, which combines a broadband superluminescent diode with fast learning algorithms to provide speed and accuracy improvements for the optimization of on-dimensional optical dipole potentials, here generated with a digital micromirror device. To characterize the setup and potential speckle patterns arising from coherence, we compare the superluminescent diode to a single-mode laser by investigating interference properties. We employ machine-learning tools to train a physics-inspired model acting as a digital twin of the optical system predicting the behavior of the optical apparatus including all its imperfections. Implementing an iterative algorithm based on iterative learning control we optimize optical potentials an order of magnitude faster than heuristic optimization methods. We compare iterative model-based “offline” optimization and experimental feedback-based “online” optimization. Our methods provide a route to fast optimization of optical potentials, which is relevant for the dynamical manipulation of ultracold gases.

**M. Calzavara, Y. Kuriatnikov, A. Deutschmann-Olek, F. Motzoi, S. Erne, A. Kugi, T. Calarco, J. Schmiedmayer, M. Prüfer, “Optimizing Optical Potentials With Physics-Inspired Learning Algorithms”, Phys. Rev. Appl. 19 (2023).**

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.044090

Related to Project A03

**Abstract:**

The theoretical understanding of scaling laws of entropies and mutual information has led to substantial advances in the study of correlated states of matter, quantum field theory and gravity. Experimentally measuring von Neumann entropy in quantum many-body systems is challenging, as it requires complete knowledge of the density matrix, which normally requires the implementation of full state reconstruction techniques. Here we measure the von Neumann entropy of spatially extended subsystems in an ultracold atom simulator of one-dimensional quantum field theories. We experimentally verify one of the fundamental properties of equilibrium states of gapped quantum many-body systems—the area law of quantum mutual information. We also study the dependence of mutual information on temperature and on the separation between the subsystems. Our work represents a step towards employing ultracold atom simulators to probe entanglement in quantum field theories.

**M. Tajik, I. Kukuljan, S. Sotiriadis, B. Rauer, T. Schweigler, F. Cataldini, J. Sabino, F. Møller, P. Schüttelkopf, Si-Cong Ji, D. Sels, E. Demler, J. Schmiedmayer, “Experimental verification of the area law of mutual information in a quantum field simulator”, Nature Phys., (2023).
**

https://www.nature.com/articles/s41567-023-02027-1

Related to Project A03

**Abstract:**

Closed quantum systems far from thermal equilibrium can show universal dynamics near attractor solutions, known as non-thermal fixed points, generically in the form of scaling behavior in space and time. A systematic classification and comprehensive understanding of such scaling solutions are tasks of future developments in non-equilibrium quantum many-body theory. In this tutorial review, we outline several analytical approaches to non-thermal fixed points and summarize corresponding numerical and experimental results. The analytic methods include a non-perturbative kinetic theory derived within the two-particle irreducible effective-action formalism, as well as a low-energy effective field theory framework. As one of the driving forces of this research field are numerical simulations, we summarize the main results of exemplary cases of universal dynamics in ultracold Bose gases. This encompasses quantum vortex ensembles in turbulent superfluids as well as recently observed real-time instanton solutions in one-dimensional spinor condensates.

**A. N. Mikheev, I. Siovitz, T. Gasenzer, “Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium”, arXiv:2304.12464 (2023).**

https://arxiv.org/abs/2304.12464

Related to Project A04

**Abstract:**

Universal scaling dynamics of a many-body system far from equilibrium signals the proximity of the time-evolution to a non-thermal fixed point. We find universal dynamics connected with rogue-wave like events in the mutually coupled magnetic components of a spinor gas which propagate in an effectively random potential. The frequency of these caustics is affected by the time varying spatial correlation length of the potential, giving rise to an additional exponent $δ_{c}≃1/3$ for temporal scaling, which is different by a factor $∼4/3$ from the exponent $β_{V}≃1/4$ characterizing the scaling of the correlation length $_{V}∼t_{βV}$ with time. As a result of the caustics, real-time instanton defects appear in the Larmor phase of the spin-1 system as vortices in space and time. The temporal correlations determining the frequency of instanton events to occur scale in time as $t_{δI}$. This suggests that the universality class of a non-thermal fixed point could be characterized by different, mutually related exponents defining the coarsening evolution in time and space, respectively. Our results have a strong relevance for understanding pattern coarsening from first principles and potential implications for dynamics ranging from the early universe to geophysical dynamics and micro physics.

**I. Siovitz, S. Lannig, Y. Deller, H. Strobel, M.K. Oberthaler, T. Gasenzer, “Universal dynamics of rogue waves in a quenched spinor Bose condensate”, arXiv:2304.09293 (2023).**

https://arxiv.org/abs/2304.09293

Related to Project A04

**Abstract:**

Numerical simulations of the full quantum properties of interacting many-body systems by means of field-theoretic Monte-Carlo techniques are often limited due to a sign problem. Here we simulate properties of a dilute two-dimensional Bose gas in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) transition by means of the Complex Langevin (CL) algorithm, thereby extending our previous CL study of the three-dimensional Bose gas to the lower-dimensional case. The purpose of the paper is twofold. On the one hand, it adds to benchmarking of the CL method and thus contributes to further exploring the range of applicability of the method. With the respective results, the universality of the equation of state is recovered, as well as the long-wave-length power-law dependence of the single-particle momentum spectrum below the BKT transition. Analysis of the rotational part of the current density corroborates vortex unbinding in crossing the transition. Beyond these measures of consistency we compute quantum corrections to the critical density and chemical potential in the weakly coupled regime. Our results show a shift of these quantities to lower values as compared to those obtained from classical field theory. It points in the opposite direction as compared to the shift of the critical density found by means of the path-integral Monte-Carlo method at larger values of the coupling. Our simulations widen the perspective for precision comparisons with experiment.

**P. Heinen, T. Gasenzer, “Simulating the Berezinskii-Kosterlitz-Thouless Transition with Complex Langevin”, arXiv:2304.05699 (2023).**

https://arxiv.org/abs/2304.05699

Related to Project A04

**Abstract:**

Having a detailed theoretical knowledge of the low-energy structure of the heavy odd-mass nucleus 197Au is of prime interest as the structure of this isotope represents an important input to theoretical simulations of collider experiments involving gold ions performed at relativistic energies. In the present article, therefore, we report on new results on the structure of 197Au obtained from state-of-the-art multi-reference energy density functional (MR-EDF) calculations. Our MR-EDF calculations were realized using the Skyrme-type pseudo-potential SLyMR1, and include beyond mean-field correlations through the mixing, in the spirit of the Generator Coordinate Method (GCM), of particle-number and angular-momentum projected triaxially deformed Bogoliubov quasi-particle states. Comparison with experimental data shows that the model gives a reasonable description of 197Au with in particular a good agreement for most of the spectroscopic properties of the 3/2+1 ground state. From the collective wave function of the correlated state, we compute an average deformation β¯(3/2+1)=0.13 and γ¯(3/2+1)=40∘ for the ground state. We use this result to construct an intrinsic shape of 197Au representing a microscopically-motivated input for precision simulations of the associated collider processes. We discuss, in particular, how the triaxiality of this nucleus is expected to impact 197Au+197Au collision experiments at ultrarelativistic energy.

**B. Bally, G. Giacalone, M. Bender, “The shape of gold”, Eur. Phys. J. A 59, 58 (2023).
**

https://link.springer.com/article/10.1140/epja/s10050-023-00955-3

Related to Project C06

**Abstract:**

We present the first direct and nonperturbative computation of the graviton spectral function in quantum gravity. This is achieved with the help of a novel Lorentzian renormalization group approach, combined with a spectral representation of correlation functions. We find a positive graviton spectral function, showing a massless one-graviton peak and a multigraviton continuum with an asymptotically safe scaling for large spectral values. We also study the impact of a cosmological constant. Further steps to investigate scattering processes and unitarity in asymptotically safe quantum gravity are indicated.

**J. Fehre, D. F. Litim, J. M. Pawlowski, M. Reichert, “Lorentzian Quantum Gravity and the Graviton Spectral Function”, Phys.Rev.Lett. 130 (2023).**

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.081501

Related to Project A02

**Abstract:**

The relationship between many-body interactions and dimensionality is integral to numerous emergent quantum phenomena. A striking example is the Bose gas, which upon confinement to one dimension (1D) obeys an infinite set of conservation laws, prohibiting thermalization and constraining dynamics. In our experiment, we demonstrate that such a 1D behavior can extend much further into the dimensional crossover toward 3D than expected. Starting from a weakly interacting Bose gas trapped in a highly elongated potential, we perform a quench to instigate the dynamics of a single density mode. Employing the theory of generalized hydrodynamics, we identify the dominant relaxation mechanism as the 1D dephasing of the relevant collective excitations of the system, the rapidities. Surprisingly, the dephasing remains dominant even for temperatures far exceeding conventional limits of one dimensionality where thermalization should occur. We attribute our observations to an emergent Pauli blocking of transverse excitations caused by the rapidities assuming fermionic statistics, despite the gas being purely bosonic. Thus, our study suggests that 1D physics is less fragile than previously thought, as it can persist even in the presence of significant perturbations. More broadly, by employing the exact Bethe ansatz solutions of the many-body system, we facilitate an interpretation of how the emergent macroscopic behavior arises from the microscopic interactions.

**F. Cataldini, F. Møller, M. Tajik, J. Sabino, S.-C. Ji, I. Mazets, T. Schweigler, B. Rauer, J.
Schmiedmayer, “Emergent Pauli Blocking in a Weakly Interacting Bose Gas”, Phys. Rev. X 12 (2022).**

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.041032

Related to Project A03

**Abstract:**

We discuss the emergence of a low-energy effective theory with quarks, mesons, diquarks and baryons at vanishing and finite baryon density from first principle QCD. The present work also includes an overview on diquarks at vanishing and finite density, and elucidates the physics of transitional changes from baryonic matter to quark matter including diquarks. This set-up is discussed within the functional renormalisation group approach with dynamical hadronisation. In this framework it is detailed how mesons, diquarks, and baryons emerge dynamically from the renormalisation flow of the QCD effective action. Moreover, the fundamental degrees of freedom of QCD, quarks and gluons, decouple from the dynamics of QCD below the respective mass gaps. The resulting global picture unifies the different low energy effective theories used for low and high densities within QCD, and allows for a determination of the respective low energy constants directly from QCD.

**K. Fukushima, J. M. Pawlowski, N. Strodthoff, “Emergent Hadrons and Diquarks”, Annals Phys. 446, 169106 (2022).**

https://www.sciencedirect.com/science/article/pii/S0003491622002093?via%3Dihub

Related to Project A02

**Abstract:**

We compute high-order baryon number fluctuations at finite temperature and density within a QCD-assisted low energy effective field theory. Quantum, thermal and density fluctuations are incorporated with the functional renormalization group approach. Quantum and in-medium fluctuations are encoded via the evolution of renormalization group flow equations. The resulting fourth- and sixth-order baryon number fluctuations meet the lattice benchmark results at vanishing density. They are consistent with experimental measurements, and in particular, the non-monotonic dependence of the kurtosis of net-baryon number distributions on the collision energy is observed in our calculations. This non-monotonicity arises from the increasingly sharpened chiral crossover with the decrease of collision energy.

**W. Fu, X. Luo, J. M. Pawlowski, F. Rennecke, R. Wen, S. Yin, “High-order baryon number
fluctuations within the fRG approach”, PoS CPOD2021, 009 (2022).**

Related to Project A02

**Abstract:**

In this work, I calculate the resolved spectra for the three stages of the *bottom-up* scenario, which are comparable to the thermal contribution, particularly at higher values of the saturation scale . Analytical solutions are obtained by including a parametrization of scaling solutions from far-from-equilibrium classical statistical lattice simulations into a small angle kinetic rate. Furthermore, a theoretically motivated ansatz is used to account for near-collinear enhancement of the low- radiation. The system is phenomenologically constrained using the charged hadron multiplicities from LHC and RHIC as in previous parametric estimates and fair agreement with the data available for photons was found. I find that for this realistic set of parameters, the contribution from the pre-equilibrium dominates the excess photons.

**O. Garcia-Montero, “Non-equilibrium photons from the bottom-up thermalization scenario”, Annals Phys. 443, 168984 (2022).**

https://www.sciencedirect.com/science/article/abs/pii/S0003491622001403?via%3Dihub

Related to Project

**Abstract:**

A central question in high-energy nuclear phenomenology is how the geometry of the quark-gluon plasma (QGP) formed in relativistic nuclear collisions is precisely shaped. In our understanding of such processes, two features are especially crucial for the determination of the QGP geometry, respectively, the nucleon size and the energy deposition scheme. This contribution reports on the (circular) evolution of such features in state-of-the-art model incarnations of heavy-ion collisions over the past seven years. Ideas for future directions of investigation are pointed out.

**G. Giacalone, “There and Sharp Again: The Circle Journey of Nucleons and Energy Deposition”, Aug. 14, arXiv:2208.06839, (2023).
**

https://arxiv.org/abs/2208.06839

Related to Project C06

**Abstract:**

Vector boson production and neutrino deep-inelastic scattering (DIS) data are crucial for constraining the strange quark parton distribution function (PDF) and more generally for flavor decomposition in PDF extractions. We extend the nCTEQ15 nuclear PDFs (nPDFs) by adding the recent and production data from the LHC in a global nPDF fit. The new nPDF set, referred to as nCTEQ15WZ, is used as a starting point for a follow-up study in which we assess the compatibility of neutrino DIS data with charged lepton DIS data. Specifically, we re-analyze neutrino DIS data from NuTeV, Chorus, and CDHSW, as well as dimuon data from CCFR and NuTeV. To scrutinize the level of compatibility, different kinematic regions of the neutrino data are investigated. Fits to the neutrino data alone and a preliminary global fit are performed and compared to nCTEQ15WZ.

**K. F. Muzakka, P. Duwentäster, T. J. Hobbs, T. Jeo, M. Klasen, K. Kovaík, A. Kusina, J. G.
Morfín, F. I. Olness, R. Ruiz, I. Schienbein, J. Y. Yu, “Impact of W and Z Production Data
and Compatibility of Neutrino DIS Data in Nuclear Parton Distribution Functions”, SciPost Phys Proc. 8, 041, (2022).
**

https://scipost.org/10.21468/SciPostPhysProc.8.041

Related to Project C05

**Abstract:**

Near the second order phase transition point, QCD with two flavours of massless quarks can be approximated by an O(4) model, where a symmetry breaking external field H can be added to play the role of quark mass.

The Lee-Yang theorem states that the equation of state in this model has a branch cut along the imaginary H axis for |Im[H]|>Hc, where Hc indicates a second order critical point.

This point, known as Lee-Yang edge singularity, is of importance to the thermodynamics of the system.

We report here on ongoing work to determine the location of Hc via complex Langevin simulations.

**F. Attanasio, M. Bauer, L. Kades, J. M. Pawlowski, “Searching for Yang-Lee zeros in O(N)
models”, PoS LATTICE2021, 223 (2022).**

Related to Project A02

**Abstract:**

We determine the chiral phase structure of (2+1)-flavor QCD in dependence of temperature and the light flavor quark mass with Dyson-Schwinger equations. Specifically, we compute the renormalized chiral condensate and its susceptibility. The latter is used to determine the (pseudo)critical temperature for general light current quark masses. In the chiral limit we obtain a critical temperature of about 141 MeV. This result is in quantitative agreement with recent functional renormalization group results in QCD and is compatible with the respective lattice results. We also compute the order parameter potential of the light chiral condensate, map out the regime in the phase diagram which exhibits quasi-massless modes, and discuss the respective chiral dynamics.

**F. Gao, J. M. Pawlowski, “Phase structure of (2+1)-flavor QCD and the magnetic equation of state”, ****Physical Review D** **105****, (2022).**

https://onlinelibrary.wiley.com/doi/full/10.1002/piuz.202370204

Related to Project A02

**Abstract:**

Path integrals with complex actions are encountered for many physical systems ranging from spin- or mass-imbalanced atomic gases and graphene to quantum chromodynamics at finite density to the nonequilibrium evolution of quantum systems. Many computational approaches have been developed for tackling the sign problem emerging for complex actions. Among these, complex Langevin dynamics has the appeal of general applicability. One of its key challenges is the potential convergence of the dynamics to unphysical fixed points. The statistical sampling process at such a fixed point is not based on the physical action and hence leads to wrong predictions. Moreover, its unphysical nature is hard to detect due to the implicit nature of the process. In the present work we set up a general approach based on a Markov chain Monte Carlo scheme in an extended state space. In this approach we derive an explicit real sampling process for generalized complex Langevin dynamics. Subject to a set of constraints, this sampling process is the physical one. These constraints originate from the detailed-balance equations satisfied by the Monte Carlo scheme. This allows us to rederive complex Langevin dynamics from a new perspective and establishes a framework for the explicit construction of new sampling schemes for complex actions.

**L. Kades, M. Gärttner, T. Gasenzer, J. M. Pawlowski, “Monte Carlo sampling of complex
actions in extended state spaces”, Phys. Rev. E 105, 045315 (2022).**

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.105.045315

Related to Project A02

**Abstract:**

We reconstruct ghost and gluon spectral functions in 2+1 flavor QCD with Gaussian process regression. This framework allows us to largely suppress spurious oscillations and other common reconstruction artifacts by specifying generic magnitude and length scale parameters in the kernel function. The Euclidean propagator data are taken from lattice simulations with domain wall fermions at the physical point. For the infrared and ultraviolet extensions of the lattice propagators as well as the low-frequency asymptotics of the ghost spectral function, we utilize results from functional computations in Yang-Mills theory and QCD. This further reduces the systematic error significantly. Our numerical results are compared against a direct real-time functional computation of the ghost and an earlier reconstruction of the gluon in Yang-Mills theory. The systematic approach presented in this work offers a promising route toward unveiling real-time properties of QCD.

**J. Horak, J. M. Pawlowski, J. Rodríguez-Quintero, J. Turnwald, J. M. Urban, N. Wink, S.
Zafeiropoulos, “Reconstructing QCD spectral functions with Gaussian processes”, Phys. Rev. D 105, (2022).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.105.036014

Related to Project A02

**Abstract:**

We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity from Euclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and the background graviton propagator. We prove that the spectral function of the latter necessarily has negative parts similar to, and for the same reasons, as the gluon spectral function. In turn, the spectral function of the dynamical graviton is positive. We argue that the latter enters cross sections and other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at the unitarity of asymptotically safe quantum gravity.

**A. Bonanno, T. Denz, J. M. Pawlowski, M. Reichert, “Reconstructing the graviton”, SciPost Phys. 12, 1 (2022).**

https://scipost.org/10.21468/SciPostPhys.12.1.001

Related to Project A02, B03, C01

**Abstract:**

A longstanding question in QCD is the origin of the mass gap in the Yang-Mills sector of QCD, i.e., QCD without quarks. In Landau gauge QCD this mass gap, and hence confinement, is encoded in a mass gap of the gluon propagator, which is found both in lattice simulations and with functional approaches. While functional methods are well suited to unravel the mechanism behind the generation of the mass gap, a fully satisfactory answer has not yet been found. In this work we solve the coupled Dyson-Schwinger equations for the ghost propagator, gluon propagator and three-gluon vertex. We corroborate the findings of earlier works, namely that the mass gap generation is tied to the longitudinal projection of the gluon self-energy, which acts as an effective mass term in the equations. Because an explicit mass term is in conflict with gauge invariance, this leaves two possible scenarios: If it is viewed as an artifact, only the scaling solution survives; if it is dynamical, gauge invariance can only be preserved if there are longitudinal massless poles in either of the vertices. We find that there is indeed a massless pole in the ghost-gluon vertex, however in our approximation with the assumption of complete infrared dominance of the ghost this pole is only present for the scaling solution. We also put forward a possible mechanism that may reconcile the scaling solution, with an infrared dominance of the ghost, with the decoupling solutions based on longitudinal poles in the three-gluon vertex as seen in the PT-BFM scheme.

**G. Eichmann, J. M. Pawlowski, J. M. Silva, “Mass generation in Landau-gauge Yang-Mills
theory”, Phys. Rev. D 104, 114016 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.114016

Related to Project A02

**Abstract:**

In local scalar quantum field theories at finite temperature correlation functions are known to satisfy certain nonperturbative constraints, which for two-point functions in particular implies the existence of a generalization of the standard Källén-Lehmann representation. In this work, we use these constraints in order to derive a spectral representation for the shear viscosity arising from the thermal asymptotic states, η0. As an example, we calculate η0 in ϕ4 theory, establishing its leading behavior in the small and large coupling regimes.

**P. Lowdon, R.-A. Tripolt, J. M. Pawlowski, D. H. Rischke, “Spectral representation of the shear viscosity for local scalar QFTs at finite temperature”, Phys. Rev. D 104, 065010 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.065010

Related to Project A02

**Abstract:**

We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter 0<ξ≤5. Extrapolating the propagators to ξ→∞, we find the expected qualitative behavior. We provide arguments that our results are quantitatively reliable at least for values ξ≲1/2 of the gauge-fixing parameter. It is shown that the correlation functions, and, in particular, the ghost propagator, change significantly with increasing gauge parameter. In turn, the ghost-gluon running coupling as well as the position of the zero crossing of the Schwinger function of the gluon propagator remain within the uncertainties of our calculation unchanged.

**M. Napetschnig, R. Alkofer, M. Q. Huber, J. M. Pawlowski, “Yang-mills propagators in linear covariant gauges from nielsen identities”, Phys. Rev. D 104, 054003 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.054003

Related to Project A02

**Abstract:**

We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is ”exact” in two ways. First, the solution of the resulting flow equation is independent of the choice of the regulator. Second, this solution coincides with that of the 2PI equations for the two-point and the four-point functions, for any selection of two-skeleton diagrams characterizing a so-called Ф-derivable approximation. The transformation of the equations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, and provides new insight on certain aspects of their renormalization. It also opens the possibility to develop approximation schemes going beyond the strict Ф-derivable ones, as well as new truncation schemes for the fRG hierarchy.

**U. Reinosa, J.-P. Blaizot, J. M. Pawlowski, “Functional renormalization group and 2PI effective action formalism”, Annals Phys. 431, 168549 (2021).**

https://www.sciencedirect.com/science/article/pii/S000349162100155X?via%3Dihub

Related to Project A02

**Abstract:**

We discuss the far-from-equilibrium evolution of ϕ3 theory in 1+1 dimensions with the temporal functional renormalization group. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behavior suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conservation. We also derive a simple exact representation of the expectation value of the energy-momentum tensor solely in terms of the propagator.

**L. Corell, A. K. Cyrol, M. Heller, J. M. Pawlowski, “Flowing with the temporal renormalization group”, Phys. Rev. D 104, 025005 (2021).
**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.025005

Related to Project B03

**Abstract:**

Hadron production in relativistic nuclear collisions is well described in the framework of the Statistical Hadronization Model (SHM). We investigate the influence on SHM predictions of hadron mass spectra for light-flavor baryons and mesons modified by the addition of about 500 new states as predicted by lattice QCD and a relativistic quark model. The deterioration of the resulting thermodynamic fit quality obtained for Pb–Pb collision data at 2.76 TeV suggests that the additional states are not suited to be naively used since also interactions among the states as well as non-resonant contributions need to be considered in the SHM approach. Incorporating these effects via the pion nucleon interaction determined from measured phase shifts leads again to excellent reproduction of the experimental data. This is a strong indication that at least the additional nucleon excited states cannot be understood and used as independent resonances.

**A. Andronic, P. Braun-Munzinger, D. Gündüz, Y. Kirchhoff, M. K. Köhler, J. Stachel, M. Winn, “Influence of modified light-flavor hadron spectra on particle yields in the statistical hadronization model”, Nuclear Physics A 1010, (2023).
**

https://www.sciencedirect.com/science/article/pii/S0375947421000415

Related to Project C05, A01, A02, C06

**Abstract:**

Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relationship between mathematical structures and applications to describe relativistic fermions is emphasized throughout.

**S. Floerchinger, “Real Clifford Algebras and Their Spinors for Relativistic Fermions ”, Universe 7, 168 (2021).
**

https://www.mdpi.com/2218-1997/7/6/168

Related to Project B03, C06

**Abstract:**

We present a comprehensive study of the quark sector of 2+1 flavor QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex in the one-loop dressed approximation. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with 2+1 dynamical quark flavors, obtained from studies with Schwinger-Dyson equations, the functional renormalization group approach, and large volume lattice simulations. The appropriate renormalization procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalization group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalization group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as (245 MeV)3. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed.

**F. Gao, J. Papavassiliou, J. M. Pawlowski, “Fully coupled functional equations for the quark sector of QCD”, Phys. Rev. D 103, 094013 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.094013

Related to Project A02

**Abstract:**

We propose a novel simulation strategy for Yang-Mills theories with a complex coupling, based on the Lefschetz thimble decomposition. We envisage that the approach developed in the present work can also be adapted to QCD at finite density and real-time simulations. Simulations with Lefschetz thimbles offer a potential solution to sign problems in Monte Carlo calculations within many different models with complex actions. We discuss the structure of generalized Lefschetz thimbles for pure Yang-Mills theories with a complex gauge coupling β and show how to incorporate the gauge orbits. We propose to simulate such theories on the union of the tangential manifolds to the relevant Lefschetz thimbles attached to the critical manifolds of the Yang-Mills action. We demonstrate our algorithm on a (1+1)-dimensional U(1) model and discuss how, starting from the main thimble result, successive subleading thimbles can be taken into account via a reweighting approach. While we face a residual sign problem, our novel approach performs exponentially better than the standard reweighting approach.

**J. M. Pawlowski, M. Scherzer, C. Schmidt, F. P. G. Ziegler, F. Ziesché, “Simulating Yang-Mills theories with a complex coupling”, Phys. Rev. D 103, 094505 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.094505

Related to Project A02

**Abstract:**

We investigate the dimensional crossover from three to two dimensions in an ultracold Fermi gas across the whole BCS-BEC crossover. Of particular interest is the strongly interacting regime as strong correlations and pair fluctuations are more pronounced in reduced dimensions. Our results are obtained from first principles within the framework of the functional renormalization group (FRG). Here, the confinement of the transverse direction is imposed by means of periodic boundary conditions. We calculate the equation of state, the gap parameter at zero temperature, and the superfluid transition temperature across a wide range of transversal confinement length scales. Particular emphasis is put on the determination of the finite-temperature phase diagram for different confinement length scales. In the end, our results are compared with recent experimental observations and we discuss them in the context of other theoretical works.

**B. M. Faigle-Cedzich, J. M. Pawlowski, C. Wetterich, “Dimensional crossover in ultracold Fermi gases from functional renormalization”, Phys. Rev. A 103, 033320 (2021).**

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.103.033320

Related to Project C01

**Abstract:**

Reliably computing the free energy in a gauge theory like QCD is a challenging and resource-demanding endeavor. We explore the possibility to obtain the associated thermodynamic anomaly from two-point functions based on a conjectured relation. This conjecture is triggered by the relation to the Tan contact in condensed matter systems. For this investigation we use state-of-the-art results for the Yang-Mills gluon two-point function from the lattice and the functional renormalization group, as well as novel Dyson-Schwinger results at finite temperature computed in the present work. This allows for a first, qualitative, test of this conjecture. The results from all methods reveal the same nontrivial temperature behavior of the subleading large momentum dependence of the gluon propagator relevant for the conjectured relation. The comparison with the expected behavior for SU(2) Yang-Mills theory is encouraging to further pursue this approach.

**O. Hajizadeh, M. Q. Huber, A. Maas, J. M. Pawlowski, “Exploring the Tan contact term in Yang-Mills theory”, Phys. Rev. D 103, 034023 (2021).**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.034023

Related to Project A02, B03, C01, C05, C06

**Abstract:**

In this contribution, we discuss the asymptotic safety scenario for quantum gravity with a functional renormalization group approach that disentangles dynamical metric fluctuations from the background metric. We review the state of the art in pure gravity and general gravity–matter systems. This includes the discussion of results on the existence and properties of the asymptotically safe ultraviolet fixed point, full ultraviolet-infrared trajectories with classical gravity in the infrared, and the curvature dependence of couplings also in gravity–matter systems. The results in gravity–matter systems concern the ultraviolet stability of the fixed point and the dominance of gravity fluctuations in minimally coupled gravity–matter systems. Furthermore, we discuss important physics properties such as locality of the theory, diffeomorphism invariance, background independence, unitarity, and access to observables, as well as open challenges.

**J. M. Pawlowski, M. Reichert, “Quantum Gravity: A Fluctuating Point of View”, Front. in
Phys. 8, 551848 (2021).**

https://www.frontiersin.org/articles/10.3389/fphy.2020.551848/full

Related to Project A02

**Abstract:**

As a result of a nontrivial mixing matrix, neutrinos cannot be simultaneously in a flavor and mass eigenstate. We formulate and discuss information entropic relations that quantify the associated quantum uncertainty. We also formulate a protocol to determine the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix from quantum manipulations and measurements on an entangled lepton-neutrino pair. The entangled state features neutrino oscillations in a conditional probability involving measurements on the lepton and the neutrino. They can be switched off by choosing a specific observable on the lepton side which is determined by the PMNS matrix. The parameters of the latter, including the CP-violating phase δ, can be obtained by guessing them and improving the guess by minimizing the remaining oscillations.

**S. Floerchinger, J.-M. Schwind, “Neutrino flavor-mass uncertainty relations and an entanglementassisted determination of the PMNS matrix”, Phys. Rev. D 102, 093001 (2020).
**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.102.093001

Related to Project C06, A06*

**Abstract:**

We present programmable two-dimensional arrays of microscopic atomic ensembles consisting of more than 400 sites with nearly uniform filling and small atom number fluctuations. Our approach involves direct projection of light patterns from a digital micromirror device with high spatial resolution onto an optical pancake trap acting as a reservoir. This makes it possible to load large arrays of tweezers in a single step with high occupation numbers and low power requirements per tweezer. Each atomic ensemble is confined to ~1 μm^{3} with a controllable occupation from 20 to 200 atoms and with (sub)-Poissonian atom number fluctuations. Thus, they are ideally suited for quantum simulation and for realizing large arrays of collectively encoded Rydberg-atom qubits for quantum information processing.

**Y. Wang, S. Shevate, T. M. Wintermantel, M. Morgado, G. Lochead, S. Whitlock, “Preparation of hundreds of microscopic atomic ensembles in optical tweezer arrays”, npj Quantum Information 6 (2020).
**

https://www.nature.com/articles/s41534-020-0285-1

Related to Project A05

**Abstract:**

We derive a simple relation between strangeness neutrality and baryon-strangeness correlations. In heavy-ion collisions, the former is a consequence of quark number conservation of the strong interactions while the latter are sensitive probes of the character of QCD matter. This relation allows us to directly extract baryon-strangeness correlations from the strangeness chemical potential at strangeness neutrality. The explicit calculations are performed within a low-energy theory of QCD with 2+1 dynamical quark flavors at finite temperature and density. Nonperturbative quark and hadron fluctuations are taken into account within the functional renormalization group. The results show the pronounced sensitivity of baryon-strangeness correlations on the QCD phase transition and the crucial role that strangeness neutrality plays for this observable.

**W. Fu, J. M. Pawlowski, and F. Rennecke, “Strangeness neutrality and baryon-strangeness correlations”, Phys. Rev. D 100, 111501 (2019).
**

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.111501

Related to Project A01, A02, B03, C05, C06

**Abstract:**

A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.

**M. Heyl, P. Hauke, P. Zoller, “Quantum localization bounds Trotter errors in digital quantum simulation”, Science Advances 5 (2019).
**

https://www.science.org/doi/10.1126/sciadv.aau8342

Related to Project B04