Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a generalized Hamiltonian dynamics in an extra time variable $\tau$ which, at large times, samples a microcanonical ensemble. In a previous work we showed that, for an interacting scalar theory in 1+1 dimensions, this framework captures genuine real time features that are inaccessible to Euclidean simulations. That original formulation suffers from two structural limitations, an ill defined non interacting limit and the lack of a direct correspondence between its correlation functions and those generated by the Feynman path integral. To solve these problems we introduced constrained symplectic quantization, a holomorphic reformulation in which fields and action are analytically continued and constraints are imposed on the intrinsic time Hamiltonian flow. The constraints select stable deterministic trajectories and they define convergent holomorphic integration cycles for the corresponding microcanonical measure. In the continuum limit we establish exact equivalence with the Feynman path integral at the level of the generating functional, thus providing a direct link between intrinsic time correlators and real time Green functions. In this contribution, we apply the method to the quantum harmonic oscillator on a real-time 1-dimensional lattice. Testing various observables, we find agreement between numerical and exact results for one- and two-point functions, and we reconstruct characteristic real-time features such as an oscillatory propagator, the discrete energy-gap spectrum, and the evolution of eigenstate probability densities. These tests provide numerical evidence that constrained symplectic quantization can sample real-time quantum observables and offers a practical route beyond Euclidean-time importance sampling.

Recent software advances now allow large-scale lattice studies of the Corrigan--Ramond large-$N_C$ limit of Yang-Mills theory coupled with a two-index antisymmetric fermion, providing a path to SUSY Yang-Mills. We are currently generating ensembles for $N_C=4,5,6$ for lattice spacings in the range $0.11 - 0.08$ fm. We report on two aspects of our work: the study of topological properties as well as estimates of discretisation effects. The first aspect is relevant since naively, fractional topological charges might be expected in our simulations. Using a gluonic definition of the topological charge combined with gradient flow, we perform an analysis of the effect of different discretisations of the kernel action, from which we identify and interpret quantitative differences between Wilson and over-improved flows such as DBW2. The second aspect is addressed by considering ratios of different reference flow times. We conclude that our current simulations might be affected by discretisation effects of order 10\%.

We present our updated results on the intrinsic width of the profile of the flux tube in (2+1)-dimensional Yang-Mills theory with SU(2) gauge group. We identify the intrinsic width as the characteristic length scale of the exponentially decaying tails of the profile of the flux tube. Inspecting a broad range of temperature, we check that this length does not depend on the length of the flux tube. Our estimations of the intrinsic width show a constant value at low temperature and a growing trend approaching the deconfinement temperature that can be understood from the universality class of the phase transition via the Svetitsky-Yaffe mapping.

The MexNICA Collaboration coordinates the activities of Mexican scientists, engineers, postdoctoral fellows and students in the Multi-Purpose Detector experiment at the Nuclotron-based Ion Collider fAcility of the Joint Institute for Nuclear Research in Dubna, Russia. Established in 2016, the collaboration brings together five Mexican institutions whose contributions span detector development as well phenomenological and theoretical studies, including modeling by means of Monte Carlo simulations. This work summarizes the main achievements of MexNICA, consisting of the development of the miniBeBe trigger detector as well of results of phenomenological investigations of the baryon-rich region in the QCD phase diagram accessible at NICA energies, and theoretical advances based on lattice QCD and effective models.

We study the strong coupling expansion of large $N$ QCD in various dimensions, reformulating the Kogut-Susskind Hamiltonian on a square lattice in terms of (constrained) one dimensional spin chain models. We study the integrability properties of the spin chain obtained this way: there is large class of integrable subsectors, but we show that the full spin chain is not integrable, at least when viewed from a description based on Bethe ansatz. We demonstrate that the spin chains no longer possess integrability due to the constraints arising from the zigzag symmetry of the confining strings. The spin chain description properly estimates the roughening transition point by extrapolating the first-order analytical results based on integrability of some subsectors. The generalization to higher dimensions are also considered, where we also find the small subsectors without the zigzag constraints to be integrable.

For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to describe this system by observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. We construct the functional integral for the quantum field theory, and discuss in detail the classical relativistic Klein-Gordon equation with interactions.

Classical tensor network and hybrid quantum-classical algorithms are promising candidates for the investigation of real-time properties of lattice gauge theories. We develop here a novel framework which enforces gauge symmetry via a quantum-link virtual rishon representation applied at intermediate steps. Crucially, the gauge and matter degrees of freedom are dynamical variables encoded in terms of qubits, enabling analysis of gauge theories in $d+1$ spacetime dimensions. We benchmark this framework in a U(1) gauge theory with and without matter fields. For $d = 1$, the multi-flavor Schwinger model with $1\leq N_f\leq3$ flavors is analyzed for arbitrary boundary conditions and nonzero topological angle, capturing signatures of the underlying Wess-Zumino-Witten conformal field theory. For $d = 2$, we extract the confining string tension in close agreement with continuum expectations. These results establish the virtual rishon framework as a scalable and robust approach for the simulation of lattice gauge theories using both classical tensor networks as well as near-term quantum hardware.

We investigate the spectrum of $T_{\Upsilon\Upsilon}$ tetraquark candidates within a coupled-channels framework. The analysis includes all $L\leq2$ combinations of $\Upsilon(1S)$, $\Upsilon(2S)$, $\eta_b(1S)$, and $\eta_b(2S)$ in the $J^P = 0^\pm, 1^\pm, 2^\pm$ sectors. The meson-meson interaction is derived from an underlying constituent quark model through the resonating group method, and the properties of the states are obtained from poles of the scattering matrix. We find a rich spectrum of resonant, and virtual, states distributed between the $\eta_b(1S)\eta_b(1S)$ and $\Upsilon(2S)\Upsilon(2S)$ thresholds. The pattern of poles exhibits approximate heavy-quark spin symmetry multiplets. Several states are dominated by a single channel and can be associated with threshold-driven structures, while higher-mass resonances show sizable mixing among channels involving radially excited bottomonia. The predicted widths range from tens to several hundred MeV. Branching ratios indicate that many states couple predominantly to final states with at least one excited bottomonium, whereas only a subset of the spectrum is expected to be visible in the $\eta_b(1S)\eta_b(1S)$, $\eta_b(1S)\Upsilon(1S)$ and $\Upsilon(1S)\Upsilon(1S)$ channels. These results provide quantitative guidance for experimental searches of fully heavy tetraquarks and offer a test of coupled-channel dynamics and heavy-quark spin symmetry in the $bb\bar b\bar b$ sector.

We report on a scale determination, scale setting, and determination of the strong coupling in the gradient flow scheme using the $N_f=2+1$ highly improved staggered quark (HISQ) ensembles generated by the HotQCD Collaboration for bare gauge couplings ranging from $\beta = 7.030$ to $8.400$. The gradient flow scales we obtain in this work are $\sqrt{t_0} = 0.14229(98)$~fm and $w_0 = 0.17190(140)$~fm. Using the decay constants of the kaon and $\eta_s$, as well as the bottomonium mass splitting from the literature, we also calculate the potential scale $r_1$, obtaining $r_1 = 0.3072(22)$~fm. We fit the flow scales to an Allton-type ansatz as a function of $\beta$, providing a polynomial expression that allows for the prediction of lattice spacings at new $beta$ values. As a secondary result, we make an attempt to determine $\Lambda_{\overline{\mathrm{MS}}}$ and use it to estimate the strong coupling in the $\overline{\mathrm{MS}}$ scheme.

Quantum simulation offers a powerful approach to studying quantum field theories, particularly (2+1)D quantum electrodynamics (QED$_3$) with Wilson fermions, which hosts a rich landscape of physical phenomena. A key challenge in lattice formulations is the proper realization of topological phases and the Chern-Simons terms, where fermion discretization plays a crucial role. In this work, we highlight the differences between staggered and Wilson fermions coupled to $\text{U}(1)$ gauge fields in the Hamiltonian formulation. We analyze why staggered fermions fail to induce (2+1)D topological phases, while Wilson fermions admit a variety of topological phases including Chern insulator and quantum spin Hall phases. Additionally, we uncover a rich phase diagram for the two-flavor Wilson fermion model in the presence of a chemical potential. Our findings resolve existing ambiguities in Hamiltonian formulations and provide a theoretical foundation for future quantum simulations of lattice field theories with topological phases. We further outline connections to experimental platforms, offering guidance for implementations on near-term quantum computing architectures. A complementary presentation of the analytical calculations, the identification of robust topological structure and response, and extensive numerical results is contained in a joint submission [1].

I review the physics of lattice fermions obeying the Ginsparg-Wilson relation. I describe their relation to domain wall fermions. I give a description of methodology for performing numerical simulations with overlap fermions. This is a chapter contributed to the on-line book ``Lattice QCD at 50 years,'' (LQCD@50), edited by Tanmoy Bhattacharya, Maarten Golterman, Rajan Gupta, Laurent Lellouch, and Steve Sharpe.

Quantum simulation of non-Abelian gauge theories requires careful handling of gauge redundancy. We address this challenge by presenting universal principles for treating gauge symmetry that apply to any quantum simulation approach, clarifying that physical states need not be represented solely by gauge singlets. Both singlet and non-singlet representations are valid, with distinct practical trade-offs, which we elucidate using analogies to BRST quantization. We demonstrate these principles within a complete quantum simulation framework based on the orbifold lattice, which enables explicit and efficient circuit constructions relevant to real-world QCD. For singlet-based approaches, we introduce a Haar-averaging projection implemented via linear combinations of unitaries, and analyze its cost and truncation errors. We also introduce an efficient simulation protocol with an additional term to the Hamiltonian that eliminates non-singlet states from the low-energy spectrum. Beyond the singlet-approach, we show how non-singlet approaches can yield gauge-invariant observables through wave packets and string excitations. This non-singlet approach is proven to be both universal and efficient. Working in temporal gauge, we provide explicit mappings of lattice Yang-Mills dynamics to Pauli-string Hamiltonians suitable for Trotterization. Classical simulations of small systems validate convergence criteria and quantify truncation and Trotter errors, showing concrete resource estimates and scalable circuit recipes for SU$(N)$ gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.