Excited-state effects lead to hard-to-quantify systematic uncertainties in lattice quantum chromodynamics (LQCD) spectroscopy calculations when computationally accessible imaginary times are smaller than inverse excitation gaps, as often arises for multi-hadron systems with signal-to-noise problems. Lanczos residual bounds address this by providing two-sided constraints on energies that do not require assumptions beyond Hermiticity, but often give very conservative systematic uncertainty estimates. Here, a more-constraining set of gap bounds is introduced for hadron spectroscopy. These bounds provide tighter constraints whose validity requires an explicit assumption about an energy gap. Exactly solvable lattice field theory correlators are used to test the utility of residual and gap bounds at finite and infinite statistics. Two-sided bounds and other analysis methods are then applied to a high-statistics LQCD calculation of nucleon-nucleon scattering at $m_\pi \sim 800$ MeV. Generalized eigenvalue problem (GEVP) and Lanczos energy estimators are compatible when applied to the same correlator data, but analyses including different interpolating operators show statistically significant inconsistencies. However, two-sided bounds from all operators are consistent. Under the assumption that the number of energy levels below $N\Delta$ and $\Delta\Delta$ thresholds is the same as for non-interacting nucleons, gap bounds are sufficient to constrain nucleon-nucleon scattering amplitudes at phenomenologically relevant precision. Lanczos methods further reveal that energy-eigenstate estimates from previously studied asymmetric correlators have not converged over accessible imaginary times. Nevertheless, data-driven examples demonstrate why assumptions are required to draw conclusions about the natures of two-nucleon ground states at these masses.

Lattice QCD has historically produced energy results interpretable as either estimates relying on implicit assumptions about asymptotic behavior or one-sided upper bounds. New Lanczos methods providing two-sided bounds with less-restrictive assumptions are introduced and quantified in a high-statistics calculation with unphysical quark masses. Two-sided bounds without spectral assumptions provide sub-percent constraints on the nucleon mass. Other bounds, which assume all states in a given energy window are resolved, provide meaningful two-sided constraints on nucleon-nucleon scattering phase shifts.

We employ the wavelet formalism of quantum field theory to study field theories in the nonperturbative Hamiltonian framework. Specifically, we make use of Daubechies wavelets in momentum space. These basis elements are characterised by a resolution and a translation index that provides for a natural nonperturbative infrared and ultraviolet truncation of the quantum field theory. As an application, we consider the $\phi^4$ theory and demonstrate the emergence of the well-known nonperturbative strong-coupling phase transition in the $m^2>0$ sector.

We propose a superconducting-circuit architecture that realizes a compact U(1) lattice gauge theory using the intrinsic infinite-dimensional Hilbert space of phase and charge variables. The gauge and matter fields are encoded directly in the degrees of freedom of the rotor variables associated with the circuit nodes, and Gauss's law emerges exactly from the conservation of local charge, without auxiliary stabilizers, penalty terms, or Hilbert-space truncation. A minimal gauge-matter coupling arises microscopically from Josephson nonlinearities, whereas the magnetic plaquette interaction is generated perturbatively via virtual matter excitations. Numerical diagonalization confirms the emergence of compact electrodynamics and coherent vortex excitations, underscoring the need for large local Hilbert spaces in the continuum regime. The required circuit parameters are within the current experimental capabilities. Our results establish superconducting circuits as a scalable, continuous-variable platform for analog quantum simulation of non-perturbative gauge dynamics.

The thermal confinement phase transition (PT) in $SU(N)$ Yang-Mills theory is first-order for $N\geq 3$, with bounce action scaling as $N^2$. Remarkably, lattice data for the action include a small coefficient whose presence likely strongly alters the PT dynamics. We give evidence, utilizing insights from softly-broken SUSY YM models, that the small coefficient originates from a deconfined phase instability just below the critical temperature. We predict the maximum achievable supercooling in $SU(N)$ theories to be a few percent, which can be tested on the lattice. We briefly discuss the potentially significant suppression of the associated cosmological gravitational wave signals.

We consider free Dirac fermions on a discretized $AdS_2$ black hole background, and analyze how curved space redshift, horizons, and the spin connection induced chiral gravitational effect shape spectral, transport, and scrambling phenomena. The system is discretized via staggered fermions followed by the Jordan-Wigner transform to encode the model in qubit degrees of freedom, whose Hamiltonian carries site dependent warp factors and bond chirality terms encoding the redshift and spin connection effects. We calculate the ground state and first excited states energies, their local charge profiles, and their half-chain entanglement entropies, showing how redshift and chirality affect the transition from criticality to a gapped regime. Probing operator growth via out-of-time-order correlators, we find that horizons and the chiral coupling accelerate scrambling, yet remain within a non-chaotic regime. Finally, we map out an integrable to ergodic crossover via level-spacing statistics and Brody fits, and introduce on-site disorder to drive a many body localization transition.

We introduce a robust numerical method for determining intersection numbers of Lefschetz thimbles in multivariable settings. Our approach employs the multiple shooting method to solve the upward flow equations from the saddle points to the original integration cycle, which also enables us to determine the signs of the intersection numbers. The method demonstrates stable and reliable performance, and has been tested for systems with up to $20$ variables, which can be further extended by adopting quadruple-precision arithmetic. We determine intersection numbers for several complex saddle points in a discretized path integral, providing new insights into the structure of real-time path integrals. The proposed method is broadly applicable to a wide range of problems involving oscillatory integrals in physics and mathematics.