Lattice quantum chromodynamics (QCD) can provide important theory inputs needed for new physics searches ranging from the muon g-2 to DUNE and other neutrino oscillation experiments. The energy spectrum of QCD, from which hadron masses as well as hadronic scattering cross sections can be derived, is determined in lattice calculations through statistical analysis and fits of imaginary-time correlation functions. This analysis is made challenging by an exponentially severe signal-to-noise problem. The problem worsens for more complex systems and prevents us from robustly predicting many properties of nucleons and nuclei directly from QCD. In this talk I will present a new method for obtaining energy spectra in lattice QCD based on applying the Lanczos algorithm to the transfer matrix. This new Lanczos method does not suffer from exponential signal-to-noise problems and provides rigorous two-sided bounds on systematic uncertainties in its determinations of QCD energy levels