Ground state energy estimation for strongly correlated quantum systems remains a central challenge in computational physics and chemistry. While tensor network methods like DMRG provide efficient solutions for one-dimensional systems, higher-dimensional problems remain difficult. Here we present a variational double bracket flow (vDBF) algorithm that leverages Pauli Propagation, a technique originally developed for classical simulation of quantum circuits, to efficiently approximate ground state energies. By combining greedy operator selection with coefficient-based fluctuation truncation and energy-variance extrapolation, we obtain sub-1% relative errors for Heisenberg and Hubbard lattices of up to 128 qubits. For a 10$$10 Heisenberg lattice (100 qubits), vDBF obtains accurate results in approximately 2 minutes on a single CPU thread, compared to over 50 hours on 64 threads for DMRG. Accurate results are also obtained for systems that are difficult for quantum Monte Carlo: on the maximally frustrated $8\times 8$ $J_1$-$J_2$ lattice and the hole-doped $8\times 8$ Hubbard model, the tightest thresholds extrapolate to within a few tenths of a percent of the best published variational estimates. We further test vDBF on the 84-qubit $\pi$-valence active space of hexabenzocoronene, where the most reliable extrapolated correlation energies agree with DMRG to within 1%. These results demonstrate that classical simulation techniques developed in the context of quantum advantage benchmarking can provide practical tools for many-body physics.