Free Snacks in Quantum Complexity

Estimating ground-state energies is a cornerstone problem in Hamiltonian complexity, and in general requires exponential resources even on quantum computers. It is in this context we analyse the recently developed Imaginary-Time Quantum Dynamical Emulation (ITQDE). This method enables estimation of spectral densities, partition functions, and low-lying gaps, but requires only minimal coherent control, modest classical post-processing, and no state preparation. Using a quadrature-based formulation, we derive scaling and stability criteria that diagnose when its estimates are reliable, and introduce a controlled smoothing that yields a principled bias-variance trade off. The resulting picture preserves the hardness of exact eigenvalue resolution but reveals a practical regime - a "free snack" - here coarse-grained spectral information is obtainable with only polynomial resources. By recasting sampling costs as explicit bounds on resolvable bandwidths, the intermediate regime between trivial and intractable complexity becomes accessible on near-term quantum hardware.