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Preparing low-energy states of many-body Hamiltonians is a central challenge in quantum computing, quantum complexity, and condensed matter physics. Existing approaches often get trapped in suboptimal states such as high-energy eigenstates or, more generally, low-variance states that resist further energy reduction. In this work, we explore a different perspective: instead of optimizing with respect to a single Hamiltonian, we leverage the fact that many systems admit families of Hamiltonians that share similar low-energy subspaces but differ at higher energies. We show that this redundancy can be turned into an algorithmic resource by establishing an energy-based uncertainty principle, which implies that these Hamiltonians cannot simultaneously admit low-variance states at higher energies. This suggests a simple strategy of alternating energy-lowering steps across such Hamiltonians to destabilize trapped states and enable continued descent. We investigate this approach numerically on several models. We also introduce a sparse variant where the uncertainty principle strengthens to yield quadratically larger variance at higher energies, leading to possibly more pronounced energy reduction. Overall, this work suggests a range of open questions at the interface of random matrix theory, local Hamiltonians and state preparation, aimed at understanding when such approaches are practical and how they can be analyzed rigorously.