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Quantum algorithms achieve Heisenberg scaling for partition function estimation with adaptive cooling, reducing query complexity to $\widetilde O(n^{5/4}/\epsilon)$ and potentially $\widetilde O(n/\epsilon)$ with advanced estimators.
Quantum Gibbs states for complex, non-commuting Hamiltonians can now be prepared efficiently by transferring guarantees from simpler sources, overcoming the limitations of prior Hamiltonian families.
Quantum many-body systems can be efficiently simulated classically when their dynamics are contractive, extending simulable time horizons significantly beyond prior work.
Quantum interferometric algorithms are no longer bottlenecked by rigid pilot-state preparation: moving from ordinary to relative algebraic decoding elevates effective code distances from $O(1)$ to $\Theta(n)$, unlocking verifiable quantum advantages in non-commuting optimization tasks.
