Topological control of quantum speed limits

Quantum Fisher Information (QFI) is a measure quantifying the sensitivity of a quantum state with respect to changes in tuning parameters in quantum metrology, and defining quantum speed limits. We show that even if the quantum state is completely dispersionless, QFI in this state remains momentum-resolved. We compute the QFI for topological phases at integer filling and demonstrate that each momentum-resolved term is fundamentally bounded by quantum geometric and topological invariants, with maximum QFI controlled by topological invariants (Chern number $|C|$). We also finds bounds on quantum speed limit which scales as $\sqrt{|C|}$ in a (dispersionless) topological phase. We conclude that quantum platforms of high Chern numbers $|C| \gg 1$, such as those featuring twisted multilayered van der Waals heterostructures, significantly enhance capacity for quantum Fisher information, and provide practical control over quantum speed limits.