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Watch this 42-minute conference talk exploring measurement-driven quantum algorithms that bridge the gap between current NISQ devices and future fault-tolerant quantum computers. Learn how classical post-processing of quantum measurement data creates powerful tools for spectral analysis, with applications ranging from eigensolver problems to matrix function evaluation. Discover multi-observable dynamic mode decomposition (MODMD), which combines signal processing with classical shadow tomography to extract multiple low-lying eigenvalues using exponentially reduced resources compared to traditional Hadamard-test circuits. Explore quantum algorithms for evaluating matrix elements through Szegö quadrature, utilizing single-ancilla quantum circuits to construct quadrature rules that achieve optimal scaling between polynomial degree and circuit count. Understand how these approaches enable flexible function evaluation after quantum execution without requiring function approximations during quantum computation, making them broadly applicable to spectral characterization tasks including Hamiltonian spectra analysis through Green's functions and Gibbs state property estimation. Examine theoretical guarantees showing exponential decay of spectral error with simulation time, and see how these methods demonstrate practical implementation on current hardware while scaling naturally toward fault-tolerant quantum computing implementations.
