Recent quantum algorithms for the Hamiltonian simulation problem, the problem of simulating the time dynamics of a quantum system, have introduced new algorithmic primitives with wide applicability. The first Hamiltonian simulation algorithms with exponentially improved dependence on precision introduced the Linear Combination of Unitaries method and Oblivious Amplitude Amplification. Later works introduced the techniques of Quantum Signal Processing and Qubitization.

These techniques are very general algorithmic primitives, much like phase estimation or amplitude amplification, and have found other applications in solving linear systems of equations, Gibbs state preparation, etc. In this talk I'll describe what problems these tools allow us to solve, and describe some applications.