Simulations of quantum mechanical systems serve as one of the most promising applications of quantum computers, with large speedups expected over the best classical techniques. At the heart of these candidate applications are Hamiltonian simulation algorithms, which engineer the unitary dynamics over a set of qubits by approximating the time evolution operator with a finite sequence of gates. After decades of innovation, this can be achieved with remarkable asymptotic efficiency. Despite this success, significant open questions remain -- notably regarding the constant-factor costs of implementation and the extension of these methods to non-unitary dynamics.

In this talk, I will show how to construct efficient quantum algorithms for Hamiltonian simulation and illustrate how they can be leveraged as a subroutine to simulate broader classes of dynamics. These include dissipative open quantum systems governed by the Lindblad equation, as well as systems governed by linear differential equations that need not be quantum at all. I will present novel algorithms with efficient asymptotic query complexity, as well as provide constant-factor bounds that significantly improve over existing art.