New Approaches to Hamiltonian Simulation

In recent years quantum simulation has made giant leaps but there still remain a number of open issues that arise. In this work I will review recent results by my collaborators and I that have changed the way that we think about simulation. We will show how extrapolation ideas can allow Trotter simulations to be performed with super-polynomially better error scaling than previously believed. I will also show how Qubitization can be adapted to work for time-dependent Hamiltonians, which is a problem that has long vexed the community. Further, I will show how methods such as Trotter formulas, qubitization and randomized product formulas can be combined into a single framework that allows simulations to be performed faster than existing methods. I will then discuss open problems and the prospect of using simulation as a computational primitive for finding exponential speedups for broad classes of problems.