Quantum computers have recently achieved significant milestones on the path to large-scale devices: error-corrected qubits with noise levels below fault-tolerance have been developed, architectures based on superconducting qubits with one tenth the physical footprint needed are being implemented, and modular photonic architectures are under construction to be completed by the end of 2027. While experimental and error-correction efforts are making significant progress, the main proposed application of future quantum computers, after 30 years of theoretical development, is still primarily the simulation of quantum mechanical systems. This talk will demonstrate two algorithms I developed to reduce the costs of key quantum simulation primitives. The first will demonstrate techniques for combining randomized and deterministic simulation methods to achieve circuit depths that are 1/18th the depth of either method individually. The second algorithm demonstrates how to prepare thermal states of a quantum system by randomly coupling the system to a single qubit that acts as an environment. This second algorithm further provides an extension of the Repeated Interactions framework for open quantum systems to arbitrary Hamiltonians, thereby providing a potential model for thermal equilibrium.