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Hamiltonian evolution is a primary use case of quantum computers, and plays a role in a wide variety of quantum algorithms. Efficiently mapping the time evolution onto quantum circuits remains an open problem because the individual terms in the Hamiltonian do not commute, leading to difficulties in constructing the exponential. The usual approach is to divide the evolution operator into small time steps, i.e. Trotterization; this either leads to a longer circuit or a larger time step when we increase the simulation time, and both lead to more larger error in todayâ€™s quantum computers. In this talk, we will discuss two new approaches to time evolution that address some of these issues. First, we show that one can leverage a Cartan decomposition of the algebra generated by a timeindependent Hamiltonian to produce a quantum circuit that has fixed depth as a function of simulation time. For certain spin models, this method generates simulation circuits with polynomial complexity, and we and apply this to the transverse field XY (TFXY) model with randomized potential to simulate Anderson localization. The second approach is based on Lie algebraic relations between individual Hamiltonian terms. If the terms satisfy 3 easytocheck operations the Trotter circuit can be compressed into a fixed depth circuit in a constructive manner. We show that this method works for time dependent 1D Kitaev Chain, XY model, transverse field Ising Model (TFIM) and TFXY models with nearest neighbor spin interactions and open boundary conditions; we use the method to generate ground state of TFIM via adiabatic state preparation (ASP) on IBM quantum computers. Time permitting, we will briefly discuss our recent application of Lie algebraic decomposition to the measurement of the electron Greenâ€™s function in the Dynamical Mean Field Theory algorithm, enabling calculation of the firstever selfconsistent DMFT phase diagram on quantum computers. 