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In this talk, I will discuss an approach to prepare the ground-states of lattice Hamiltonians in condensed matter physics and nuclear physics using an adaptive variational quantum simulation algorithm. I will start with a brief introduction of variational quantum eigensolvers (VQE), and then focus on applying ADAPT-VQE, a version of VQE that dynamically constructs a wavefunction ansatz, to two examples- the Heisenberg XXZ model and the Schwinger model.