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One of the main motivations for quantum computers is their ability to efficiently simulate the dynamics of quantum systems. Since the mid1990s, many algorithms have been developed to simulate Hamiltonian dynamics on a quantum computer, with applications to problems such as simulating spin models and quantum chemistry. While it is now well known that quantum computers can efficiently simulate Hamiltonian dynamics, ongoing work has improved the performance and expanded the scope of such simulations. In this talk I will first introduce the area of quantum computing and quantum complexity, and then describe a simple and very efficient quantum algorithm for simulating Hamiltonian dynamics on a quantum computer [1]. This quantum algorithm approximates the truncated Taylor series of the evolution operator and can simulate the time evolution of a wide variety of physical systems. The cost of this algorithm is only logarithmic on the inverse of the desired precision, and can be shown to be optimal. Such a cost also represents an exponential improvement over previously known methods for Hamiltonian simulation in the literature, and solves a longstanding open question. Roughly speaking, doubling the number of digits of accuracy of the simulation only doubles the complexity. The new algorithm and its analysis are highly simplified due to a technique developed by us for implementing linear combinations of unitary operations, to directly apply the truncated Taylor series. [1] D. Berry, A. Childs, R. Cleve, R. Kothari, and R.D. Somma, Phys. Rev. Lett. 114, 090502 (2015). Host: Mila Adamska 