This academic term, some colleagues at Stanford and I are running a journal club on Hamiltonian simulation — the problem of how to use a quantum computer to simulate the time evolution of a physical system. Hamiltonian simulation is a hot topic in research, in part because it's believed that simulating certain systems on quantum computers will allow us to probe aspects of those systems that we don't know how to access with traditional laboratory experiments. The earliest approach to this problem, and one that is still practically useful for certain applications, makes use of the Trotter decomposition and its generalization the Trotter-Suzuki decomposition . These are algorithms for decomposing a time evolution operator that acts simultaneously on the entire quantum system, into a sequence of time evolution operators that act locally on only a few physical sites at a time. Specifically, given a time-independent Hamiltonian $H = \sum_j h_j,$ we would like to find a way to approx