  Universal quantum simulation may provide insights into those many-body
systems that cannot be described classically, and that cannot be efficiently
simulated with current technology. The Trotter formula, which decomposes a
desired unitary time evolution of the simulator into a stroboscopic sequence of
repeated elementary evolutions, is a key algorithmic component which makes
quantum simulation of dynamics tractable. The Trotter number $n$ sets the
timescale on which a computer running this algorithm is switched from one
elementary evolution to another. In the ideal case, the precision of the
simulation can be arbitrarily controlled by increasing $n$. We study a more
realistic scenario where each gate is applied imperfectly. The resultant
tradeoff in errors leads to an ultimate limit on the precision of the
simulation. We calculate the optimum Trotter number $n^*$ that achieves this
limit, which is the minimum statistical distance from the actual simulation to
the ideal one.
