Optimal Trotterization in universal quantum simulators under faulty control

Abstract: 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.