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 time scale 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 trade-off 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.