  Quantum simulation promises to address many challenges in fields ranging from
quantum chemistry to material science, and high-energy physics, and could be
implemented in noisy intermediate-scale quantum devices. A challenge in
building good digital quantum simulators is the fidelity of the engineered
dynamics given a finite set of elementary operations. Here we present a
framework for optimizing the order of operations based on a geometric picture,
thus abstracting from the operation details and achieving computational
efficiency. Based on this geometric framework, we provide two alternative
second-order Trotter expansions, one with optimal fidelity at a short time
scale, and the second robust at a long time scale. Thanks to the improved
fidelity at different time scale, the two expansions we introduce can form the
basis for experimental-constrained digital quantum simulation.
