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Matrix exponentiation (ME) is widely used in various fields of science andengineering. For example, the unitary dynamics of quantum systems is describedby exponentiation of Hamiltonian operators. However, despite a significantattention, the numerical evaluation of ME remains computationally expensive,particularly for large dimensions. Often this process becomes a bottleneck inalgorithms requiring iterative evaluation of ME. Here we propose a method forapproximating ME of a single operator with a bounded coefficient into a productof certain discrete operators. This approach, which we refer to as RapidExponentiation using Discrete Operators (REDO), is particularly efficient foriterating ME over large numbers. We describe REDO in the context of a quantumsystem with a constant as well as a time-dependent Hamiltonian, although inprinciple, it can be adapted in a more general setting. As concrete examples,we choose two applications. First, we incorporate REDO in optimal quantumcontrol algorithms and report a speed-up of several folds over a wide range ofsystem size. Secondly, we propose REDO for numerical simulations of quantumdynamics. In particular, we study exotic quantum freezing with noisy drivefields.

Rapid Exponentiation using Discrete Operators: Applications in Optimizing Quantum Controls and Simulating Quantum Dynamics