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Digital quantum simulation (DQS) of continuous-variable quantum systems inthe position basis requires efficient implementation of diagonal unitariesapproximating the time evolution operator generated by the potential energyfunction. In this work, we provide efficient implementations suitable forpotential functions approximable by piecewise polynomials, with either uniformor adaptively chosen subdomains. For a fixed precision of approximation, weshow how adaptive grids can significantly reduce the total gate count at thecost of introducing a small number of ancillary qubits. We demonstrate thecircuit construction with both physically motivated and artificially designedpotential functions, and discuss their generalizations to higher dimensions.

Digital quantum simulation of Schr\"odinger dynamics using adaptive approximations of potential functions