Methods for generating quantum computing circuits by distributing approximation errors in a quantum algorithm are described. A method includes decomposing a quantum algorithm into quantum circuits. The method includes using at least one processor, automatically performing a step-wise decomposition of the quantum algorithm until the quantum algorithm is fully decomposed into the quantum circuits, where the automatically performing the step-wise decomposition results in a set of approximation errors and a set of parameters to instantiate at least a subset of the quantum circuits corresponding to the quantum algorithm, such that an overall approximation error caused by the automatically performing the step-wise decomposition is maintained below a specified threshold approximation error.