State-of-the-art algorithms in computational science and engineering (CSE) are widely known and established. In this regard, recent advancements in quantum computing promise an improvement by offering a speedup over classical computing methods [1]. Leveraging this advantage requires translating established CSE frameworks into quantum gates. Our contribution presents an algorithm for learning operators in quantum computing, which can be exemplarily used to solve linear problems with quantum hardware, e.g., Ax=b. Specifically, the operator A is mapped onto ladders of fixed qubit gates [2,3]. This mapping is based on solving an optimization problem subject to the unitary quantum constraint [4}. Using a hierarchical optimization strategy, we do not compute Riemann derivatives, cf. [4], but compute the derivatives by backpropagation [3]. The approach is not only applicable to sparse operators A from structured grid discretizations, but also to operators from unstructured discretizations. Moreover, it considers the connectivity and gate properties of the target quantum hardware during the learning process, and is also suitable for general tensor network frameworks, which allows to efficiently approximate large operators. The study discusses, by way of example, the learning of an operator that occurs in computational fluid dynamics when analyzing the inviscid flow around an airfoil modeled by a panel method [5]. The complexity and the accuracy of the framework are evaluated based on the relative error as well as the success probabilities, and the computational effort. Due to its general applicability, the approach can be easily adapted to implement other non-unitary dynamics and to optimize existing gate sequences. For example, it can be readily applied to many-body Hamiltonians in quantum chemistry or as an alternative to Trotterization approaches.