Galerkin-based reduced order models are an important tool to make the computational costs of multi-query tasks such as optimization and uncertainty quantification affordable. In contrast to other data-driven approaches, Galerkin-based ROMs come with rigorous theoretical accuracy guarantees and relatively small demand for training data. The adoption of Galerkin-based ROMs in practice, however, is often hindered by their intrusive nature: The construction of the ROM operators requires access to the operators of the full order model (FOM). This access is often not possible for software used in industry. To address this practical problem, operator inference has been proposed to approximate the intrusive ROMs in a non-intrusive way [1]. In this approach, the ROM operators are computed as the solution to a least-squares problem based on FOM snapshot data. However, as described in [2], this least-squares problem can be ill-conditioned. In fact, the rank of the corresponding least-squares matrix is often significantly smaller than the matrix dimension (when considering singular values close to machine precision as zero). So far, this rank deficiency has usually been addressed by adding a regularization term or generating more FOM snapshot data. Regularization, however, inevitably imposes a limitation on the accuracy of the ROM operator approximation. More snapshot data, on the other hand, does not guarantee the least-squares matrix to have full rank and low condition number, even if the number of snapshots is orders larger than the matrix dimension. To reliably obtain a well-conditioned least-squares matrix without deteriorating the ROM operator accuracy, we suggest a novel approach of FOM snapshot data generation. This approach is optimal in the number of generated snapshots and enables to reconstruct the intrusive ROM operators accurately up to machine precision. [1] Peherstorfer, Benjamin, and Karen Willcox. "Data-driven operator inference for nonintrusive projection-based model reduction." Computer Methods in Applied Mechanics and Engineering 306 (2016): 196-215. [2] Peherstorfer, Benjamin. "Sampling low-dimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference." SIAM Journal on Scientific Computing 42.5 (2020): A3489-A3515.