of quantum-inspired matrix-product state (MPS) techniques that, under approximations of low entanglement, represent the quantum register efficiently and provide new classical (and quantum) algorithms for interpolating, differentiating, Fourier transforming or solving differential equations of such distributions. These techniques work efficiently because of an implicit renormalization where different qubits work with different length scales, in a way that lends itself to efficient interpolation and compression. This idea, with strong parallelisms to the 2D quantum image processing world [@latorre2005], gives rise to a performance improvement over earlier techniques based on tensor trains [@grasedyck2013; @bachmayr2016] or MPS encodings of mode expansions [@iblisdir2007].

This paper is structured in three parts. The bulk of the work is preceded by a summary (Section \[sec:summary\]) of the main results and heuristic algorithms that are developed in this work. The first section addresses the representation of discretized functions in quantum registers. It presents state-of-the-art techniques (Sect. \[sec:gr-construct\]) and new discretizations (Sect. \[sec:other-discretizations\]) on an equal footing. We argue that these samplings produce to weakly entangled multi-qubit states (Sect. \[sec:bounds\]). This prediction is confirmed numerically for common distributions in 1, 2 and 3 dimensions, using exact simulations of up to 28 qubits and MPS simulations of up to 36 