store $2^m$ complex numbers as the components in of the quantum register wavefunction, $\ket{\psi} =\sum_{s=0}^{2^m-1}\psi_s\ket{s}.$ A quantum algorithm creates, manipulates and probes these amplitudes to solve a concrete problem. In this work we discuss algorithms where the amplitudes $\psi(s)$ encode a smooth function defined over some a volume in $\mathbb{R}^N.$ A common example is storing probability distributions in the quantum register [@grover2002] and developing algorithms to extract expected values [@montanaro2015] or conditional probabilities [@woerner2019]. With this, it becomes possible to perform valuations of complex financial assets [@rebentrost2018; @stamatopoulos2019], VaR estimates [@egger2019], and other sophisticated interrogations. Using a similar encoding, one may also address radically different problems, such as solving partial differential equations with finite differences [@cao2013; @fillion2018; @costa2019].

In this work we discuss how efficient it is to encode discretized functions in a quantum register. We find that for certain distributions—smooth differentiable functions with bounded derivatives or bounded spectrum—, the accuracy of the discretization increases exponentially with the number of qubits, while the bipartite entanglement grows slowly or even remains bounded with the problem size. This implies such distributions may be constructed with quasi-local operations and polynomial resources on quantum computers. But it also opens an exciting possibility: a family 