work we argue that both representations are exponentially efficient in many ways. First, the quantum register demands only a logarithmically growing number of qubits $Nm$ to store an exponential amount of weights $2^{Nm}$ in a discretized function. Second, we also need an exponentially small number of qubits $m\sim-\log(\varepsilon)$ to reduce the discretization error below a given tolerance $\varepsilon.$ Third and finally, we find that for smooth, differentiable functions with bounded derivatives, these states have a small amount of entanglement. Indeed, for many distributions of interest we obtain the scaling $\mathcal{O}(N)$ with the dimension of the problem. We conjecture that this behavior is due to an implicit renormalization that happens in the quantum register, where some bits $s_1^1,s_2^1\ldots$ carry information over long wavelengths and large features, and the least significant bits $s_1^m,s_2^m\ldots$ are efficiently approximated with low-entanglement interpolation methods over those qubits (Sect. \[sec:interpolation\]).

These findings suggest that many useful functions and problems can be constructed in a quantum register with polynomial resources, due to the bounded entanglement (see Sect. \[sec:bounds\]). However, the same results open the field of *quantum-inspired numerical analysis,* which combines the quantum computing encoding of functions with tensor-network representations and algorithms to manipulate them (see Sect. \[sec:quantum-inspired-algorithms\]). 