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There are quantum algorithms for finding a function $f$ satisfying a set ofconditions, such as solving partial differential equations, and these achieveexponential quantum speedup compared to existing classical methods, especiallywhen the number $d$ of the variables of $f$ is large. In general, however,these algorithms output the quantum state which encodes $f$ in the amplitudes,and reading out the values of $f$ as classical data from such a state can be sotime-consuming that the quantum speedup is ruined. In this study, we propose ageneral method for this function readout task. Based on the functionapproximation by a combination of tensor network and orthogonal functionexpansion, we present a quantum circuit and its optimization procedure toobtain an approximating function of $f$ that has a polynomial number of degreesof freedom with respect to $d$ and is efficiently evaluable on a classicalcomputer. We also conducted a numerical experiment to approximate afinance-motivated function to demonstrate that our method works.

Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion