Qubit encoding for a mixture of localized functions

One of the crucial generic techniques for quantum computation is amplitude encoding. Although several approaches have been proposed, each of them often requires exponential classical-computational cost or an oracle whose explicit construction is not provided. Given the growing demands for practical quantum computation, we develop moderately specialized encoding techniques that generate an arbitrary linear combination of localized complex functions. We demonstrate that ${n}_{\mathrm{loc}}$ discrete Lorentzian functions as an expansion basis set lead to efficient probabilistic encoding, whose computational time is $O(max({n}_{\mathrm{loc}}^{2}log{n}_{\mathrm{loc}},{n}_{\mathrm{loc}}^{2}log{n}_{q},{n}_{q}))$ for ${n}_{q}$ data qubits equipped with ${log}_{2}{n}_{\mathrm{loc}}$ ancillae. Furthermore, amplitude amplification in combination with amplitude reduction renders it deterministic analytically with controllable errors, and the computational time is reduced to $O(max({n}_{\mathrm{loc}}^{3/2}log{n}_{\mathrm{loc}},{n}_{\mathrm{loc}}^{3/2}log{n}_{q},{n}_{q})).$ We estimate required resources for applying our scheme to quantum chemistry in real space. We also show the results on real superconducting quantum computers to confirm the validity of our techniques.