  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 eficient probabilistic encoding, whose
computational time is $\mathcal{O}( \max ( n_{\mathrm{loc}}^2 \log
n_{\mathrm{loc}},n_{\mathrm{loc}}^2 \log n_q, n_q ))$ 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
$\mathcal{O}( \max ( n_{\mathrm{loc}}^{3/2} \log n_{\mathrm{loc}},
n_{\mathrm{loc}}^{3/2} \log n_q, n_q )).$ 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.
