Qubit encoding for a mixture of localized functions

TL;DR

This paper introduces efficient quantum encoding methods for linear combinations of localized functions, reducing computational complexity and enabling practical quantum computations, with demonstrated results on superconducting quantum computers.

Contribution

Develops specialized quantum encoding techniques for localized functions that are more efficient and deterministic, improving upon existing amplitude encoding methods.

Findings

Probabilistic encoding has complexity $ ext{O}( ext{max}(n_{ ext{loc}}^2 ext{log} n_{ ext{loc}}, n_{ ext{loc}}^2 ext{log} n_q, n_q))$

Deterministic encoding reduces complexity to $ ext{O}( ext{max}(n_{ ext{loc}}^{3/2} ext{log} n_{ ext{loc}}, n_{ ext{loc}}^{3/2} ext{log} n_q, n_q))$

Validated techniques on real superconducting quantum computers

Abstract

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.