Solving the encoding bottleneck: of the HHL algorithm, by the HHL algorithm

TL;DR

This paper presents a modified HHL algorithm that enables approximate quantum state preparation in polylogarithmic time, overcoming the encoding bottleneck and preserving exponential speedup for solving linear systems.

Contribution

It introduces a method to prepare initial quantum states efficiently within the HHL algorithm, maintaining exponential speedup, which was previously hindered by state preparation complexity.

Findings

State preparation can be done in polylogarithmic time.

The modified HHL preserves exponential speedup.

Applicable to other quantum algorithms requiring fast state preparation.

Abstract

The Harrow-Hassidim-Lloyd (HHL) algorithm offers exponential speedup for solving the quantum linear-system problem. But some caveats for the speedup could be hard to met. One of the difficulties is the encoding bottleneck, i.e., the efficient preparation of the initial quantum state. To prepare an arbitrary $N$-dimensional state exactly, existing state-preparation approaches generally require a runtime of $O(N)$, which will ruin the speedup of the HHL algorithm. Here we show that the states can be prepared approximately with a runtime of $O(poly(\log N))$ by employing a slightly modified version of the HHL algorithm itself. Thus, applying this approach to prepare the initial state of the original HHL algorithm can preserve the exponential speedup advantage. It can also serve as a standalone solution for other applications demanding fast state preparation.