Worst-case Harrow-Hassidim-Lloyd algorithm with average-case correct quantum Fourier transform

TL;DR

This paper demonstrates that the Harrow-Hassidim-Lloyd quantum algorithm can achieve reliable worst-case performance if the quantum Fourier transform is correct on average, extending previous average-case verification protocols.

Contribution

It provides a strengthened protocol for worst-case performance guarantees of the HHL algorithm based on average-case correctness of the QFT.

Findings

HHL algorithm can be executed with good worst-case performance under average-case QFT correctness.

The protocol extends the applicability of average-case to worst-case performance guarantees.

The approach applies across three distinct quantum computational scenarios.

Abstract

In [\href{https://quantum-journal.org/papers/q-2022-12-07-872/}{Quantum 6, 872, 2022}], Linden and de Wolf proposed a lightweight protocol for verifying the average-case correct behavior of the quantum Fourier transform (QFT). They proved that good average-case QFT performance suffices for good worst-case performance in several quantum tasks. Here we provide another application of this worst-case-to-average-case reduction, using a strengthened Linden-de Wolf protocol. We show that, across three distinct scenarios, the Harrow-Hassidim-Lloyd algorithm can be executed with provably good worst-case performance, assuming only that the QFT is correct on average.