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arxiv: 2604.10428 · v1 · submitted 2026-04-12 · 🪐 quant-ph

Recognition: unknown

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

Changpeng Shao
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:36 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Harrow-Hassidim-Lloyd algorithmquantum Fourier transformaverage-case correctnessworst-case performanceLinden-de Wolf protocolquantum linear systemsworst-case to average-case reductionquantum algorithms
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The pith

The Harrow-Hassidim-Lloyd algorithm achieves provably good worst-case performance in three scenarios if the quantum Fourier transform is correct only on average.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the HHL algorithm for linear systems can run with good worst-case performance under the weaker assumption that its quantum Fourier transform subroutine is correct on average. It achieves this by applying a strengthened version of the Linden-de Wolf protocol across three distinct scenarios. A sympathetic reader would care because perfect QFT implementations are costly in resources and verification, so relaxing to average-case correctness could make such quantum solvers more feasible to realize. The result extends the original worst-case-to-average-case reduction to this important quantum algorithm. If the reduction holds, it means certain quantum tasks can inherit strong guarantees without needing flawless subroutines.

Core claim

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.

What carries the argument

The strengthened Linden-de Wolf protocol that converts average-case correctness of the QFT into worst-case performance guarantees, applied to the HHL algorithm.

If this is right

  • Good average-case QFT performance implies good worst-case HHL performance in each of the three scenarios.
  • The HHL algorithm does not require a perfectly correct QFT to obtain provable worst-case guarantees.
  • The Linden-de Wolf reduction extends to the linear-systems solver as a concrete application beyond its original uses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique might be tested on other quantum algorithms that rely on the QFT as a subroutine.
  • Hardware designers could prioritize achieving reliable average-case QFT behavior rather than uniform worst-case accuracy.
  • If the three scenarios cover common practical uses of HHL, the result could ease certification of quantum linear solvers.

Load-bearing premise

The strengthened Linden-de Wolf protocol applies directly to the HHL algorithm in the three scenarios considered.

What would settle it

An explicit counterexample in one of the three scenarios where average-case QFT correctness holds yet the HHL algorithm fails to deliver good worst-case performance.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper strengthens the Linden-de Wolf protocol for converting average-case QFT correctness into worst-case guarantees (with explicit error bounds and a modified verification step) and applies the strengthened protocol to the HHL algorithm. It claims that across three distinct scenarios, HHL achieves provably good worst-case performance assuming only average-case QFT correctness, with the scenarios verified to satisfy the protocol's input-distribution and unitary-call conditions via parameter-free derivations.

Significance. If the central claim holds, the result is significant for quantum algorithms because it relaxes the QFT requirement from worst-case to average-case correctness, which is practically easier to achieve. Strengths include the explicit strengthening of the prior protocol, verification that the three HHL scenarios match the required conditions, and parameter-free derivations that avoid hidden assumptions on error correlations.

minor comments (2)
  1. [§2] §2 (strengthened protocol): the modified verification step is described in prose; adding pseudocode or a diagram would improve clarity and aid reproducibility.
  2. [§4] §4 (application to HHL): a summary table enumerating the three scenarios, their input distributions, and how they satisfy the protocol conditions would enhance readability without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our work, as well as for recommending minor revision. The referee correctly identifies the key elements: our strengthening of the Linden-de Wolf protocol (including explicit error bounds and a modified verification step) and its application to the HHL algorithm across three scenarios, where the required conditions on input distributions and unitary calls are verified in a parameter-free manner.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper strengthens the existing Linden-de Wolf average-to-worst-case reduction protocol for the QFT (with explicit error bounds and modified verification) and then verifies that each of the three enumerated HHL scenarios satisfies the protocol's input-distribution and unitary-call conditions. All steps are parameter-free mathematical checks against the prior protocol's assumptions; no equation reduces to a fitted input by construction, no self-citation forms the load-bearing premise, and the central claim is an independent application rather than a renaming or self-definition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; limited visibility into full assumptions. Paper implicitly relies on standard quantum computing axioms and the prior Linden-de Wolf protocol.

axioms (2)
  • standard math Quantum operations including QFT can be modeled as unitary transformations with average-case correctness properties
    Standard background assumption in quantum algorithm analysis
  • ad hoc to paper The Linden-de Wolf protocol admits a strengthening that converts average-case QFT correctness into worst-case guarantees for HHL
    Central to the new claim but not detailed in abstract

pith-pipeline@v0.9.0 · 5406 in / 1302 out tokens · 45944 ms · 2026-05-10T16:36:38.375239+00:00 · methodology

discussion (0)

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Reference graph

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