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A Brief Note on a Recent Claim About NP-Hard Problems and BQP

Michael C. Chavrimootoo

TL;DR

The paper targets the question of whether NP-hard problems can lie in $BQP$ and critiques Czerwinski's claim that no NP-hard set is in $BQP$. It analyzes the main argument and identifies three flaws—one major and two minor—that undermine the purported proof, including a misdefinition of foundational sets and a misinterpretation of a key theorem. The major flaw hinges on a noncomputability assertion that, in fact, corresponds to an exponential-time computable process, invalidating the central claim. Consequently, the purported consequences such as ${ m NP}\not\subseteq {\rm BQP}$ or ${\rm P} \neq {\rm NP}$ do not follow from Czerwinski's work, and the note clarifies why the claimed separation remains unestablished by that paper.

Abstract

This short note outlines some of the issues in Czerwinski's paper [Cze23] claiming that NP-hard problems are not in BQP. We outline one major issue and two minor issues, and conclude that their paper does not establish what they claim it does.

A Brief Note on a Recent Claim About NP-Hard Problems and BQP

TL;DR

The paper targets the question of whether NP-hard problems can lie in and critiques Czerwinski's claim that no NP-hard set is in . It analyzes the main argument and identifies three flaws—one major and two minor—that undermine the purported proof, including a misdefinition of foundational sets and a misinterpretation of a key theorem. The major flaw hinges on a noncomputability assertion that, in fact, corresponds to an exponential-time computable process, invalidating the central claim. Consequently, the purported consequences such as or do not follow from Czerwinski's work, and the note clarifies why the claimed separation remains unestablished by that paper.

Abstract

This short note outlines some of the issues in Czerwinski's paper [Cze23] claiming that NP-hard problems are not in BQP. We outline one major issue and two minor issues, and conclude that their paper does not establish what they claim it does.
Paper Structure (7 sections, 2 theorems, 2 equations)

This paper contains 7 sections, 2 theorems, 2 equations.

Table of Contents

  1. Overview
  2. Preliminaries
  3. Main Analysis
  4. First Minor Issue
  5. Second Minor Issue
  6. The Major Issue
  7. Conclusion

Key Result

Theorem 1

Let $M$ be an arbitrary TM. A quantum computer or OTM with oracle $L(M)$ cannot decide whether $1^n \in D_M$ faster than with a black box search.

Theorems & Definitions (2)

  • Theorem 1: cze:t:np-hard-bqp
  • Theorem 2: cze:t:np-hard-bqp