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Dequantizability from inputs

Tae-Won Kim, Byung-Soo Choi

TL;DR

This work investigates dequantizability of quantum algorithms under the sparse-access input model. It proposes a sparsity-based verification framework that ties the possibility of classical dequantization to bounds on s_p norms and the Frobenius norm of the input matrix, while examining inner-model choices for quantum-access data structures. The authors present a main theorem asserting a quantitative relationship between conditioning, sparsity, and eigenvalues that delineates when dequantization can be expected, and they supply a corollary with a concrete 1-sparse Hermitian input to illustrate unanticipated dequantizability behavior. The results emphasize that sparsity alone is not sufficient for quantum advantage and highlight the need for careful input-structure analysis and potential reformulations in terms of signal processing to understand QML speedups and limitations.

Abstract

By comparing constructions of block encoding given by [1-4], we propose a way to extract dequantizability from advancements in dequantization techniques that have been led by Tang, as in [5]. Then we apply this notion to the sparse-access input model that is known to be BQP-complete in general, thereby conceived to be un-dequantizable. Our goal is to break down this belief by examining the sparse-access input model's instances, particularly their input matrices. In conclusion, this paper forms a dequantizability-verifying scheme that can be applied whenever an input is given.

Dequantizability from inputs

TL;DR

This work investigates dequantizability of quantum algorithms under the sparse-access input model. It proposes a sparsity-based verification framework that ties the possibility of classical dequantization to bounds on s_p norms and the Frobenius norm of the input matrix, while examining inner-model choices for quantum-access data structures. The authors present a main theorem asserting a quantitative relationship between conditioning, sparsity, and eigenvalues that delineates when dequantization can be expected, and they supply a corollary with a concrete 1-sparse Hermitian input to illustrate unanticipated dequantizability behavior. The results emphasize that sparsity alone is not sufficient for quantum advantage and highlight the need for careful input-structure analysis and potential reformulations in terms of signal processing to understand QML speedups and limitations.

Abstract

By comparing constructions of block encoding given by [1-4], we propose a way to extract dequantizability from advancements in dequantization techniques that have been led by Tang, as in [5]. Then we apply this notion to the sparse-access input model that is known to be BQP-complete in general, thereby conceived to be un-dequantizable. Our goal is to break down this belief by examining the sparse-access input model's instances, particularly their input matrices. In conclusion, this paper forms a dequantizability-verifying scheme that can be applied whenever an input is given.
Paper Structure (7 sections, 11 theorems, 43 equations)

This paper contains 7 sections, 11 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Main Ideas
  3. Inner models of quantum-access
  4. Dequantizability
  5. Main results
  6. Conclusions and Outlook
  7. Ordering on $s_p$: Hölder inequality

Key Result

Lemma 1

There is a matrix $A\in\mathbb{C}^{m\times n}$ stored in a quantum-accessible data structure such that no quantum algorithm performs $s_p$-normalizing state preparation of $A$ in $\textrm{\normalfont{polylog}}(mn/\epsilon)$ time.

Theorems & Definitions (22)

  • Lemma 1: Informal, Lemma \ref{['transdisproof']}
  • Lemma 2: Informal, Lemma \ref{['deq']}
  • Lemma 3: Informal, Lemma \ref{['lem:undeq']}
  • Corollary 4: Informal, Corollary \ref{['cor:mainresult']}
  • Lemma 5: csg19, gslw19
  • Lemma 6: negation of csg19
  • proof
  • Definition 7
  • Definition 8: chm21
  • Lemma 9
  • ...and 12 more