Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Worst-Case to Average-Case Reduction for Matrix-Vector Multiplication

Divesh Aggarwal, Dexter Kwan

TL;DR

This paper addresses quantum worst-case to average-case reductions for Matrix-Vector Multiplication, proposing a new reduction that adapts hardness self-amplification to the quantum setting. The method leverages a block-wise decomposition, a direct-product type amplification, and a quantum concatenation/extraction toolkit to assemble subproblem solutions into a full matrix-vector product, guarded by a MvPV verifier. The main contributions are a quantum reduction with overhead $O(\alpha^{-2})$ and total oracle cost $O((Q(n)+n^{3/2})\alpha^{-2}\mathrm{poly}\log(1/\alpha))$, together with a modular framework for concatenation and extraction of subvectors/matrices. This work simplifies quantum worst-case to average-case reductions relative to additive-combinatorics-based approaches and broadens the potential for fine-grained quantum reductions in cryptography and data-structure problems.

Abstract

Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.

Quantum Worst-Case to Average-Case Reduction for Matrix-Vector Multiplication

TL;DR

This paper addresses quantum worst-case to average-case reductions for Matrix-Vector Multiplication, proposing a new reduction that adapts hardness self-amplification to the quantum setting. The method leverages a block-wise decomposition, a direct-product type amplification, and a quantum concatenation/extraction toolkit to assemble subproblem solutions into a full matrix-vector product, guarded by a MvPV verifier. The main contributions are a quantum reduction with overhead and total oracle cost , together with a modular framework for concatenation and extraction of subvectors/matrices. This work simplifies quantum worst-case to average-case reductions relative to additive-combinatorics-based approaches and broadens the potential for fine-grained quantum reductions in cryptography and data-structure problems.

Abstract

Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.
Paper Structure (18 sections, 15 theorems, 42 equations, 3 figures)

This paper contains 18 sections, 15 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Contribution
  4. Proof Overview
  5. Matrix-vector Multiplication (MvM) and Direct Product Theorem
  6. Comparison to Prior Work
  7. Preliminaries
  8. Union Bound
  9. Quantum Unitary Oracles
  10. Average-case quantum algorithms
  11. Sampler
  12. Hardness Self-Amplification
  13. Matrix-vector Product Verification (MvPV)
  14. Concatenation of Matrices
  15. Concatenation of matrices via linear combination of unitaries
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let ${\mathbb{F}} = {\mathbb{F}}_p$ be a prime field, $n\in{\mathbb{N}}$, and $\alpha := \alpha(n) \in (0, 1]$. Suppose that there exists a quantum algorithm ALG that has oracle access to a matrix $M\in {\mathbb{F}}^{n\times n}$ and a vector $v$, makes $Q(n)$ queries, and satisfies Then, for every constant $\delta > 0$, there exists a worst-case algorithm $\sf ALG'$ that makes $O(\alpha^{-2})$ q

Figures (3)

  • Figure 1: Concatenation of submatrices to compute the full matrix-vector product.
  • Figure 2: $M'\overline{v} = Mv$, where $M\in {\mathbb{F}}^{d\times d}$ and $v\in{\mathbb{F}}^d$ is embedded in the second block i.e. $i=2$.
  • Figure 3: $M^{(i,j)}$ and $v^{(j)}$ for $k=4$.

Theorems & Definitions (26)

  • Theorem 1.1: Informally stated; see \ref{['thm:main']}
  • Lemma 3.1: Union Bound
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5: Direct Product Lemma
  • Lemma 3.6: Direct Product Theorem for Verifiable Function
  • Lemma 3.7: Noisy quantum MvPV
  • Lemma 4.1
  • proof
  • ...and 16 more