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How to Design a Quantum Streaming Algorithm Without Knowing Anything About Quantum Computing

John Kallaugher, Ojas Parekh, Nadezhda Voronova

TL;DR

The quantum sketch and its proof of correctness are designed to be accessible to a reader with no background in quantum computation, relying on only a small number of self-contained quantum postulates.

Abstract

A series of work [GKK+08, Kal22, KPV24] has shown that asymptotic advantages in space complexity are possible for quantum algorithms over their classical counterparts in the streaming model. We give a simple quantum sketch that encompasses all these results, allowing them to be derived from entirely classical algorithms using our quantum sketch as a black box. The quantum sketch and its proof of correctness are designed to be accessible to a reader with no background in quantum computation, relying on only a small number of self-contained quantum postulates.

How to Design a Quantum Streaming Algorithm Without Knowing Anything About Quantum Computing

TL;DR

The quantum sketch and its proof of correctness are designed to be accessible to a reader with no background in quantum computation, relying on only a small number of self-contained quantum postulates.

Abstract

A series of work [GKK+08, Kal22, KPV24] has shown that asymptotic advantages in space complexity are possible for quantum algorithms over their classical counterparts in the streaming model. We give a simple quantum sketch that encompasses all these results, allowing them to be derived from entirely classical algorithms using our quantum sketch as a black box. The quantum sketch and its proof of correctness are designed to be accessible to a reader with no background in quantum computation, relying on only a small number of self-contained quantum postulates.

Paper Structure

This paper contains 40 sections, 34 theorems, 88 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Quantum Streaming Model
  3. Sketching
  4. Other Quantum Streaming Models
  5. Related Work
  6. Our Contributions
  7. How to Read This Paper
  8. A Quantum Sketch for Pair Sampling
  9. Streaming Applications
  10. Boolean Hidden Matching
  11. Triangle Counting
  12. Maximum Directed Cut
  13. Quantum Pair Sketch
  14. Implementing the Pair Sketch
  15. Quantum Preliminaries
  16. ...and 25 more sections

Key Result

Lemma 0

For any sequence of $\mathtt{query\_one}$ and $\mathtt{query\_pair}$ operations applied to $\mathcal{Q}_T$, interleaved in any way with $\mathtt{update}$ operations, there is a unique $T'$ such that the sketch will become $\mathcal{Q}_{T'}$ after these operations if none of them destroy the sketch.

Theorems & Definitions (62)

  • Lemma 0
  • Theorem 1
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 7
  • proof
  • ...and 52 more