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On the Need for (Quantum) Memory with Short Outputs

Zihan Hao, Zikuan Huang, Qipeng Liu

TL;DR

This work establishes the first separation between computation with bounded and unbounded space, for problems with short outputs, both in the classical and the quantum setting, based on a novel ``two-oracle recording'' technique.

Abstract

In this work, we establish the first separation between computation with bounded and unbounded space, for problems with short outputs (i.e., working memory can be exponentially larger than output size), both in the classical and the quantum setting. Towards that, we introduce a problem called nested collision finding, and show that optimal query complexity can not be achieved without exponential memory. Our result is based on a novel ``two-oracle recording'' technique, where one oracle ``records'' the computation's long outputs under the other oracle, effectively reducing the time-space trade-off for short-output problems to that of long-output problems. We believe this technique will be of independent interest for establishing time-space tradeoffs in other short-output settings.

On the Need for (Quantum) Memory with Short Outputs

TL;DR

This work establishes the first separation between computation with bounded and unbounded space, for problems with short outputs, both in the classical and the quantum setting, based on a novel ``two-oracle recording'' technique.

Abstract

In this work, we establish the first separation between computation with bounded and unbounded space, for problems with short outputs (i.e., working memory can be exponentially larger than output size), both in the classical and the quantum setting. Towards that, we introduce a problem called nested collision finding, and show that optimal query complexity can not be achieved without exponential memory. Our result is based on a novel ``two-oracle recording'' technique, where one oracle ``records'' the computation's long outputs under the other oracle, effectively reducing the time-space trade-off for short-output problems to that of long-output problems. We believe this technique will be of independent interest for establishing time-space tradeoffs in other short-output settings.
Paper Structure (29 sections, 41 theorems, 139 equations, 10 figures)

This paper contains 29 sections, 41 theorems, 139 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Discussions and open questions
  4. Proof of space.
  5. Does quantum advantage require space?
  6. Distinguishing between quantum and classical space?
  7. Technical overview
  8. Preliminaries
  9. Compress oracle and our model of computation
  10. Standard oracle.
  11. Phase oracle.
  12. Compressed phase oracle
  13. Hadamard Oracle
  14. Finding collision implies many entries in the compressed oracle
  15. l-Nested collision finding and upper bounds
  16. ...and 14 more sections

Key Result

Theorem 1.1

For $\ell\geq 2$, any quantum algorithm that solves the $\ell$-Nested Collision Finding problem with constant probability with space $S=\Omega(\log N)=O\left(N^{\frac{1}{(\ell + 1)(2 \ell + 1)}} \right)$ and $T$ oracle queries must satisfy $S^{\frac{\ell+1}{2}}T=\Omega\left(N^{\frac{1}{2(\ell+1)}}N_

Figures (10)

  • Figure 1: Conversion from a space-bounded computation to a time-segmented computation with space constraints applied only during transitions.
  • Figure 2: A roadmap in establishing our main results.
  • Figure 3: The LHS represents computation with oracle access to both $H, G$. The RHS represents computation with only oracle access to $H$, whereas having access to an (exponentially large) write-only memory $G$.
  • Figure 4: Challenge an algorithm for finding a random $\ell$-tuple.
  • Figure 5: An overview of this section.
  • ...and 5 more figures

Theorems & Definitions (83)

  • Theorem 1.1: Main quantum theorem
  • Theorem 1.2: Separation between space-bounded/unbounded quantum computation
  • Theorem 1.3: Main classical theorem
  • Theorem 1.4: Separation between space-bounded/unbounded classical computation
  • Lemma 2.1: Informal
  • Lemma 3.1: Chebyshev Bound
  • Lemma 3.2: Grover Search, grover1996fast
  • Lemma 3.3: Jensen's Inequality, JensenInequality
  • Lemma 4.1: zhandry2019record
  • Lemma 4.2: zhandry2019record
  • ...and 73 more