Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pseudo-deterministic Quantum Algorithms

Hugo Aaronson, Tom Gur, Jiawei Li

TL;DR

It is shown that for any total problem $R, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., $D(R) = \tilde O(psQ(R)^5)$.

Abstract

We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism: - We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is $O(1)$ but is maximally hard for pseudo-deterministic quantum algorithms ($Ω(N)$ query complexity). - We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms ($O(\log(N))$ vs. $Θ(\sqrt{N})$), while the randomized query complexity is $O(1)$. Complementing these separations, we show that for any total problem $R$, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., $D(R) = \tilde O(psQ(R)^5)$. On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, $k$-sum, and graph collision.

Pseudo-deterministic Quantum Algorithms

TL;DR

It is shown that for any total problem D(R) = \tilde O(psQ(R)^5)$.

Abstract

We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism: - We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is but is maximally hard for pseudo-deterministic quantum algorithms ( query complexity). - We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms ( vs. ), while the randomized query complexity is . Complementing these separations, we show that for any total problem , pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., . On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, -sum, and graph collision.
Paper Structure (37 sections, 37 theorems, 39 equations, 4 algorithms)

This paper contains 37 sections, 37 theorems, 39 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Query complexity separations
  4. Hardness of quantum pseudo-determinism.
  5. Power of quantum pseudo-determinism.
  6. Fundamental properties of quantum pseudo-determinism.
  7. Deterministic emulation of total problem.
  8. General pseudo-deterministic quantum query upper bound.
  9. Making quantum algorithms pseudo-deterministic.
  10. Beyond the query complexity model
  11. Search problems with quantum inputs.
  12. The white-box setting.
  13. Techniques
  14. A maximal separation between randomized and pseudo-deterministic quantum algorithms.
  15. Exponential pseudo-deterministic quantum advantage.
  16. ...and 22 more sections

Key Result

Theorem 1.1

There exists a search problem ${R}$ whose classical randomized complexity is $O(1)$ and whose pseudo-deterministic quantum query complexity is $\Omega(N)$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1: Pseudo-deterministic algorithm
  • Definition 2.2: Query complexity measures
  • Definition 2.3: Find1
  • ...and 73 more