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Basic quantum subroutines: finding multiple marked elements and summing numbers

Joran van Apeldoorn, Sander Gribling, Harold Nieuwboer

TL;DR

It is shown how to find all quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory.

Abstract

We show how to find all $k$ marked elements in a list of size $N$ using the optimal number $O(\sqrt{N k})$ of quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory. Previous algorithms either incurred a factor $k$ overhead in the gate complexity, or had an extra factor $\log(k)$ in the query complexity. We then consider the problem of finding a multiplicative $δ$-approximation of $s = \sum_{i=1}^N v_i$ where $v=(v_i) \in [0,1]^N$, given quantum query access to a binary description of $v$. We give an algorithm that does so, with probability at least $1-ρ$, using $O(\sqrt{N \log(1/ρ) / δ})$ quantum queries (under mild assumptions on $ρ$). This quadratically improves the dependence on $1/δ$ and $\log(1/ρ)$ compared to a straightforward application of amplitude estimation. To obtain the improved $\log(1/ρ)$ dependence we use the first result.

Basic quantum subroutines: finding multiple marked elements and summing numbers

TL;DR

It is shown how to find all quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory.

Abstract

We show how to find all marked elements in a list of size using the optimal number of quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory. Previous algorithms either incurred a factor overhead in the gate complexity, or had an extra factor in the query complexity. We then consider the problem of finding a multiplicative -approximation of where , given quantum query access to a binary description of . We give an algorithm that does so, with probability at least , using quantum queries (under mild assumptions on ). This quadratically improves the dependence on and compared to a straightforward application of amplitude estimation. To obtain the improved dependence we use the first result.
Paper Structure (15 sections, 26 theorems, 91 equations)

This paper contains 15 sections, 26 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Finding multiple marked elements in a list
  3. Improved quantum summing algorithm
  4. Mean estimation algorithms.
  5. Approximate counting algorithms.
  6. Preliminaries
  7. Notation and assumptions
  8. Computational model
  9. Various quantum subroutines
  10. Fast Grover search for multiple items, without quantum memory
  11. Deterministic Grover for multiple elements
  12. Coupon collecting Grover
  13. Grover for multiple elements, fast
  14. Improved query complexity for approximate summation
  15. Classical lower bound for approximate counting

Key Result

Theorem 1.1

Let $x \in \{0,1\}^N$ with $\lvert x\rvert = k \geq 2$, and let $\rho \in (0,1)$ be such that $k \in \Omega(\log(k/\rho)^3)$ (e.g. $\rho = \Omega(1/\!\mathop{\mathrm{poly}}\nolimits(k))$). Then we can find, with probability $\geq 1 - \rho$, all $k$ indices $i \in [N]$ for which $x_i = 1$ using $O\lp

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2: Informal version of \ref{['thm:fast-summing']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Amplitude estimation bhmt:countingj
  • proof
  • Theorem 2.4: bhmt:countingj
  • Theorem 2.5: bhmt:countingj
  • Lemma 2.6
  • proof
  • ...and 31 more