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Sharp Error Bounds on Quantum Boolean Summation in Various Settings

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Marek Kwas, Henryk Wozniakowski

Quantum Physics · Submitted 10 Mar 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting.

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arXiv IDquant-ph/0303049 (version 1)
Pith IDOCF24SUJ, Pith's permanent record of this paper and its review
CitationHenryk Wozniakowski, Marek Kwas. "Sharp Error Bounds on Quantum Boolean Summation in Various Settings." arXiv:quant-ph/0303049 (2003). https://pith.science/paper/OCF24SUJ
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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.71 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.02 · states quantitative result 0.98 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to $M^{-1}^$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. background 0.81 0.18 0.04 asserted 0.97
s1 We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. contribution 1.00 0.92 0.31 asserted 1.00
s2 In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ and show they are sharp for large $M$ and $NM^{-1}$. result 0.99 0.93 0.46 asserted 1.00
s3 In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. result 0.99 0.94 0.24 asserted 1.00
s4 This bound is optimal, as recently shown in [10]. interpretation 0.70 0.05 0.09 asserted 1.00
s5 For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to $M^{-1}^$. result 0.86 0.74 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to $M^{-1}^$.

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