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Sampling Fourier Transforms on Different Domains

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Lisa Hales, Sean Hallgren

Quantum Physics · Submitted 21 Dec 1998

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

The authors' main claim In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can be efficiently approximated by transforming over ${\mathbb Z}_q$ for any q in a large range.

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arXiv IDquant-ph/9812060 (version 1)
Pith IDUEYEGS37, Pith's permanent record of this paper and its review
CitationLisa Hales, Sean Hallgren. "Sampling Fourier Transforms on Different Domains." arXiv:quant-ph/9812060 (1998). https://pith.science/paper/UEYEGS37
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,618 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can be efficiently approximated by transforming over ${\mathbb Z}_q$ for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.

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 isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. contribution 0.97 0.87 0.11 asserted 0.99
s1 In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can be efficiently approximated by transforming over ${\mathbb Z}_q$ for any q in a large range. result 0.97 0.93 0.76 asserted 1.00
s2 Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. contribution 0.48 0.87 0.18 asserted 1.00
s3 In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions. contribution 0.56 0.90 0.06 asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

What was examined, and what was not

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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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 isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can be efficiently approximated by transforming over ${\mathbb Z}_q$ for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.

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