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Density theorems with applications in quantum signal processing.

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Title: Density theorems with applications in quantum signal processing.
Authors: Sarkar, Rahul1 (AUTHOR) rsarkar@stanford.edu, Yoder, Theodore J.2 (AUTHOR) ted.yoder@ibm.com
Source: Journal of Computational & Applied Mathematics. Oct2023, Vol. 430, pN.PAG-N.PAG. 1p.
Subjects: Signal processing, Polynomial approximation, Heuristic algorithms, Continuous functions, Polynomials, Approximation algorithms
Abstract: We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain [ 0 , 1 ] is physically desired, these polynomial families are defined by bound constraints not just in [ 0 , 1 ] , but also with additional bound constraints outside [ 0 , 1 ]. One might wonder then if these additional constraints inhibit their approximation properties within [ 0 , 1 ]. The main result of this paper is that this is not the case — the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function f : [ 0 , 1 ] → [ 0 , 1 ] in the supremum norm, provided f also matches any polynomial in the family at 0 and 1. We additionally study the specific problem of approximating the step function on [ 0 , 1 ] (with the step from 0 to 1 occurring at x = 1 2 ) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 163549104
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  Data: Density theorems with applications in quantum signal processing.
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  Data: <searchLink fieldCode="AR" term="%22Sarkar%2C+Rahul%22">Sarkar, Rahul</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> rsarkar@stanford.edu</i><br /><searchLink fieldCode="AR" term="%22Yoder%2C+Theodore+J%2E%22">Yoder, Theodore J.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> ted.yoder@ibm.com</i>
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  Data: <searchLink fieldCode="DE" term="%22Signal+processing%22">Signal processing</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomial+approximation%22">Polynomial approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Heuristic+algorithms%22">Heuristic algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Continuous+functions%22">Continuous functions</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+algorithms%22">Approximation algorithms</searchLink>
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  Data: We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain [ 0 , 1 ] is physically desired, these polynomial families are defined by bound constraints not just in [ 0 , 1 ] , but also with additional bound constraints outside [ 0 , 1 ]. One might wonder then if these additional constraints inhibit their approximation properties within [ 0 , 1 ]. The main result of this paper is that this is not the case — the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function f : [ 0 , 1 ] → [ 0 , 1 ] in the supremum norm, provided f also matches any polynomial in the family at 0 and 1. We additionally study the specific problem of approximating the step function on [ 0 , 1 ] (with the step from 0 to 1 occurring at x = 1 2 ) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Computational & Applied Mathematics is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.cam.2023.115243
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      – Code: eng
        Text: English
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        PageCount: 1
        StartPage: N.PAG
    Subjects:
      – SubjectFull: Signal processing
        Type: general
      – SubjectFull: Polynomial approximation
        Type: general
      – SubjectFull: Heuristic algorithms
        Type: general
      – SubjectFull: Continuous functions
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Approximation algorithms
        Type: general
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      – TitleFull: Density theorems with applications in quantum signal processing.
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            NameFull: Sarkar, Rahul
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            NameFull: Yoder, Theodore J.
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            – D: 01
              M: 10
              Text: Oct2023
              Type: published
              Y: 2023
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              Value: 430
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