Complementary polynomials in quantum signal processing
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915340934447104 |
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| author | Berntson, Bjorn K. Sünderhauf, Christoph |
| author_facet | Berntson, Bjorn K. Sünderhauf, Christoph |
| contents | Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of $Q$ in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complementary polynomials in quantum signal processing Berntson, Bjorn K. Sünderhauf, Christoph Quantum Physics Mathematical Physics Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of $Q$ in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided. |
| title | Complementary polynomials in quantum signal processing |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2406.04246 |