Two exact quantum signal processing results
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909612134891520 |
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| author | Berntson, Bjorn K. Sünderhauf, Christoph |
| author_facet | Berntson, Bjorn K. Sünderhauf, Christoph |
| contents | Quantum signal processing (QSP) is a framework for implementing certain polynomial functions via quantum circuits. To construct a QSP circuit, one needs (i) a target polynomial $P(z)$, which must satisfy $\lvert P(z)\rvert\leq 1$ on the complex unit circle $\mathbb{T}$ and (ii) a complementary polynomial $Q(z)$, which satisfies $\lvert P(z)\rvert^2+\lvert Q(z)\rvert^2=1$ on $\mathbb{T}$. We present two exact mathematical results within this context. First, we obtain an exact expression for a certain uniform polynomial approximant of $1/x$, which is used to perform matrix inversion via quantum circuits. Second, given a generic target polynomial $P(z)$, we construct the complementary polynomial $Q(z)$ exactly via integral representations, valid throughout the entire complex plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two exact quantum signal processing results Berntson, Bjorn K. Sünderhauf, Christoph Quantum Physics Quantum signal processing (QSP) is a framework for implementing certain polynomial functions via quantum circuits. To construct a QSP circuit, one needs (i) a target polynomial $P(z)$, which must satisfy $\lvert P(z)\rvert\leq 1$ on the complex unit circle $\mathbb{T}$ and (ii) a complementary polynomial $Q(z)$, which satisfies $\lvert P(z)\rvert^2+\lvert Q(z)\rvert^2=1$ on $\mathbb{T}$. We present two exact mathematical results within this context. First, we obtain an exact expression for a certain uniform polynomial approximant of $1/x$, which is used to perform matrix inversion via quantum circuits. Second, given a generic target polynomial $P(z)$, we construct the complementary polynomial $Q(z)$ exactly via integral representations, valid throughout the entire complex plane. |
| title | Two exact quantum signal processing results |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2505.10710 |