Two exact quantum signal processing results

• URL: http://arxiv.org/abs/2505.10710v1
• Date: Thu, 15 May 2025 21:13:23 GMT
• Title: Two exact quantum signal processing results
• Authors: Bjorn K. Berntson, Christoph Sünderhauf,
• Abstract summary: Quantum signal processing (QSP) is a framework for implementing certain functions via quantum circuits.<n>To construct a QSP circuit, one needs a target $P(z)$, which must satisfy $lvert P(z)rvertleq 1 on the complex unit circle $mathbb$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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