Product Decomposition of Periodic Functions in Quantum Signal Processing

Jeongwan Haah

Microsoft Quantum and Microsoft Research, Redmond, Washington, USA

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We consider an algorithm to approximate complex-valued periodic functions $f(e^{i\theta})$ as a matrix element of a product of $SU(2)$-valued functions, which underlies so-called quantum signal processing. We prove that the algorithm runs in time $\mathcal O(N^3 \mathrm{polylog}(N/\epsilon))$ under the random-access memory model of computation where $N$ is the degree of the polynomial that approximates $f$ with accuracy $\epsilon$; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.

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