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Product Decomposition of Periodic Functions in Quantum Signal Processing

We consider an algorithm to approximate complex-valued periodic functions $f(e^{iθ})$ 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 $\mathcal O(N^3 \mathrm{polylog}(N/ε))$ under the random-access memory model of computation where $N$ is the degree of the polynomial that approximates $f$ with accuracy $ε$; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.