  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.
