  Symmetric quantum signal processing provides a parameterized representation
of a real polynomial, which can be translated into an efficient quantum circuit
for performing a wide range of computational tasks on quantum computers. For a
given polynomial $f$, the parameters (called phase factors) can be obtained by
solving an optimization problem. However, the cost function is non-convex, and
has a very complex energy landscape with numerous global and local minima. It
is therefore surprising that the solution can be robustly obtained in practice,
starting from a fixed initial guess $\Phi^0$ that contains no information of
the input polynomial. To investigate this phenomenon, we first explicitly
characterize all the global minima of the cost function. We then prove that one
particular global minimum (called the maximal solution) belongs to a
neighborhood of $\Phi^0$, on which the cost function is strongly convex under
the condition ${\left\lVert f\right\rVert}_{\infty}=\mathcal{O}(d^{-1})$ with
$d=\mathrm{deg}(f)$. Our result provides a partial explanation of the
aforementioned success of optimization algorithms.
