On the energy landscape of symmetric quantum signal processing

TL;DR

This paper analyzes the complex energy landscape of symmetric quantum signal processing, characterizing global minima and explaining why optimization algorithms reliably find solutions despite non-convexity.

Contribution

It explicitly characterizes all global minima and proves the existence of a strongly convex neighborhood around a particular global minimum, explaining optimization success.

Findings

Explicit characterization of all global minima.

Identification of a strongly convex neighborhood around a global minimum.

Proof that optimization algorithms can reliably find solutions.

Abstract

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.