Abstract
The recently developed quantum signal processing (QSP) and its generalization called quantum singular value transformation (QSVT) provide a "grand unification" of a large class of quantum algorithms. Mathematically, QSP provides unitary representation of a general set of polynomials, parameterized by a set of "phase factors". Given phase factors, it is easy to compute the polynomial. However, for a given polynomial, it is a considerably harder task to find the corresponding phase factors. A few years ago, finding 30 such phase factors for the Hamiltonian simulation problem was considered to be numerically challenging. In this talk, I will propose an optimization based approach, which is able to stably find phase factors of length > 10,000 for a wide range of tasks, using standard double precision arithmetic operations. To understand the energy landscape, I will describe an algorithm to identify (using exact arithmetic operations) all global minima of this optimization problem. If time allows, I will also introduce a challenging problem of finding phase factors for approximating a special complex function, and how to use it to define a new quantum benchmark based on the Hamiltonian simulation problem.