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Description
Quantum signal processing (QSP) refers to the task of implementing a given function f on a quantum signal, which typically corresponds to the eigenvalues of an input Hermitian matrix H. QSP is employed in many modern fault-tolerant quantum algorithms, including those for Hamiltonian simulation, matrix inversion, solving differential equations and optimization. The quantum circuits proposed in the literature for implementing QSP for a polynomial function have optimal size. However, given a general function, computing the description of a circuit that implements a polynomial approximation of the function with optimal error scaling requires calculation of certain rotation angles on a classical computer, which limits the overall complexity of QSP. In this work, we use ideas from the theory of interpolating polynomials to construct a simple circuit for implementing QSP without angle finding. This circuit enables implementation of QSP for any continuous black-box function f with nearly optimal complexity, including the classical operations required for computing a description of the circuit.
