Quantum Signal Processing and Quantum Singular Value Transformation on U ( N )

Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on U ( N ) , which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on U ( 2 ) in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate N -interval decision achieving O ( d ) query complexity with a log 2 ⁡ N improvement over iterative U ( 2 ) -QSP requiring O ( d log 2 ⁡ N ) queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.