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

• URL: http://arxiv.org/abs/2408.01439v2
• Date: Fri, 4 Oct 2024 01:34:25 GMT
• Title: Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$
• Authors: Xi Lu, Yuan Liu, Hongwei Lin,
• Abstract summary: Quantum signal processing and quantum value transformation are powerful tools to implement transformations of block-encoded matrices on quantum computers. We propose a framework which realizes multiples simultaneously from a block-encoded input. We also give an algorithm to construct the quantum circuit that gives the desired transformation.
• Score: 8.264300525515097
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 also perform a comprehensive analysis on achievable polynomials and give a recursive algorithm to construct the quantum circuit that gives the desired polynomial transformation. As two example applications, we propose a framework to realize bi-variate polynomial functions, and study the quantum amplitude estimation algorithm with asymptotically optimal query complexity.

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