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

TL;DR

This paper generalizes quantum signal processing and singular value transformation to $U(N)$, enabling simultaneous polynomial transformations, with applications including bi-variate functions, efficient decision algorithms, and amplitude estimation at the Heisenberg limit.

Contribution

It introduces a framework for quantum signal processing on $U(N)$, allowing multiple polynomial transformations and providing recursive algorithms for circuit construction.

Findings

Achieves $O(d)$ query complexity for $N$-interval decision with $ ext{log}_2 N$ improvement.

Demonstrates a quantum amplitude estimation algorithm reaching the Heisenberg limit without adaptive measurements.

Provides a comprehensive characterization of achievable polynomial matrices on $U(N)$.

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 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.