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  "path": "/papers/q-2026-03-27-2048/",
  "publishedAt": "2026-03-27T11:55:02.000Z",
  "site": "https://quantum-journal.org",
  "tags": [
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    "https://doi.org/10.22331/q-2026-03-27-2048"
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  "textContent": "Quantum 10, 2048 (2026).\n\nhttps://doi.org/10.22331/q-2026-03-27-2048\n\nQuantum 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.",
  "title": "Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$"
}