Abstract
This research extends quantum singular value transformation (QSVT) for general bounded operators embedded in unitary operators on possibly infinite-dimensional Hilbert spaces. Through in-depth mathematical exploration, we have achieved a refined operator-theoretic understanding of QSVT, leading to a more streamlined approach. One of the key discoveries is that polynomial transformations in QSVT inherently apply to the entire operator, rather than being contingent on the selection of a specific basis. We expect that this research will pave the way for applying these insights to a broader range of problems in quantum information processing and provide analytical tools for quantum dynamics, such as quantum walks.