Abstract
Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an -regular block encoding, namely, a block encoding whose -th power reproduces the -th power of the encoded matrix for every . We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an -regular one using only ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix.