Quantum Eigenvalue Transformations for Arbitrary Matrices

• URL: http://arxiv.org/abs/2604.19688v1
• Date: Tue, 21 Apr 2026 17:11:03 GMT
• Title: Quantum Eigenvalue Transformations for Arbitrary Matrices
• Abstract summary: We propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues.<n>We show that applying QSP to any unitary with this property is equivalent to applying a degree at most $n$ to the block-encoded $0 matrix.<n>We also provide a simple construction that transforms any block encoding into an $n$-regular one using only $O(log n)$ ancillary operations.
• Score: 0.19116784879310025
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $n$-regular block encoding, namely, a block encoding whose $k$-th power reproduces the $k$-th power of the encoded matrix for every $0 < k < n$. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most $n$ to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an $n$-regular one using only $O(\log n)$ 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.

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