Hermitian Matrix Function Synthesis without Block-Encoding

Implementing arbitrary functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advance quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix, by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding by expressing the target Hermitian matrix as a symmetric combination of unitary conjugates, enabling polynomial synthesis via GQSP circuits applied to each unitary component. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead, and opens new pathways for quantum algorithm design for functions of Hermitian matrices.