Quantum estimation of Chernoff bound and Hoeffding bound

Determining the closeness of two quantum states with minimal error probability is a central problem in quantum information theory. The optimal exponents characterizing the error probability in the symmetric and asymmetric settings are given by the quantum Chernoff bound (QCB) and the quantum Hoeffding bound (QHB), respectively. In this work, we introduce the first algorithms for estimating the QCB and QHB of two unknown states. We propose a unitary encoding for diagonal density matrix based on the copy matrix, which enables unbiased estimation of nonlinear matrix functions via quantum singular value transformation (QSVT). For general density matrices, we combine QSVT with the Hadamard test to estimate both QCB and QHB. Numerical simulations demonstrate that the algorithm converges to the optimal value within a limited number of iterations. Furthermore, we establish the first integration of QSVT with convex optimization, yielding a hybrid quantum-classical framework for quantum estimation.