Improved Quantum Algorithms for Eigenvalues Finding and Gradient Descent

Abstract: Block encoding is a key ingredient in the recently developed quantum singular value transformation (QSVT) framework, which provides a unifying description for many quantum algorithms. Initially introduced to simplify and optimize resource utilization in various problems, such as searching, amplitude estimation, and Hamiltonian simulation, it is reasonable to expect that the capabilities of QSVT extend beyond these applications and offer untapped potential for designing new quantum algorithms. In this article, we affirm this perspective by leveraging block encoding to substantially enhance two previously proposed quantum algorithms: largest eigenvalue estimation and quantum gradient descent. Unlike previous works that rely on sophisticated approaches, our findings demonstrate that even just elementary operations within the unitary block encoding framework can eliminate major scaling factors present in their original counterparts. This results in significantly more efficient quantum algorithms capable of tackling target computational problems with remarkable efficiency. Furthermore, we illustrate how our proposed method can be extended to other contexts, including matrix inversion and multiple eigenvalue estimation.