  Block encoding is a key ingredient in the recently developed quantum signal
processing that forms a unifying framework for quantum algorithms. Initially
showcased for simplifying and optimizing resource utilization in several
problems, such as searching, amplitude estimation, and Hamiltonian simulation,
the capabilities of the quantum signal processing go beyond these and offer
untapped potential for devising new quantum algorithms. In this article, we
utilize block encoding to substantially enhance two previously proposed quantum
algorithms: largest eigenvalue estimation and quantum gradient descent. Unlike
previous works that involve sophisticated procedures, our findings, using the
unitary block encoding, demonstrate that even with elementary operations, these
new quantum algorithms can eliminate major scaling factors present in their
original counterparts. This yields much more efficient quantum algorithms
capable of tackling complex computational problems with remarkable efficiency.
Furthermore, we show how to extend our proposed method to different contexts,
including matrix inversion and multiple eigenvalues estimation.

Quantum Physics