The problem of finding the smallest eigenvalue of a Hermitian matrix (also called the ground state energy) has wide applications in quantum physics. In this talk, I will first briefly introduce the mathematical setup of quantum algorithms, and discuss how to use textbook quantum algorithms to tackle this problem. I will then introduce a new quantum algorithm that can significantly and provably reduce the circuit depth for solving this problem (the reduction can be around two orders of magnitude). This algorithm reduces the requirement on the maximal coherent time for the quantum computer, and can therefore be suitable for early fault-tolerant quantum devices. No prior knowledge on quantum algorithms is necessary for understanding most parts of the talk. (Joint work with Zhiyan Ding)