Explore an advanced quantum computing lecture on solving linear systems using a discrete adiabatic theorem approach. Delve into the development of an asymptotically optimal quantum algorithm with linear complexity in the condition number, matching known lower bounds. Examine the rigorous proof of the discrete adiabatic theorem, its application to quantum linear systems, and the algorithm's simplified implementation. Investigate the constant factors, gate count complexities, and potential applications. Compare this method to existing suboptimal approaches and understand its advantages in terms of precision and efficiency.