The original estimate
Pfenning and Ford applied quantum inequalities, limits from quantum field theory on how much negative energy can be concentrated and for how long, to the Alcubierre bubble. The wall can then be only a few hundred Planck lengths thick, and the total energy needed to maintain the bubble becomes, in their words, physically unattainable.
Shrinking the bill
Van Den Broeck kept a large interior behind a microscopic outer surface. His geometry needs a negative mass of the order of a few solar masses together with a comparable amount of positive energy, and it satisfies the quantum inequality. Bobrick and Martire later presented optimizations that reduce the negative energy of the Alcubierre metric by two orders of magnitude.
Positive energy, but not small
The slower-than-light warp drive of Fuchs et al. (2024) satisfies all energy conditions by combining the warp shift with a shell of ordinary matter. That removes the need for negative energy but requires a massive shell. It is a proof of principle, not a fuel budget.