Comment by ryandrake

With a traditional rocket, I believe you'd need decreasing energy to maintain the same acceleration as the flight progressed, since you are carrying along with you and burning the fuel, and so the total mass (payload + fuel) that needs to be accelerated is constantly decreasing.

Of course there's the pesky problem that for every N kg of mass you want to accelerate at 1G for that kind of a trip, you're probably going to need somewhere on the order of N billion kg of fuel to burn.