x1 and x2 are given as the solutions of x^2 - x + 1 = 0

So if we substitute x with x1 we get x1^2 - x1 + 1 = 0

=> x1^2 - x1 = -1

Similarly with x2, x2^2 - x2 + 1 = 0

=> x2^2 - x2 = -1

The required sum is (x1^2-x1)^1000 + (x2^2-x2)^1000

=> -1^1000 + -1^1000

=> 1 + 1

=> 2

The required sum is 2.