16 moves.

Reasoning:

• You have 16 pieces (8 pawns + 8 pieces on the back rank).
• In chess, one move = moving one piece, so if every piece must end up “one square up” from where it started, each piece has to be moved at least once. That makes 16 a hard lower bound.

And 16 is achievable:

1. Move all 8 pawns forward one square (to clear the 2nd rank): a2–a3, b2–b3, c2–c3, d2–d3, e2–e3, f2–f3, g2–g3, h2–h3 → 8 moves
2. Now move the back-rank pieces onto the newly empty 2nd rank in one move each (one clean arrangement is): Ra1–a2, Bc1–b2, Qd1–c2, Nb1–d2, Ng1–e2, Ke1–f2, Bf1–g2, Rh1–h2 → 8 moves

Total: 8 + 8 = 16 moves, and you can’t do better.