  We introduce a quantum extension of dynamic programming, a fundamental
computational method for efficiently solving recursive problems using memory.
Our innovation lies in showing how to coherently generate unitaries of
recursion steps using memorized intermediate quantum states. We find that
quantum dynamic programming yields an exponential reduction in circuit depth
for a large class of fixed-point quantum recursions, including a known
recursive variant of the Grover's search. Additionally, we apply quantum
dynamic programming to a recently proposed double-bracket quantum algorithm for
diagonalization to obtain a new protocol for obliviously preparing a quantum
state in its Schmidt basis, providing a potential pathway for revealing
entanglement structures of unknown quantum states.
