Quantum Kravchuk Transform using $\mathfrak{su}(2)$ fast-forwarding

Abstract: We present a quantum algorithm for the Kravchuk transform that scales logarithmically in both the dimension and the inverse of the error parameter. The quantum Kravchuk transform maps computational basis states to states with amplitudes proportional to Kravchuk functions. We achieve this by combining two key techniques: the structural relationship between the Kravchuk transform and the Lie algebras $\mathfrak{su}(2)$, and a recent fast-forwarding simulation method for $\mathfrak{su}(2)$ operators in the oscillator representation. More precisely, we first establish the map from Kravchuk transform in computational basis to $\mathfrak{su}(2)$ in Fock basis. Then built on this connection, we apply the fast-forwarding to achieve an efficient quantum Kravchuk transform.