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.

翻译：我们提出了一种用于克拉夫丘克变换的量子算法，其在维度和误差参数的逆上均呈现对数尺度缩放。该量子克拉夫丘克变换将计算基态映射到振幅正比于克拉夫丘克函数的态。我们通过结合两项关键技术来实现这一目标：克拉夫丘克变换与李代数 $\mathfrak{su}(2)$ 之间的结构关系，以及最近提出的在谐振子表示中对 $\mathfrak{su}(2)$ 算符进行快速推进模拟的方法。更确切地说，我们首先建立了从计算基中的克拉夫丘克变换到福克基中 $\mathfrak{su}(2)$ 的映射。接着，基于这一联系，我们应用快速推进方法以实现高效的量子克拉夫丘克变换。