The paper presents a new quantum algorithm for the element distinctness problem, which involves determining whether a set of numbers contains any repeated elements. The algorithm achieves an $O(N^{2/3})$ query complexity, improving upon the previous best-known quantum algorithm by Buhrman et al., which required $O(N^{3/4})$ queries. This result matches the lower bound established by Aaronson and Shi. The algorithm combines quantum search on graphs and quantum walks, reducing the problem to searching a graph where vertices represent subsets of the input set, and marked vertices represent pairs of equal elements. The quantum walk is designed to gather amplitudes at marked vertices, leading to a high probability of measuring a marked state after $O(N^{2/3})$ steps. The paper also extends the algorithm to the $k$-distinctness problem, where the goal is to find $k$ equal elements among $N$ items, achieving an $O(N^{k/(k+1)})$ query complexity. Additionally, the paper discusses the use of $d$-wise independent functions to optimize the algorithm's running time and space complexity.The paper presents a new quantum algorithm for the element distinctness problem, which involves determining whether a set of numbers contains any repeated elements. The algorithm achieves an $O(N^{2/3})$ query complexity, improving upon the previous best-known quantum algorithm by Buhrman et al., which required $O(N^{3/4})$ queries. This result matches the lower bound established by Aaronson and Shi. The algorithm combines quantum search on graphs and quantum walks, reducing the problem to searching a graph where vertices represent subsets of the input set, and marked vertices represent pairs of equal elements. The quantum walk is designed to gather amplitudes at marked vertices, leading to a high probability of measuring a marked state after $O(N^{2/3})$ steps. The paper also extends the algorithm to the $k$-distinctness problem, where the goal is to find $k$ equal elements among $N$ items, achieving an $O(N^{k/(k+1)})$ query complexity. Additionally, the paper discusses the use of $d$-wise independent functions to optimize the algorithm's running time and space complexity.