A Note on the Quantum Collision and Set Equality Problems

The results showing a quantum query complexity of $\Theta(N^{1/3})$ for the collision problem do not apply to random functions. The issues are two-fold. First, the $\Omega(N^{1/3})$ lower bound only applies when the range is no larger than the domain, which precludes many of the cryptographically interesting applications. Second, most of the results in the literature only apply to $r$-to-1 functions, which are quite different from random functions. Understanding the collision problem for random functions is of great importance to cryptography, and we seek to fill the gaps of knowledge for this problem. To that end, we prove that, as expected, a quantum query complexity of $\Theta(N^{1/3})$ holds for all interesting domain and range sizes. Our proofs are simple, and combine existing techniques with several novel tricks to obtain the desired results. Using our techniques, we also give an optimal $\Omega(N^{1/3})$ lower bound for the set equality problem. This new lower bound can be used to improve the relationship between classical randomized query complexity and quantum query complexity for so-called permutation-symmetric functions.