Approximate Degrees of Multisymmetric Properties with Application to Quantum Claw Detection

The claw problem is central in the fields of theoretical computer science as well as cryptography. The optimal quantum query complexity of the problem is known to be $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ for input functions $f\colon [F]\to Z$ and $g\colon [G]\to Z$. However, the lower bound was proved when the range $Z$ is sufficiently large (i.e., $|{Z}|=\Omega(FG)$). The current paper proves the lower bound holds even for every smaller range $Z$ with $|{Z}|\ge F+G$. This implies that $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ is tight for every such range. In addition, the lower bound $\Omega\left(\sqrt{G}+F^{1/3}G^{1/6}M^{1/6}\right)$ is provided for even smaller range $Z=[M]$ with every $M\in [2,F+G]$ by reducing the claw problem for $|{Z}|= F+G$. The proof technique is general enough to apply to any $k$-symmetric property (e.g., the $k$-claw problem), i.e., the Boolean function $\Phi$ on the set of $k$ functions with different-size domains and a common range such that $\Phi$ is invariant under the permutations over each domain and the permutations over the range. More concretely, it generalizes Ambainis's argument [Theory of Computing, 1(1):37-46] to the multiple-function case by using the notion of multisymmetric polynomials.