A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem 论文

摘要

We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$. In this problem an oracle computes a function f on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast, the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of $D_N$ is essentially the abelian quantum Fourier transform. This yields the name of a group representation of $D_N$, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of $D_N$. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.