Hidden Subgroup States are Almost Orthogonal

TL;DR

This paper demonstrates that quantum states for hidden subgroups are nearly orthogonal in both Abelian and noncommutative groups, enabling an exponential speedup in oracle calls for subgroup identification, though with exponential time complexity.

Contribution

It extends the understanding of hidden subgroup states to noncommutative groups and presents a quantum algorithm with exponential oracle call efficiency.

Findings

Quantum states for different subgroups are almost orthogonal.

Quantum algorithms require exponentially fewer oracle calls than classical methods.

The proposed quantum algorithm operates in linear oracle call complexity but exponential time.

Abstract

It is well known that quantum computers can efficiently find a hidden subgroup $H$ of a finite Abelian group $G$. This implies that after only a polynomial (in $\log |G|$) number of calls to the oracle function, the states corresponding to different candidate subgroups have exponentially small inner product. We show that this is true for noncommutative groups also. We present a quantum algorithm which identifies a hidden subgroup of an arbitrary finite group $G$ in only a linear (in $\log |G|$) number of calls to the oracle function. This is exponentially better than the best classical algorithm. However our quantum algorithm requires an exponential amount of time, as in the classical case.