Quantum Fourier Sampling is Guaranteed to Fail to Compute Automorphism Groups of Easy Graphs

TL;DR

This paper demonstrates that the quantum hidden subgroup approach cannot solve certain large classes of graph automorphism problems, indicating fundamental limitations of this method for graph isomorphism challenges.

Contribution

It systematically constructs large classes of easy graph automorphism problems where the hidden subgroup approach is provably doomed to fail, highlighting its limitations.

Findings

Hidden subgroup approach fails for arbitrarily large easy graph automorphism problems.

Failure extends to graph isomorphism problems due to their relation.

Suggests need for alternative quantum algorithms for these problems.

Abstract

The quantum hidden subgroup approach is an actively studied approach to solve combinatorial problems in quantum complexity theory. With the success of the Shor's algorithm, it was hoped that similar approach may be useful to solve the other combinatorial problems. One such problem is the graph isomorphism problem which has survived decades of efforts using the hidden subgroup approach. This paper provides a systematic approach to create arbitrarily large classes of classically efficiently solvable graph automorphism problems or easy graph automorphism problems for which hidden subgroup approach is guaranteed to always fail irrespective of the size of the graphs no matter how many copies of coset states are used. As the graph isomorphism problem is believed to be at least as hard as the graph automorphism problem, the result of this paper entails that the hidden subgroup approach is also guaranteed to always fail for the arbitrarily large classes of graph isomorphism problems. Combining these two results, it is argued that the hidden subgroup approach is essentially a dead end and alternative quantum algorithmic approach needs to be investigated for the graph isomorphism and automorphism problems.