Faster isomorphism testing of strongly regular graphs
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum Computation and Lattice Problems
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Hidden translation and orbit coset in quantum computing
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Fast quantum algorithms for computing the unit group and class group of a number field
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem
SIAM Journal on Computing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
A polynomial quantum algorithm for approximating the Jones polynomial
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Limitations of quantum coset states for graph isomorphism
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
Journal of the ACM (JACM)
The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts
SIAM Journal on Computing
The Symmetric Group Defies Strong Fourier Sampling
SIAM Journal on Computing
Quantum algorithms for Simon's problem over nonabelian groups
ACM Transactions on Algorithms (TALG)
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It is known that any quantum algorithm for graph isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across $\Omega(n\log n)$ coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve” starts with coset states and works its way down toward representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for graph isomorphism. Specifically, we consider the natural reduction of graph isomorphism to the HSP over the wreath product $S_n\wr\mathbb{Z}_2$. Using a recently proved bound on the irreducible characters of $S_n$, we show that no algorithm in this family can solve graph isomorphism in less than $\mathrm{e}^{\Omega(\sqrt{n})}$ time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time $\mathrm{e}^{O(\sqrt{n\log n})}$.