Weak FourierSchur sampling, the hidden subgroup problem, and the quantum collision problem
Abstract
Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.
 Publication:

arXiv eprints
 Pub Date:
 September 2006
 DOI:
 10.48550/arXiv.quantph/0609110
 arXiv:
 arXiv:quantph/0609110
 Bibcode:
 2006quant.ph..9110C
 Keywords:

 Quantum Physics
 EPrint:
 21 pages