Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Details Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Publishing date

2006-09-14

URL

arxiv.org/abs/quant-ph/0609110v1

Journals

Proc. 24th Symposium on Theoretical Aspects of Computer Science (STACS 2007), Lecture Notes in Computer Science 4393, pp. 598-

Science

Physics

Field

Quantum Physics

Additional info

21 pages

Type

Scientific paper

Digital Object Identifier

10.1007/978-3-540-70918-3_51

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.

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