2 research outputs found
Hidden Translation and Translating Coset in Quantum Computing
We give efficient quantum algorithms for the problems of Hidden Translation
and Hidden Subgroup in a large class of non-abelian solvable groups including
solvable groups of constant exponent and of constant length derived series. Our
algorithms are recursive. For the base case, we solve efficiently Hidden
Translation in , whenever is a fixed prime. For the induction
step, we introduce the problem Translating Coset generalizing both Hidden
Translation and Hidden Subgroup, and prove a powerful self-reducibility result:
Translating Coset in a finite solvable group is reducible to instances of
Translating Coset in and , for appropriate normal subgroups of
. Our self-reducibility framework combined with Kuperberg's subexponential
quantum algorithm for solving Hidden Translation in any abelian group, leads to
subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in
any solvable group.Comment: Journal version: change of title and several minor update
Optimal phase estimation in quantum networks
We address the problem of estimating the phase phi given N copies of the
phase rotation u(phi) within an array of quantum operations in finite
dimensions. We first consider the special case where the array consists of an
arbitrary input state followed by any arrangement of the N phase rotations, and
ending with a POVM. We optimise the POVM for a given input state and fixed
arrangement. Then we also optimise the input state for some specific cost
functions. In all cases, the optimal POVM is equivalent to a quantum Fourier
transform in an appropriate basis. Examples and applications are given.Comment: 9 pages, 2 figures; this is an extended version of
arXiv:quant-ph/0609160. v2: minor corrections in reference