77 research outputs found
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
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.Comment: 21 page
Symmetric functions of qubits in an unknown basis
Consider an n qubit computational basis state corresponding to a bit string
x, which has had an unknown local unitary applied to each qubit, and whose
qubits have been reordered by an unknown permutation. We show that, given such
a state with Hamming weight |x| at most n/2, it is possible to reconstruct |x|
with success probability 1 - |x|/(n-|x|+1), and thus to compute any symmetric
function of x. We give explicit algorithms for computing whether or not |x| is
at least t for some t, and for computing the parity of x, and show that these
are essentially optimal. These results can be seen as generalisations of the
swap test for comparing quantum states.Comment: 6 pages, 3 figures; v2: improved results, essentially published
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A Survey of Quantum Property Testing
The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow “far” from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions. Contents
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