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We study the hardness of the dihedral hidden subgroup problem. It is known that lattice problems reduce to it, and that it reduces to random subset sum with density $> 1$ and also to quantum sampling subset sum solutions. We examine a decision version of the problem where the question asks whether the hidden subgroup is trivial or order two. The decision problem essentially asks if a given vector is in the span of all coset states. We approach this by first computing an explicit basis for the coset space and the perpendicular space. We then look at the consequences of having efficient unitaries that use this basis. We show that if a unitary maps the basis to the standard basis in any way, then that unitary can be used to solve random subset sum with constant density $>1$. We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density $>1$ but approaching $1$ as the problem size increases. This strengthens the previous hardness result that implementing the optimal POVM in a specific way is as hard as quantum sampling subset sum solutions.
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The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k = 3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achive on two restrictions of the problem,namely (i) when the records values are over a binary alphabet and k = 3, and (ii) when the records have length at most 8 and k = 4, showing that these restrictions of the problem are APX-hard.
When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -- that is, correlations for which the number of measurements and number of measurement outcomes are fixed -- such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.
Suppose a Boolean function $f$ is symmetric under a group action $G$ acting on the $n$ bits of the input. For which $G$ does this mean $f$ does not have an exponential quantum speedup? Is there a characterization of how rich $G$ must be before the function $f$ cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions $G$ which are quantum intolerant in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a well-shuffling property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where $G=S_n$. Our main application is for graph symmetries: we show that any Boolean function $f$ defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power $6$ relationship between its randomized and quantum query complexities, even if $f$ is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).
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Adrien Friggeri
, Eric Fleury (ENS / LIPn Laboratoire de lInformatique du Parallelisme / INRIA Grenoblen Rh^one-Alpes
2011
We show that the problem of finding a set with maximum cohesion in an undirected network is NP-hard.
Recently, it has been stated that single-world interpretations of quantum theory are logically inconsistent. The claim is derived from contradicting statements of agents in a setup combining two Wigners-friend experiments. Those statements stem from applying the measurement-update rule subjectively, i.e., only for the respective agents own measurement. We argue that the contradiction expresses the incompatibility of collapse and unitarity - resulting in different formal descriptions of a measurement - and does not allow to dismiss any specific interpretation of quantum theory.
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