The quantum query complexity of the abelian hidden subgroup problem

AbstractSimon, in his FOCS’94 paper, was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon’s problem from the point of view of quantum query complexity and give here a first non-trivial lower bound on the query complexity of a hidden subgroup problem, namely Simon’s problem. More generally, we give a lower bound which is optimal up to a constant factor for any abelian group

The Quantum Query Complexity of the Abelian Hidden Subgroup Problem.

Simon in his FOCS'94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon's problem from the point of view of quantum query complexity and give here a first nontrivial lower bound on the query complexity of a hidden subgroup problem, namely Simon's problem. Our bound is optimal up to a constant factor. We also show how, as a consequence, this gives us the query complexity of the Abelian hidden subgroup problem, up to a constant factor. At last we expose some elementary facts about complexity in weaker query models.Dans son article de FOCS’94, Simon fut le premier à montrer un cas où le calcul quantique permet une accélération exponentielle par rapport au calcul classique. Il s’agissait d’un problème qui fait partie d’une classe de problèmes aujourd’hui très étudiés, les problèmes de sous-groupes cachés. Nous étudions le problème de Simon du point de vue de la complexité en requêtes quantiques. Nous donnons une première borne inférieure non triviale sur cette complexité et montrons comment on obtient en conséquence la complexité en requêtes quantiques du problème du sous-groupe caché Abélien, à un facteur constant près. Nous présentons enfin quelques résultat élémentaires de complexité pour des modèles de requêtes plus faible

A quantum view on convex optimization

In this dissertation we consider quantum algorithms for convex optimization. We start by considering a black-box setting of convex optimization. In this setting we show that quantum computers require exponentially fewer queries to a membership oracle for a convex set in order to implement a separation oracle for that set. We do so by proving that Jordan's quantum gradient algorithm can also be applied to find sub-gradients of convex Lipschitz functions, even though these functions might not even be differentiable. As a corollary we get a quadraticly faster algorithm for convex optimization using membership queries. As a second set of results we give sub-linear time quantum algorithms for semidefinite optimization by speeding up the iterations of the Arora-Kale algorithm. For the problem of finding approximate Nash equilibria for zero-sum games we then give specific algorithms that improve the error-dependence and only depend on the sparsity of the game, not it's size. These last results yield improved algorithms for linear programming as a corollary. We also show several lower bounds in these settings, matching the upper bounds in most or all parameters