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Explicit relation between all lower bound techniques for quantum query complexity
 In Proceedings of 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013), 2013. arXiv:1209.2713. 17
"... The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof ..."
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The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
Lower Bounds on Quantum Query and Learning Graph Complexities
, 2014
"... In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph co ..."
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In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, nonadaptive learning graphs. We express both adaptive and nonadaptive learning graph complexities of Booleanvalued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learning graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)
General Terms
"... We prove a tight quantum query lower bound Ω(nk/(k+1)) for the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is sufficiently large. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complex ..."
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We prove a tight quantum query lower bound Ω(nk/(k+1)) for the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is sufficiently large. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complex
Optimal parallel quantum query algorithms
"... We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number o ..."
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We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Θ((n/p)2/3) pparallel queries for element distinctness and Θ((n/p)k/(k+1)) for ksum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical pparallel complexity are polynomially related for all total functions when p is not too large.