Optimal Parallel Quantum Query Algorithms

Authors: Stacey Jeffery Frederic Magniez Ronald de Wolf

Publish Date: 2016/09/08

Volume: 79, Issue: 2, Pages: 509-529

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Abstract

We study the complexity of quantum query algorithms that make p queries in parallel in each timestep This model is in part 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 Theta n/p2/3 pparallel queries for element distinctness and Theta n/pk/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 query complexity are polynomially related for all total functions f when p is small compared to f’s block sensitivityPartially supported by the French ANR Blanc project ANR12BS02005 RDAM a Vidi grant from the Netherlands Organization for Scientific Research NWO ERC Consolidator grant QPROGRESS the European Commission IST STREP projects Quantum Computer Science QCS 255961 Quantum Algorithms QALGO 600700 and the US ARO An extended abstract of this paper appeared in the Proceedings of the 22nd European Symposium on Algorithms ESA’14 pp 592–604The proof is a straightforward adaptation of the proof of 15 Theorem 9 but we repeat it here for completeness Let w SJSJin mathcal E p and theta SJMSJin mathcal E p Min mathcal C be an optimal solution to the primal formulation of mathrmLGCpparallel mathcal CFirst we use a variation of the adversary bound from 17 that allows the duplication of row and column indices Concretely rows and columns of Gamma are now indexed by x a and y b respectively where xin f11 yin f10 and a and b belong to some finite sets Then with slight abuse of notation Delta J is now defined such that Delta Jxayb=1 if x Jne y J and Delta Jxayb=0 otherwise Specifically in our case rows of Gamma will be indexed by x M for some xin f11 and Min mathcal C and columns will simply be indexed by yin f10Second Gamma will be the submatrix of a larger matrix widetildeGamma defined below whose rows are indexed by the elements of qntimes mathcal C and whose columns are indexed by qn Then Delta J is naturally extended to all xyin qn and Min mathcal C by tildeDelta JxMy=1 if x Jne y J and tildeDelta JxMy=0 otherwise Since Gamma circ Delta J is a submatrix of widetildeGamma circ tildeDelta J we will have Gamma circ Delta Jle widetildeGamma circ tildeDelta J Hence it suffices to upper bound the latter norm

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