Applications of Adversary Method in Quantum Query Algorithms
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Latvijas Universitāte
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eng
Abstract
Disertācijā es izmantoju nesen izstrādātu kvantu vaicājumu sarežģītības precīzu raksturojumu - adversary metodi - lai konstruētu jaunus kvantu algoritmus un apakšējos novērtējumus. Rezultāti ir sekojoši: tika izstrādāta jauna tehnika kvantu algoritmu konstruēšanai: mācīšanas grafi; mācīšanas grafi tika izmantoti lai uzlabotu kvantu vaicājumu sarežģītību trijstūra atrašanas un k-atšķirīguma problēmām; tika pierādīti precīzi apakšējie novērtējumi k-sumas un trijsūra summas problēmām; tika uzbūvēti kvantu algoritmi dažu apakšgrafu meklēšanas problēmām, kas ir optimāli vaicājumu, laik un atmiņas ziņā;tika izstrādāts kvantu klejošanas vispārinājums, kas savieno grafa elektriskās īpašības ar kvantu klejošanas soļu skaitu. Tas tika izmantots, lai izstrādātu laika-efektīvu kvantu algoritmu 3-atšķirīguma problēmai.
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows:we develop a new technique for the construction of quantum algorithms: learning graphs. We use learning graphs to improve quantum query complexity of the triangle detection and the k-distinctness problems. We prove toght lower bounds for the k-sun and the triangle sum problems. We construct quantum algorithms for some subgraph finding problems that are optimal in terms of query, time and space compexities. We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows:we develop a new technique for the construction of quantum algorithms: learning graphs. We use learning graphs to improve quantum query complexity of the triangle detection and the k-distinctness problems. We prove toght lower bounds for the k-sun and the triangle sum problems. We construct quantum algorithms for some subgraph finding problems that are optimal in terms of query, time and space compexities. We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.