Parametrizētu algoritmu paātrinājumi kvantu datoram

Abstract

Darbā tiek aplūkots, kā uz kvantu datora iespējams paātrināt zināmus parametrizētus algoritmus problēmām “Closest string”, “Cluster vertex deletion”, “Cluster editing”, “Vertex cover” un “Longest path” problēmu risinājumiem, ka arī to uzlabošana kvantu datoram, izmantojot kvantu algoritmus meklēšanas koka apstaigāšanai un dinamiskai programmēšanai. Ir parādīts, kā izveidot kvantu algoritmus “Cluster vertex deletion” problēmas risinājumam ar laika sarežģītību 𝑂(1.7321^𝑘 * √𝑘 * 𝑛^3) un “Vertex cover” problēmai ar laiku 𝑂(1.175^𝑘 * 𝑘^𝑂(1) + 𝑛√𝑚), kas ir labāk, nekā labākajiem zināmajiem algoritmiem ar laiku 𝑂(1.9102^𝑘 * (𝑛 + 𝑚)) un 𝑂(1.2738^𝑘 + 𝑘𝑛), attiecīgi. Ka arī ir parādīts, kā izveidot kvantu algoritmus “Closest string” problēmai ar laiku 𝑂(𝑘𝐿 + k^2 * 𝑑^(3/2) * (√(d + 1))^𝑑), “Cluster editing” problēmai ar laiku 𝑂(1.7321^𝑘 * √𝑘 * 𝑛^3) un “Longest path” problēmai ar laiku 𝑂((1.7274√𝑒)^𝑘 * 𝑛^𝑂(1)), kas nav labāk, nekā ātrākiem pazīstamiem algoritmiem ar laiku 𝑂(𝐿𝑘 + 𝑘𝑑(|Σ| − 1)^𝑑 ⋅ 2^3.25𝑑), 𝑂(1.62^𝑘 + 𝑚 + 𝑛)) un 𝑂(2^𝑘 ⋅ 𝑝𝑜𝑙𝑦(𝑛, 𝑘)), attiecīgi.

In this work few parametrized algorithms for solving “Closest string”, “Cluster vertex deletion”, “Cluster editing”, “Vertex cover” and “Longest path” problems are described, as well as their improvements for quantum computer, using quantum backtracking algorithm for solution three and quantum algorithm for dynamic programming. There are shown, how to create quantum algorithms for “Vertex cover” problem solving with time complexity 𝑂(1.175^𝑘 * 𝑘^𝑂(1) + 𝑛√𝑚) and for “Cluster vertex deletion” with time 𝑂(1.7321^𝑘 * √𝑘 * 𝑛^3), improving best known classical algorithms with time 𝑂(1.2738^𝑘 + 𝑘𝑛) and 𝑂(1.9102^𝑘 * (𝑛 + 𝑚)) respectively. Also, there are shown, how to create quantum algorithms for “Closest string” with time 𝑂(𝑘𝐿 + k^2 * 𝑑^(3/2) * (√(d + 1))^𝑑), for “Cluster editing” with time 𝑂(1.7321^𝑘 * √𝑘 * 𝑛^3) and for “Longest path” with time 𝑂((1.7274√𝑒)^𝑘 * 𝑛^𝑂(1)), comparing to best known classical algorithms with time 𝑂(𝐿𝑘 + 𝑘𝑑(|Σ| − 1)^𝑑 ⋅ 2^(3.25𝑑)), (1.62^𝑘 + 𝑚 + 𝑛)) and 𝑂(2^𝑘⋅ 𝑝𝑜𝑙𝑦(𝑛, 𝑘)) respectively.