| Co-Investigator(Kenkyū-buntansha) |
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate, 総合科学部, 助教授 (90192509)
IKEDA Takeshi Okayama University of Science, Faculty of Mathematics, Assistant, 理学部, 助手 (40309539)
SAKATA Toshio Kyushu Institute of Design, Faculty of Design, Professor, 教授 (20117352)
OHE Takashi Okayama University of Science, Mathematical Information Science, Instructor, 総合情報学部, 助教授 (90258210)
KAJIMOTO Hiroshi Nagasaki University, Faculty of Education, Associate Professor, 教育学部, 助教授 (50194741)
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| Research Abstract |
There are many applications of Groebner basis to various fields. The starting point of our research is to calculate Markov basis for the set of contingency tables, which have non-negative integers as their elements and are fixedwith some given row and column sums. Since Markov basis is closely related to Groebner basis, we need only to calculate Groebner basis for the ideal of a polynomial ring over integer defined by these tables. As our results of this research line, we calculated the Groebner basis of 3x3x4, 3x4x4, 4x4x4, etc for three way contingency tables byuse of an algorithmwith toric ideals. Also, we proved some lemma of echelon form and lattice basis coming from contingency tables. And then, we performed random walks on the set of these contingency tables, and obtained results such as p-value for given real data. This is sufficient to solve the Sturmfels conjecture.
As for our quantum computer research, we made a program of emulating quantum computers by aclassical computer, which were used to simulate an algorithm of the order-finding problem. And, we calculated the optimal solutions for this problem by use of the integer programmingmethod!It leads us to some prediction about the optimal solutions in general case. Moreover, we did aresearch of quantum randomwalks on the set of contingency tables by defining unitary matricesfor this quantum algorithm.
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