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Quantum  queries on permutations

Quantum  queries on permutations. Taisia Mischenko-Slatenkova, Agnis Skuskovniks , Alina Vasilieva, Ruslans Tarasovs and Rusins Freivalds. “COMPUTER SCIENCE APPLICATIONS AND ITS RELATIONS TO QUANTUM PHYSICS”, project of the European Social Fund Nr. 2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044.

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Quantum  queries on permutations

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  1. Quantum  queries on permutations Taisia Mischenko-Slatenkova, Agnis Skuskovniks, Alina Vasilieva, Ruslans Tarasovs and Rusins Freivalds “COMPUTER SCIENCE APPLICATIONS AND ITS RELATIONS TO QUANTUM PHYSICS”, project of the European Social Fund Nr. 2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044 INVESTING IN YOUR FUTURE

  2. Introduction • Query algorithms • Decision trees • Quantum Querry algorithms • Querry algorithm complexity • Number of queries performed • Exact quantum query algorithms • Acceptingprbability is always 0 or 1 • Promise for Quantum algorithms • Domain of correctness of the algoritm is explicitly restricted Decision tree from:http://manuals.matrix.squizsuite.net/other-cms-assets/chapters/decision-tree

  3. StudiedFunctions Problem domain – permutations! • For example: • 5 symbol permutation 43210 has function variables: Functions value Determines permutation properties • For example: • 3-permutation is even (opposite to odd) • There is a 1 query Quantum algorithm for this problem • Classical query algorithm needs 2 queries • This can be generalize to: 2m-permutation QQ=m; [2m+1 -> QQ=m+1]

  4. Our results for 5-permutations • A group GR with operation “multiplication of permutations” • GR is not a commutative and is not cyclic • Function: • Membership in the group GR of • 5-permutation • Result: Solution with 2 queries • 4 Deterministic queries are needed classically

  5. Structure the query/answer pairs in a table • Ask all 20 question pairs in parallel • If result is “far away” (Distance={2,3}) then multiply to -1 • If result is “close” (Distance={1,2}) then multiply to +1 • Example: 03241 • Example: 20314 • If all the signs are equal – belongs to GR • If signs are different then • Half the signs are + and half – • Does not belong to GR

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