Application of quantum algorithms to the study of permutations and group automorphisms
- Department of Mathematics, Graduate Center of the City University of New York, 365 Fifth Avenue, New York, New York 10016 (United States)
- Research Center for Quantum Information, Slovak Academy of Sciences, Dubravska cesta 9, 845 11 Bratislava (Slovakia)
We discuss three applications of efficient quantum algorithms to determining properties of permutations and group automorphisms. The first uses the Bernstein-Vazirani algorithm to determine an unknown homomorphism from Z{sub p-1}{sup m} to Aut(Z{sub p}) where p is prime. The remaining two make use of modifications of the Grover search algorithm. The first finds the fixed point of a permutation or an automorphism (assuming it has only one besides the identity). It can be generalized to find cycles of a specified size for permutations or orbits of a specified size for automorphisms. The second finds which of a set of permutations or automorphisms maps one particular element of a set or group onto another. This has relevance to the conjugacy problem for groups. We show how two of these algorithms can be implemented via programmable quantum processors. This approach opens new perspectives in quantum information processing when both the data and the programs are represented by states of quantum registers. In particular, quantum programs that specify control over data can be treated using methods of quantum information theory.
- OSTI ID:
- 21011191
- Journal Information:
- Physical Review. A, Vol. 76, Issue 1; Other Information: DOI: 10.1103/PhysRevA.76.012324; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
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