Hidden Symmetry Subgroup Problems

We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an or...

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Vydané v:SIAM journal on computing Ročník 42; číslo 5; s. 1987 - 2007
Hlavní autori: Decker, Thomas, Ivanyos, Gábor, Santha, Miklos, Wocjan, Pawel
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Predmet:
Algebra
Algorithms
Computer science
Polynomials
Symmetry
ISSN:0097-5397, 1095-7111
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Abstract We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms. [PUBLICATION ABSTRACT]
AbstractList We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms. [PUBLICATION ABSTRACT]
Author Santha, Miklos
Decker, Thomas
Ivanyos, Gábor
Wocjan, Pawel
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Cites_doi 10.1137/S0097539703440678
10.1137/S009753970139450X
10.1137/S0097539795293172
10.1137/S0097539705447177
10.1109/TIT.1983.1056666
10.1109/TC.2002.1032628
10.1137/S0097539796300933
10.1137/S0097539703436345
10.1016/j.ipl.2004.01.024
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