Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  gives polynomial time quantum algorithms for computing the ideal class group (CGP) under the Generalized Riemann Hypothesis and solving the principal ideal problem (PIP) in number fields of  arbitrary  degree. These are are fundamental problems in number theory and they are connected to many unproven conjectures in both analytic and algebraic number theory. Previously the best known algorithms by Hallgren [20] only allod to solve these problems in quantum polynomial time for number fields of  constant  degree. In a recent breakthrough, Eisentrager et al. [11] shod how to compute the unit group in arbitrary fields, thus opening the way to the resolution of CGP and PIP in the general case. For example, Biasse and Song [3] pointed out how to directly apply this result to solve PIP in classes of cyclotomic fields of arbitrary degree.   The methods  introduce in this paper run in quantum polynomial time in arbitrary classes of number fields. They can be applied to solve other problems in computational number theory as ll including computing the ray class group and solving relative norm equations. They are also useful for ongoing cryptanalysis of cryptographic schemes based on ideal lattices [5, 10].   Our algorithms generalize the quantum algorithm for computing the (ordinary) unit group [11].  first show that CGP and PIP reduce naturally to the computation of  S -unit groups, which is another fundamental problem in number theory. Then  show an efficient quantum reduction from computing  S -units to the continuous hidden subgroup problem introduced in [11]. This step is our main technical contribution, which involves careful analysis of the metrical properties of lattices to prove the correctness of the reduction. In addition,  show how to convert the output into an exact compact representation, which is convenient for further algebraic manipulations.
['Jean-François Biasse', 'Fang Song 0001']
I am writing a research paper on a idea Efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree number fields
[{"from": "human", "value": "I am writing a research paper on a idea Efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree number fields"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  gives polynomial time quantum algorithms for computing the ideal class group (CGP) under the Generalized Riemann Hypothesis and solving the principal ideal problem (PIP) in number fields of  arbitrary  degree. These are are fundamental problems in number theory and they are connected to many unproven conjectures in both analytic and algebraic number theory. Previously the best known algorithms by Hallgren [20] only allod to solve these problems in quantum polynomial time for number fields of  constant  degree. In a recent breakthrough, Eisentrager et al. [11] shod how to compute the unit group in arbitrary fields, thus opening the way to the resolution of CGP and PIP in the general case. For example, Biasse and Song [3] pointed out how to directly apply this result to solve PIP in classes of cyclotomic fields of arbitrary degree.   The methods  introduce in this paper run in quantum polynomial time in arbitrary classes of number fields. They can be applied to solve other problems in computational number theory as ll including computing the ray class group and solving relative norm equations. They are also useful for ongoing cryptanalysis of cryptographic schemes based on ideal lattices [5, 10].   Our algorithms generalize the quantum algorithm for computing the (ordinary) unit group [11].  first show that CGP and PIP reduce naturally to the computation of  S -unit groups, which is another fundamental problem in number theory. Then  show an efficient quantum reduction from computing  S -units to the continuous hidden subgroup problem introduced in [11]. This step is our main technical contribution, which involves careful analysis of the metrical properties of lattices to prove the correctness of the reduction. In addition,  show how to convert the output into an exact compact representation, which is convenient for further algebraic manipulations."}]