Probabilistic Bounds for Generating Finite Nilpotent Groups Require Fewer Than Elements with Probability for Rank or Chain Length

The challenge of determining how many random elements are needed to generate a finite nilpotent group has long occupied mathematicians, and a new study by Ziyuan Dong, Xiang Fan, and Tengxun Zhong, all from Sun Yat-sen University, alongside Daowen Qiu, offers a significant advance in understanding this problem. The researchers establish a novel probabilistic bound on the number of elements required, proving that a group can be generated with high probability if a specific condition relating to the group’s rank or chain length is met. This finding improves upon previous requirements and represents a substantial refinement in the field, offering practical benefits for algorithms used in cryptography and computational number theory. The work provides a more accurate estimation of iteration counts for solving the finite Abelian hidden subgroup problem and reduces the computational overhead of Regev’s factoring algorithm, paving the way for more efficient computations in these areas.

The research demonstrates that for any nilpotent group G and a desired success probability of 1, ε, the probability of generating the group with k random elements is greater than or equal to 1, ε when k is greater than or equal to the group’s rank plus the ceiling of log base 2 of 2 divided by ε. Alternatively, the same success probability is achieved when k is greater than or equal to the group’s chain length plus the ceiling of log base 2 of 1 divided by ε. These bounds represent a sharpening of previously known requirements, improving upon earlier conditions, and are nearly optimal, with the group chain length bound potentially reducible by only 1 and the group rank bound by at most 2.

This precision delivers substantial benefits for quantum algorithms, specifically the finite Abelian hidden subgroup problem (AHSP). By using these improved bounds, scientists can reduce the number of iterations needed to solve the AHSP, achieving an exponential improvement over previous methods. Furthermore, the research optimizes iteration counts in Regev’s quantum factoring algorithm, demonstrating that only the group’s rank plus 2 random elements are sufficient to generate the group with a given success probability, a reduction from the previously established requirement of the group’s rank plus 4. This refinement directly reduces the quantum circuit repetition count, improving the efficiency of the algorithm while maintaining identical success probability. The work establishes a foundational tool for analyzing probabilistic algorithms and has significant implications for advancements in quantum computing.

Efficient Group Generation For Quantum Algorithms

Scientists have established new probabilistic bounds on the number of elements required to generate finite nilpotent groups. The research demonstrates that a random set of elements will generate a finite nilpotent group with high probability if the size of the set exceeds a specific threshold related to either the group’s rank or its chain length. This refinement of existing criteria has direct implications for quantum algorithms. The team demonstrated that these bounds can improve the efficiency of algorithms used in quantum computation, notably reducing the number of iterations needed for the finite Abelian hidden subgroup problem and decreasing the repetitions required in Regev’s factoring algorithm. This represents a step towards optimizing quantum computations by providing a more precise understanding of the resources needed for successful execution.

Quantum Speed-Up Through Group Generation

Researchers have made a significant advance in understanding how to efficiently generate finite nilpotent groups using random elements. The team established a new probabilistic bound, demonstrating that a group can be generated with high probability if the number of random elements exceeds a specific threshold related to the group’s rank or chain length. This improvement represents a step towards optimizing quantum computations by providing a more precise understanding of the resources needed for successful execution. The team proved that these bounds lead to a reduction in the number of iterations required for the finite Abelian hidden subgroup problem, and a corresponding decrease in repetitions needed for Regev’s factoring algorithm. The work establishes a foundational tool for analyzing probabilistic algorithms and has significant implications for advancements in quantum computing.