# Summary of 'An Efficient Quantum Factoring Algorithm'

Author: Oded Regev

The paper presents an efficient quantum factoring algorithm that can be used to factorize n-bit integers. The algorithm involves running a quantum circuit with ˜O(n3/2) gates for √n + 4 times, and then using a polynomial-time classical post-processing step. The correctness of the algorithm is based on a number-theoretic assumption similar to those used in subexponential classical factorization algorithms. The author demonstrates that quantum circuits of size ˜O(n3/2) are sufficient for factoring integers, which is an improvement over previous algorithms that required larger circuit sizes. The number of qubits in the quantum circuit is O(n3/2), which is higher than the qubit requirement in optimized implementations of Shor’s algorithm. However, the depth of the quantum circuit is smaller than Shor’s algorithm, making it more feasible for implementation. The paper also discusses the potential implications of the algorithm in practice. It is highlighted that the analysis is asymptotic and the algorithm may not be efficient for small values of n. The algorithm may benefit from optimizations in fast integer multiplication and the use of smaller qubit counts, similar to optimizations used in Shor’s algorithm. However, it is currently unclear if these optimizations can be applied to the proposed algorithm. The author concludes by stating that the algorithm provides an improvement over Shor’s algorithm in terms of circuit size. However, it remains to be seen if the algorithm can be practically implemented and if it can provide an improvement over Shor’s algorithm for small values of n. The analysis in the paper is based on asymptotics, and it is unclear if hidden constants in the algorithm would make it inefficient for small values of n. In summary, the paper presents an efficient quantum factoring algorithm that uses a quantum circuit with ˜O(n3/2) gates and a classical post-processing step. The algorithm provides an improvement over previous algorithms in terms of circuit size, but its practicality and potential improvements for small values of n remain to be seen.