Shor's algorithm

Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. It is one of the few known quantum algorithms with compelling potential applications and strong evidence of superpolynomial speedup compared to best known classical (non-quantum) algorithms. However, beating classical computers will require quantum computers with millions of qubits due to the overhead caused by quantum error correction.

Shor proposed multiple similar algorithms for solving the factoring problem, the discrete logarithm problem, and the period-finding problem. "Shor's algorithm" usually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem.

On a quantum computer, to factor an integer ${\displaystyle N}$, Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in ${\displaystyle \log N}$. It takes quantum gates of order ${\displaystyle O\!\left((\log N)^{2}(\log \log N)(\log \log \log N)\right)}$ using fast multiplication, or even ${\displaystyle O\!\left((\log N)^{2}(\log \log N)\right)}$ using the asymptotically fastest multiplication algorithm currently known due to Harvey and van der Hoeven, thus demonstrating that the integer factorization problem is in complexity class BQP. Shor's algorithm is asymptotically faster than the most scalable classical factoring algorithm, the general number field sieve, which works in sub-exponential time: ${\displaystyle O\!\left(e^{1.9(\log N)^{1/3}(\log \log N)^{2/3}}\right)}$.