Shor's Algorithm

Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm which runs on a quantum computer) for integer factorization formulated in 1994. Informally it solves the following problem: Given an integer N, find its prime factors.

On a quantum computer, to factor an integer N, Shor's algorithm runs in polynomial time (the time taken is polynomial in log N, which is the size of the input). Specifically it takes time O((log N)3), demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is thus in the complexity class BQP. This is exponentially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time — about O(e1.9 (log N)1/3 (log log N)2/3). The efficiency of Shor's algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by squarings.

If a quantum computer with a sufficient number of qubits were to be constructed, Shor's algorithm could be used to break public-key cryptography schemes such as the widely used RSA scheme. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor in polynomial time. However, Shor's algorithm shows that factoring is efficient on a quantum computer, so a sufficiently large quantum computer can break RSA. It was also a powerful motivator for the design and construction of quantum computers and for the study of new quantum computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.

In 2001, Shor's algorithm was demonstrated by a group at IBM, who factored 15 into 3 × 5, using an NMR implementation of a quantum computer with 7 qubits. However, some doubts have been raised as to whether IBM's experiment was a true demonstration of quantum computation, since no entanglement was observed. Since IBM's implementation, several other groups have implemented Shor's algorithm using photonic qubits, emphasizing that entanglement was observed. In 2012, the factorization of 15 was repeated. Also in 2012, the factorization of 21 was achieved, setting the record for the largest number factored with a quantum computer.