Which is the highest number factorized by QC in a non-specific experiment?

Question

Since the original experimental contribution using the Shor's factoring algorithm to factorize the integer 15 some experiments have been performed in order to calculate the largest factorized number. But most of the experiments are particularly designed for a specific number ($N$) and not a general approach which could be used for any $<N$ integer. Example.

I am wondering which is, at the moment, the largest number that has been experimentally factorized in a general procedure by a quantum algorithm.

Answer 1

The answer is $N = 200\,099$.

Shor's algorithm is not the only way to factorize integers. In fact, it is also possible to factorize integers with an optimization approach. This approach even allows for integers with more than two prime-factors to be composed.

See this paper from D-Wave, Prime factorization using quantum annealing and computational algebraic geometry, in which the explain their approach and show the results of factoring multiple composite numbers, among which $N=200\,099$.

Answer 2

For Shor's algorthm: Every experiment has been designed for the specific number being factored. The largest number factored without cheating was 15 which is the smallest non-trivial semi-prime on which to apply Shor's algorithm. Major changes would be needed in the experiment (including in the number of qubits) in order to factor 21, for example. IBM's 50-qubit machine can implement Shor's algorithm on larger numbers, but the noise is so bad that you will only get the correct factors if you're very lucky, and that's why it hasn't been done yet.

For the annealing algorithm: 376289 has been factored with D-Wave's 2048-qubit annealer, and this is not a specific experiment but a general algorithm on an easily programmable machine, but we do not know how this will scale. A very crude upper limit to the number of qubits needed to factor RSA-230 is 5.5 billion qubits (but this can be brought down significantly by better compilers), while Shor's algorithm can do it with 381 qubits.