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Factoring large integers using a quantum computer is an outstanding researchproblem that can illustrate true quantum advantage over classical computers.Exponential time order is required in order to find the prime factors of aninteger by means of classical computation. However, the order can bedrastically reduced by converting the factorization problem to an optimizationone and solving it using a quantum computer. Recent works involving boththeoretical and experimental approaches use Shor's algorithm, adiabatic quantumcomputation and quantum annealing principles to factorize integers. However,our work makes use of the generalized Grover's algorithm as proposed by Liu,with an optimal version of classical algorithm/analytic algebra. We utilize thephase-matching property of the above algorithm for only amplitude amplificationpurposes to avoid an inherent phase factor that prevents perfect implementationof the algorithm. Here we experimentally demonstrate the factorization of twobi-primes, 4088459 and 966887 using IBM's 5- and 16-qubit quantum processors,hence making those the largest numbers that have been factorized on a quantumdevice. Using the above 5-qubit processor, we also realize the factorization ofa tri-prime integer 175, which had not been achieved to date. We observe goodagreement between experimental and theoretical results with high fidelities.The difficulty of the factorization experiments has been analyzed and it hasbeen concluded that the solution to this problem depends on the level ofsimplification chosen, not the size of the number factored. In principle, ourresults can be extended to factorize any multi-prime integer with minimumquantum resources.

Exact search algorithm to factorize large biprimes and a triprime on IBM quantum computer