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The number 551 looks harmless, but finding the two primes that multiply to give it is no easy task. A classical computer tries options one by one, like someone searching for a key on a keyring. A quantum processor works differently: it tries all possibilities at once, but you need a way to steer it toward the right answer.

Usually, this requires complex circuits or lengthy classical calculations to set up the quantum device. The new method does away with that help. The quantum system itself, step by step, closes in on the solution, receiving feedback from its own measurements. It's like a game of hot-and-cold: each step tells you whether you're getting closer or farther.

The experiment used a quantum register of three qubits — quantum analogues of bits. It factored 551 into 19 and 29. Then the method was tested on a computer for the numbers 9 167 and 2 106 287, using virtual quantum systems of 5 and 9 qubits. Even with control errors, the method remained accurate.

This approach removes the main bottleneck: the need to precompute parameters on a classical computer. If scaled up, quantum devices could crack cryptographic keys that are currently considered secure. For now, it's just a proof of principle, but the road is open.

Original abstract

as written by the authors arXiv:2601.16116

Prime factorization on quantum processors is typically implemented either via circuit-based approaches such as Shor's algorithm or through Hamiltonian optimization methods based on adiabatic, annealing, or variational techniques. While Shor's algorithm demands high-fidelity quantum gates, Hamiltonian optimization schemes, with prime factors encoded as degenerate ground states of a problem Hamiltonian, generally require substantial classical post-processing to determine control parameters. We propose an all-quantum, measurement-based feedback approach that iteratively steers a quantum system toward the target ground state, eliminating the need for classical computation of drive parameters once the problem Hamiltonian is determined and realized. As a proof of principle, we experimentally factor the biprime 551 using a three-qubit NMR quantum register and numerically analyze the robustness of the method against control field-errors. We further demonstrate scalability by numerically implementing the FALQON factorization of larger biprimes, 9,167 and 2,106,287, using 5 and 9 qubits, respectively.