Prime factorization on quantum processors is typically implemented either via circuit-based approaches such as Shor's algorithm or through analog 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, generally require substantial classical postprocessing to determine control parameters of the driving field. We propose a measurement-based feedback approach that iteratively steers a quantum system towards the target ground state. Akin to an all-quantum implementation, the present approach completely bypasses the computationally expensive classical optimization of the control parameters. As a proof of principle, we experimentally factor the biprime 551 using a three-qubit nuclear magnetic resonance 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 9167 and 2 106 287 using 5 and 9 qubits, respectively.

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• Quantum algorithms & computation
• Quantum circuits
• Quantum computation
• Quantum feedback