A proposal for factorization using Kerr nonlinearities between three harmonic oscillators

Using three coupled harmonic oscillators, we propose a quantum approach without using qubits to efficiently factorize an integer with only three steps. These three harmonic oscillators are coupled together via nonlinear interactions. This factoring method can be applied even if two harmonic oscillators are prepared in mixed states. As simple examples, we show how to factorize N = 15 and 35 using this approach. For finding factors, the required time evolution of the system exponentially shortens for stronger nonlinear strengths and for higher-order nonlinearity. The effect of dissipation of the harmonic oscillators on the performance of this method is also discussed. Our results show that an alternative form of quantum computing can be exploited for factorization, without using qubits.