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A scheme to encode arbitrarily long integer pairs on degenerate opticalparametric oscillations multiplexed in time is proposed. The classicalentanglement between the polarization directions and the phases of theoscillating pulses, regarded as two computational registers, furnishes theinteger correlations within each pair. We show the major algorithmic steps,modular exponentiation and discrete Fourier transform, of Shor's quantumfactoring algorithm can be executed in the registers as pulse interferencesunder the assistance of external logics. The factoring algorithm is thusrendered equivalent to a semi-classical optical-path implementation that isscalable and decoherence-free. The sought-after multiplicative order, fromwhich the prime factors are deduced, is identified from a two-dimensionalfringe image generated by four-hole interference measured at the end of thepath.

Scalable semi-classical implementation of Shor factoring using time-multiplexed degenerate optical parametric oscillators