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arXiv:quant-ph/0305008 · Physical Review A

Simulation of quantum random walks using the interference of a classical field

Hyunseok Jeong, Mauro Paternostro, M. S. Kim

Published Jan 16, 2004• 89 citations
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Abstract

We suggest a theoretical scheme for the simulation of quantum random walks on a line using beam splitters, phase shifters, and photodetectors. Our model enables us to simulate a quantum random walk using of the wave nature of classical light fields. Furthermore, the proposed setup allows the analysis of the effects of decoherence. The transition from a pure mean-photon-number distribution to a classical one is studied varying the decoherence parameters.

DOI
10.1103/physreva.69.012310

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We suggest a theoretical scheme for the simulation of quantum random walks on a line using beam splitters, phase shifters, and photodetectors. Our model enables us to simulate a quantum random walk using of the wave nature of classical light fields. Furthermore, the proposed setup allows the analysis of the effects of decoherence. The transition from a pure mean-photon-number distribution to a classical one is studied varying the decoherence parameters.
In the optical quantum random walk system, phase fluctuation and beam splitter fluctuation are two unavoidable decoherence factors. These two factors degrade the performance of quantum random walk by destroying coherence, and even degrade it into a classical one. We propose a scheme for the simulation of quantum random walk using phase shifters, tunable beam splitters, and photodetectors. This proposed scheme enables us to analyze the effect of phase fluctuation and beam splitter fluctuation on two-photon quantum random walk. Furthermore, it is helpful to guide the control of phase fluctuation and beam splitter fluctuation in the experiment.
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We investigate quantum walks in multiple dimensions with different quantum coins. We augment the model by assuming that at each step the amplitudes of the coin state are multiplied by random phases. This model enables us to study in detail the role of decoherence in quantum walks and to investigate the quantum-to-classical transition. We also provide classical analog of the quantum random walks studied. Interestingly enough, it turns out that the classical counterparts of some quantum random walks are classical random walks with a memory and biased coin. In addition random phase shifts ``simplify'' the dynamics (the cross-interference terms of different paths vanish on average) and enable us to give a compact formula for the dispersion of such walks.
Quantum walks (QW) offer a speed-up advantage over random walks in quantum search applications. We present an experimental study of the transition from quantum-to-classical random walk using an emulation of the decoherence process for polarization qubits that exploits maximally non-separable spin-orbit modes of an intense laser beam for the first, to the best of our knowledge, time. We are able to continuously control the input polarization mode in an all-optical quantum walk circuit to observe transitions associated with quantum, quantum stochastic, and classical random walk distributions. The results are in agreement with theoretical expectations.