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Chinese Science Bulletin

Optical implementation of quantum random walks using weak cross-Kerr media

Yan Li

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Weak cross-Kerr media provides additional degrees of freedom of qubits in quantum information processing.In this paper,by exploiting weak cross-Kerr nonlinearity,we propose an optical implementation scheme of one-dimensional quantum random walks. The random walks are described by the interaction of single photons with cross-Kerr media.The proposed scheme can also be used to implement one-dimensional quantum random walks on an infinite line.

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Weak cross-Kerr media provides additional degrees of freedom of qubits in quantum information processing.In this paper,by exploiting weak cross-Kerr nonlinearity,we propose an optical implementation scheme of one-dimensional quantum random walks. The random walks are described by the interaction of single photons with cross-Kerr media.The proposed scheme can also be used to implement one-dimensional quantum random walks on an infinite line.
A quantum walk is the quantum analog of the classical random walk, which provides a powerful tool for quantum computation and quantum simulation. However, the current experimental realization of the quantum walk based on photonic systems has been restricted to evolution in one dimension or small scale in two dimensions so far, and the growing demands cannot be fulfilled for further application of certain quantum computation tasks. Here, we theoretically propose a feasible scheme of a two-dimensional quantum walk with infinite steps by exploiting the orbital angular momentum of the photons and the phase information of the auxiliary coherent state induced by cross-Kerr nonlinearity. The numerical simulation shows that a proper parameter setting guarantees the walking for infinite steps in a two-dimensional grid with nearly $100\%$100% success probability.
We present a method for implementing a weak optical Kerr interaction (single-mode Kerr Hamiltonian) in a measurement-based fashion using the common set of universal elementary interactions for continuous-variable quantum computation. Our scheme is a conceptually distinct alternative to the use of naturally occurring, weak Kerr nonlinearities or specially designed nonlinear media. Instead, we propose to exploit suitable off-line prepared quartic ancilla states together with beam splitters, squeezers, and homodyne detectors. For perfect ancilla states and ideal operations, our decompositions for obtaining the measurement-based Kerr Hamiltonian lead to a realization with near-unit fidelity. Nonetheless, even by using only approximate ancilla states in the form of superposition states of up to four photons, high fidelities are still attainable. Our scheme requires four elementary operations and its deterministic implementation corresponds to about 10 ancilla-based gate teleportations. We test our measurement-based Kerr interaction against an ideal Kerr Hamiltonian by applying them both to weak coherent states and single-photon superposition states.
With the assistance of weak cross-Kerr nonlinearities, we present a preparation scheme of four-photon polarization-entangled decoherence-free states, which can be used to construct the minimal optical decoherence-free subspaces where a logical qubit is fully protected against collective decoherence. To complete the preparation task, one spatial entanglement process, two polarization entanglement processes, and one detecting process are applied. The fulfillments of the above processes are contributed by a cross-Kerr nonlinear interaction between the signal photons and a coherent state via Kerr media. Exploiting the available single-photon resource and simple linear optics elements, this scheme is feasible and desirable to be extended to the construction of multiphoton decoherence-free states against the collective decoherence.
Quantum information processing via hyperentanglement can improve the quantum channel capacity for long-distance quantum communication and large-scale quantum communication networks. Some important works have been presented in quantum remote state preparation with hyperentangled states. However, the protocols for quantum operation remote implementation are based on quantum systems encoded in one degree of freedom. We present a quantum operation remote implementation protocol for arbitrary single-qubit operations with two-photon hyperentangled states via cross-Kerr nonlinearities. Moreover, we discuss the scheme for parallel remote implementation of partially unknown quantum operations via two-photon hyperentangled states.
High-dimensional entangled states have attracted attention because of their strong nonlocality and powerful capability for quantum information processing. By the methods presented in this paper, arbitrary forms-entangled qudits including symmetric and asymmetric forms could be generated with weak cross-Kerr nonlinearity. These schemes are heralded by the use of single-photon detectors. If all the detectors do not register any single photons, the generation is a success with the probability 1/nM determined by dimension n and partite M. Furthermore, these schemes work well even with the common photon number nonresolving detectors; therefore they are feasible with the current experimental technology.