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Journal of the Optical Society of America B

Nearly deterministic Fredkin gate based on weak cross-Kerr nonlinearities

Li Dong, Yan-Fang Lin, Jun-Xi Wang, Qingyang Li, Hong-Zhi Shen, Haikuan Dong, Yuan‐Peng Ren, Xiao‐Ming Xiu, Yajun Gao, Choo Hiap Oh

Published Jan 20, 2016• 21 citations
Publisher

Abstract

A Fredkin gate is an indispensable logic gate for the applications of the scalable quantum network. Weak cross-Kerr nonlinearities serve as the intermediary between photons and coherent states and thus provide the indirect interaction between photons. By means of weak cross-Kerr nonlinearities, we propose a construction scheme of a nearly deterministic Fredkin gate, where polarization modes of photons carry quantum bits. By virtue of an X homodyne measurement on the coherent state, the appropriate classical feed-forward operations are executed according to the different measurement outcomes. Applying one swap gate and two combination gates and other assistant simple optical elements, the nearly deterministic Fredkin gate is successfully constructed. The unique features of photons as quantum information carriers with high velocity and low decoherence, combined with simple elements and mature operation techniques, afford the possibility to the scheme proposed here which takes aim at feasible quantum logic circuits for practical and scalable quantum computation and quantum communication.

DOI
10.1364/josab.33.000253

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A Fredkin gate is an indispensable logic gate for the applications of the scalable quantum network. Weak cross-Kerr nonlinearities serve as the intermediary between photons and coherent states and thus provide the indirect interaction between photons. By means of weak cross-Kerr nonlinearities, we propose a construction scheme of a nearly deterministic Fredkin gate, where polarization modes of photons carry quantum bits. By virtue of an X homodyne measurement on the coherent state, the appropriate classical feed-forward operations are executed according to the different measurement outcomes. Applying one swap gate and two combination gates and other assistant simple optical elements, the nearly deterministic Fredkin gate is successfully constructed. The unique features of photons as quantum information carriers with high velocity and low decoherence, combined with simple elements and mature operation techniques, afford the possibility to the scheme proposed here which takes aim at feasible quantum logic circuits for practical and scalable quantum computation and quantum communication.
A scheme is proposed to implement the optical quantum Fredkin gate with weak nonlinearities and feed-forward. The distinctive feature of the present scheme is that the present Fredkin gate has four sets of possible output ports, and the total success probability for each set of the output ports is near 1/4 without ancillary single-photon. The present scheme requires nonlinear strength of the order of 10−3 and mean photon number of the probe coherent beam of the order of 106 so that the discrimination error probability does not exceed 10−4. These features show that it is still possible to operate in the regime of weak cross-Kerr nonlinearities, and the amplitude of the probe coherent beam is physically reasonable with current technology. These facts make us more confident in the feasibility of the proposed scheme.
Abstract By harnessing weak cross‐Kerr nonlinearities, it proposes two deterministic hyperparallel quantum control gates, including the hyperparallel controlled‐NOT (hyper‐CNOT) gate and hyperparallel Fredkin (hyper‐Fredkin) gate for photonic systems in the polarization and spatial degrees of freedom (DoFs), which are composed of two procedures for the implementations of the first polarized control (CNOT and Fredkin) gates and the second spatial control (CNOT and Fredkin) ones in turn. Moreover, compared with the two two‐photon CNOT gates (three‐photon Fredkin gates) in one DoF, the hyper‐CNOT (hyper‐Fredkin) gate is provided with more straightforward quantum circuits, lower computational costs of resource overhead (without the assistance of single photons or entangled states), less cross‐Kerr nonlinear interactions between three photons and the coherent states, and reduces the effect caused by photon‐loss noise. The success probabilities of two quantum control gates are approximated unit by performing the corresponding classical feed‐forward operations based on the different measuring outcomes of the X‐homodyne detectors to be aimed at the coherent states, and they are robust against the photon loss as well, which are feasible with the current technology and convenient in practical applications.
In this paper, with the weak cross-Kerr nonlinearity, we first present a special experimental scheme called controlled-path gate with which the realization of all possible bipartite positive-operator-value measurements of two-photon polarization states may be nearly deterministic. Following the same technique, the realization of quantum control gates, including the controlled-NOT gate, Fredkin gate, Toffoli gate, arbitrary controlled-U gate, and even arbitrary multicontrolled-U gate, are proposed. The corresponding probabilities are 1∕21∕21∕2, 1∕81∕81∕8, 2∕232∕232∕23, etc., respectively. Only the coherent states are required but not any ancilla photons, and no coincidence measurement are required, which results in these gates being scalable. The structures of these gates are very simple, therefore we think they are feasible with the current experimental technology in optics.
On the basis of the probe coherent state and weak cross-Kerr nonlinearities, we present a scheme of a nearly deterministic Controlled-NOT gate. In this construction, feed-forward methods, quantum nondemolition detectors and several optical elements are applied. It is a potentially practical quantum gate with certain features. First, the lack of auxiliary photons is allowable, which decreases consumption of resources. Secondly, employment of the signal photon from either of target output ports and three quantum nondemolition detectors enable the success probability to approach unit and judge whether the signal photons lose or not. Thirdly, the displacement measurement is adopted, and thus the Controlled-NOT gate works against photon loss of the probe coherent state. Finally, in order to circumvent the effect of dephasing, the monochromatic signal photons are exploited.
The decoherence effect caused by the coupling between the system and the environment undoubtedly leads to the errors in efficient implementations of two (or three) qubit logical gates in quantum information processing. Fortunately, decoherence-free subspace (DFS) introduced can effectively decrease the influence of decoherence effect. In this paper, we propose some schemes for setting up a family of quantum control gates, including controlled-NOT (CNOT), Toffoli, and Fredkin gates for two or three logical qubits by means of cross-Kerr nonlinearities in DFS. These three logical gates require neither complicated quantum computational circuits nor auxiliary photons (or entangled states). The success probabilities of three logical gates are approximate 1 by performing the corresponding classical feed-forward operations based on the different measuring results of the X-homodyne detectors, and their fidelities are robust against the photon loss with the current technology. The proposed logical gates rely on only simple linear-optics elements, available single-qubit operations, and mature measurement methods, making our proposed gates be feasible and efficient in practical applications.