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Modern Physics Letters B

Implementing Toffoli gate via weak cross-Kerr nonlinearity and classical feedback

Qian Sun, Liu Ye

Published Apr 9, 2015• 7 citations
Publisher

Abstract

We design an efficient optical circuit to realize a Toffoli gate via weak cross-Kerr nonlinearity and classical feedback. To obtain a high-fidelity gate, we probe different path detectors to minimize the influence from the environmental noise. Compared to previous proposals, our scheme does not need external auxiliary photons and is easy to implement in experiment.

DOI
10.1142/s0217984915500323

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We design an efficient optical circuit to realize a Toffoli gate via weak cross-Kerr nonlinearity and classical feedback. To obtain a high-fidelity gate, we probe different path detectors to minimize the influence from the environmental noise. Compared to previous proposals, our scheme does not need external auxiliary photons and is easy to implement in experiment.
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.
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.
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.
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 Toffoli gate is an especially important case of the quantum gates. In this paper, we show how to construct Toffoli gates using several additional (ancilla) photons, linear optical instruments and cross-Kerr nonlinearities. Then they are expanded to implement Extended Toffoli gates (ETOF). Our work shows only one n-IPC (n-input parity check) gate plus at most 2m CNOT or NCNOT gates can implement arbitrary ETOF (n, m), indicating the feasibility and advantage of modularizing quantum gates with photons.