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International Journal of Theoretical Physics

Efficient Nonlocal M-Control and N-Target Controlled Unitary Gate Using Non-symmetric GHZ States

Libing Chen, H. Lu

Published Nov 27, 2017• 4 citations
Publisher
DOI
10.1007/s10773-017-3602-5

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We present a scheme for implementing locally a nonlocal N -target controlled–controlled gate with unit probability of success by harnessing two ( N +1)-qubit Greenberger–Horne–Zeilinger (GHZ) states as quantum channel and N qutrits as catalyser. The quantum network that implements this nonlocal ( N +2)-body gate is built entirely of local single-body and two-body gates, and has only (3 N +2) two-body gates. This result suggests that both the computational depth of quantum network and the quantum resources required to perform this nonlocal gate might be significantly reduced. This scheme can be generalized straightforwardly to implement a nonlocal N -target and M -control qubits gate.

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We show how a nonlocal N-qubit controlled unitary gate can be implemented locally and effectively by using non-symmetric quantum channels. We construct respectively two quantum networks for realizing conclusively this nonlocal quantum gate. The first one hires (N–2) symmetric qubit-qubit Bell states and a non-symmetric qubit-qudit Bell state as quantum channels. The basic idea of this scheme is to use (N–2) additional levels of this qudit to “hide” certain computational states of (N–1) nonlocal control states from the conditional dynamics, which results in an effective nonlocal N-qubit controlled unitary gate design. In this scheme, however, either the number of the additional levels or that of 1-qudit gates needs to increase with N. The other one can improve significantly the local implementation of this nonlocal gate if we harness (N–1) non-symmetric qubit-qutrit Bell states as quantum channels. This scheme uses respectively (N–1) qutrits’s additional levels to expose one and only one initial computational state of (N–1) nonlocal control states to the conditional dynamics. In comparison with the first one, the procedure is greatly simplified, and the total gate time is reduced. The fact that the quantum network that does the proposed implementation is built entirely of local single-body and two-body gates, and has only (3N–4) two-body gates is notable.
Quantum computers are very efficient in terms of speed and security. Decoherence and architectural complexity restrict the control of sensitive quantum information as the number of qubits in a quantum computer increases. Therefore, it is more convenient to make a device with multiple quantum processors with small number of qubits instead of making a device with a large number of qubits quantum processors. The implementation of controlled unitary gates is a problem in such nonlocal systems. The methods in the literature for this problem use entangled qubit pairs, classical communication channels and classical bits. The existing methods perform some unitary operations on the target state for reconstruction after sending information through classical communication channels and applying quantum measurements on control states. In this study, a generalized method for nonlocal implementation of multi-qubit controlled unitary quantum gates with quantum channel is proposed. The proposed method can implement any controlled gate on the control and target qubits that are far from each other in terms of location. The method does not require classical channels and classical bits, any extra unitary operation for reconstruction. The proposed method is both more secure and uses less resources for operations than the other hybrid methods in the literature. Comparisons with existing studies are given in terms of required entangled qubit pairs, classical channels and bits, extra unitary operations for reconstruction the target state, and the advantages of the proposed method are revealed.
We present a scheme for implementing locally a nonlocal N -target controlled–controlled gate with unit probability of success by harnessing two ( N +1)-qubit Greenberger–Horne–Zeilinger (GHZ) states as quantum channel and N qutrits as catalyser. The quantum network that implements this nonlocal ( N +2)-body gate is built entirely of local single-body and two-body gates, and has only (3 N +2) two-body gates. This result suggests that both the computational depth of quantum network and the quantum resources required to perform this nonlocal gate might be significantly reduced. This scheme can be generalized straightforwardly to implement a nonlocal N -target and M -control qubits gate.