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Zhongguo kexue. Wulixue Lixue Tianwenxue

Quantum networks for implementing locally and conclusively a nonlocal qudit Toffoli gate: Designing and optimizing

Hong LU, LiBing CHEN

Published Sep 1, 2016• 8 citations
Publisher

Abstract

Local implementation of a nonlocal d-dimensional quantum Toffoli gate is considered. We construct respectively two quantum networks for realizing this nonlocal quantum gate conclusively. The first one is based on Eisert’s insight but follow a different pathway which results in a nonlocal d-dimensional Toffoli gate design. This scheme involves a local d-dimensional three-body interaction, which does not appear naturally in physical system. The other one can improve significantly the local implementation of this nonlocal gate if the target node harnesses a qutrit as a catalyser. In its simplest form this quantum network that does the improved scheme is built entirely of local single-body and two-body elementary gates, and with only 3 two-body elementary gates at the target node. The latter is simpler but conclusive, and more efficient but with less resource, which will make it more feasible with the current experimental technology and more suitable for large-scale quantum network.

DOI
10.1360/sspma2016-00174

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Local implementation of a nonlocal d-dimensional quantum Toffoli gate is considered. We construct respectively two quantum networks for realizing this nonlocal quantum gate conclusively. The first one is based on Eisert’s insight but follow a different pathway which results in a nonlocal d-dimensional Toffoli gate design. This scheme involves a local d-dimensional three-body interaction, which does not appear naturally in physical system. The other one can improve significantly the local implementation of this nonlocal gate if the target node harnesses a qutrit as a catalyser. In its simplest form this quantum network that does the improved scheme is built entirely of local single-body and two-body elementary gates, and with only 3 two-body elementary gates at the target node. The latter is simpler but conclusive, and more efficient but with less resource, which will make it more feasible with the current experimental technology and more suitable for large-scale quantum network.
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.
Although a Toffoli gate can be equivalently implemented using several single-qubit and two-qubit gates, it will consume much resource, two of which are nonlocal controlled-NOT (CNOT) gates acting on two non-adjacent nodes, especially in distributed quantum computation (DQC). We, for the first time, employ an ancillary qubit to construct a nonlocal Toffoli gate for DQC in linear network topology. The ancillary-qubit-based scheme needs fewer qubits, quantum gates, and entanglement states than that based on quantum teleportation scheme and the entanglement swapping scheme. We also analyze the performance of the three proposed schemes under different application scenarios, and present their pros and cons. Our work will help to implement DQC in noisy intermediate-scale quantum (NISQ) era.
We present quantum networks for a nnn-qubit controlled gate ${C}^{n\ensuremath{-}1}(U)$ which use a higher-dimensional (qudit) ancilla as a catalyzer. In its simplest form the network has only nnn two-particle gates (qubit-qudit)---this is the minimum number of two-body interactions needed to couple all n+1n+1n+1 subsystems (nnn qubits plus one ancilla). This class of controlled gates includes the generalized Toffoli gate ${C}^{n\ensuremath{-}1}(X)$ on nnn qubits, which plays an important role in several quantum algorithms and error correction. A particular example implementing this model is given by the dispersive limit of a generalized Jaynes-Cummings Hamiltonian of an effective spin sss interacting with a cavity mode.
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.
Remote implementation of nonlocal quantum gates is one of the central tasks in quantum communication. Researchers devote much interest in remote implementation of nonlocal quantum gates via quantum entangled channel. Remote implementation protocols of two-qubit controlled-not gate, three-qubit Toffoli gate have been presented. We present two schemes for remote implementation of nonlocal Toffoli gate via partially entangled quantum channel against collective noise. The first is used to remote implementing of nonlocal Toffoli gate via partially entangled channel against collective-dephasing noise, and the second is used to remote implementing of nonlocal Toffoli gate against collective-rotation noise. The schemes are tolerant of collective-dephasing noise and collective-rotating noise. The agents can implement the Toffoli gate nonlocally with partially entangled quantum channel by introducing an auxiliary qubit and performing general evolution. The schemes are more convenient in application since they only requires partially entangled quantum states for nonlocal implementation of quantum operations.
Abstract In distributed quantum computation, quantum remote-controlled gates are used frequently and applied on separate nodes or subsystems of a network. One of the universal and well-known controlled gates is the n-qubit controlled-NOT gate, especially Toffoli gate for the case of three qubits, which are frequently used to synthesize quantum circuits. In this paper, we considered a more general case, an n-qubit controlled-U gate, and present a general protocol for implementing these gates remotely with minimum required resources. Then, the proposed method is applied to implement a Toffoli gate in bipartite and tripartite systems. In this method, we considered cases in which a group of qubits belongs to one subsystem of the network. Then, we improved its consumption resources.