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Zhongguo kexue. Wulixue Lixue Tianwenxue

Probabilistic remote implementation of nonlocal Toffoli gate with pure entangled states against collective noise

Ping Zhou, Xian-Fang Jiao

Published May 30, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1360/sspma-2019-0004

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We present a systematic simple method to implement a generalized quantum control-NOT (CNOT) gate on two d -dimensional distributed systems. First, we show how the nonlocal generalized quantum CNOT gate can be implemented with unity fidelity and unity probability by using a maximally entangled pair of qudits as a quantum channel. We also put forward a scheme for probabilistically implementing the nonlocal operation with unity fidelity by employing a partially entangled qudit pair as a quantum channel. Analysis of the scheme indicates that the use of partially entangled quantum channel for implementing the nonlocal generalized quantum CNOT gate leads to the problem of 'the general optimal information extraction'. We also point out that the nonlocal generalized quantum CNOT gate can be used in the entanglement swapping between particles belonging to distant users in a communication network and distributed quantum computer.
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 propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a puritlcation protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We investigate the problem of quantum remote implementation of a single-qubit rotation operation using three-qubit entangled state. Firstly, we utilize the entanglement property of maximally entangled Greenberger–Horne–Zeilinger (GHZ) state to design a theoretical scheme for implementing the operation remotely with unit fidelity and unit probability. Then, we put forward two schemes for conclusive implementing the non-local single-qubit rotation with unit fidelity by employing a partially entangled pure GHZ state as quantum channel. The features of these schemes are that a third side is included, who may participate the process of quantum remote implementation as a supervisor. Furthermore, when the quantum channel is partially entangled, the third side can rectify the state distorted by imperfect quantum channel. In addition to the GHZ class state, the W class state can also be used to remotely implement the same operation probabilistically. The probability of successful implementation using the W class state is always less than that using the GHZ class state.
We propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a purification protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We investigate the minimal resources that are required in the local implementation of nonlocal quantum gates in a distributed quantum computer. Both classical communication requirements and entanglement consumption are investigated. We present general statements on the minimal resource requirements and present optimal procedures for a number of important gates, including controlled-NOT (CNOT) and Toffoli gates. We show that one bit of classical communication in each direction is both necessary and sufficient for the nonlocal implementation of the quantum CNOT, while in general two bits in each direction is required for the implementation of a general two-bit quantum gate. In particular, the state swapper requires this maximum classical communication overhead. Extensions of these ideas to multiparty gates are presented.