Hyper-parallel nonlocal CNOT operation with hyperentanglement assisted by cross-Kerr nonlinearity

Implementing CNOT operation nonlocally is one of central tasks in distributed quantum computation. Most of previously protocols for implementation quantum CNOT operation only consider implement CNOT operation in one degree of freedom(DOF). In this paper, we present a scheme for nonlocal implementation of hyper-parallel CNOT operation in polarization and spatial-mode DOFs via hyperentanglement. The CNOT operations in polarization DOF and spatial-mode DOF can be remote implemented simultaneously with hyperentanglement assisited by cross-Kerr nonlinearity. Hyper-parallel nonlocal CNOT gate can enhance the quantum channel capacity for distributed quantum computation and long-distance quantum communication. We discuss the experiment feasibility for hyper-parallel nonlocal gate. It shows that the protocol for hyper-parallel nonlocal CNOT operation can be realized with current technology.

Implementing quantum operation on remote quantum system is one of central tasks in distributed quantum computation. It is shown that universal quantum gate can be constructed by single-qubit gates and two-qubit controlled-not(CNOT) gates, or constructed by single-qubit gates and three-qubit Toffoli gates 42 . Remote implementation of quantum operations has attached much interest [43][44][45][46][47][48][49][50][51][52][53][54] . On the one hand, theoretical schemes for remote implementation of quantum operations, especially single-qubit operations, two-qubit CNOT operations and three-qubit gates, have been presented via different quantum channels. In 1999, Gottesman and Chuang studied university quantum computation via quantum gate teleportation 43 . The CNOT operation can be teleported from acting on local qubits to acting on remote qubits by using entangled state χ | 〉 = | 〉 + | 〉 | 〉 + | 〉 + | 〉 | 〉 [( 00 11 ) 00 (01 10 ) 11 ] 1 2 as the quantum channel. In 2000, Eisert et al. discussed the minimal resources required in remote implementation of nonlocal quantum operations 44 . They showed that quantum CNOT operation can be nonlocal implemented via two bits of classical communication and one ebit of quantum entanglement (a maximally entangled state of two qubits). Jiang et al. presented a scheme for deterministic remote implementation of nonlocal coupling gates between different registers 45 . In 2013, Wang et al. proposed a scheme for teleportation of quantum CNOT gate via quantum dots 46 . In 2015, Hu et al. proposed a protocol for deterministic remote implementation of nonlocal Toffoli operation among distant solid-state qubits 47 . In 2018, Lv et al. presented a scheme for multiparty joint remote implementation of an arbitrary single-qubit operation via single-qubit measurements and quantum entangled channel 48 . On the other hand, remote implementation of Results nonlocal cnot operation in the polarization Dof. To present the principle of our protocol for hyper-parallel CNOT gate clearly, we first present the protocol for nonlocal implementation of CNOT operation in polarization DOF via cross-Kerr nonlinearity, then present the protocol for nonlocal implementation of CNOT operation in spatial-mode DOF.
Similar to the case for nonlocal remote implementation of CNOT operation in one DOF, the agent Alice has the photon A 1 whose polarization state and spatial-mode state are arbitrary single-qubit states The agent Bob has photon B 1 whose polarization state and spatial-mode state are arbitrary single-qubit states Here | 〉 a 4 , | 〉 b 4 are two spatial modes of photon B 1 . The complex coefficients α 2 , β 2 , γ 2 , δ 2 satisfy the normalization conditions: α β | | + | | = 1 . The two agents want to implement nonlocal CNOT operation on photons A 1 , B 1 in polarization DOF by using polarization state of photon A 1 as the control qubit, and implement nonlocal CNOT operation on photons A 1 , B 1 in spatial-mode DOF by using the spatial-mode state of photon A 1 as the control qubit.
To parallel implement nonlocal CNOT operations in polarization and spatial-mode DOFs, the two agents Alice and Bob share a two-photon four-qubit hyperentangled state ψ | 〉 A B 2 2 as the quantum channel 67 . Here  The quantum circuit for nonlocal implementation of CNOT operation in polarization DOF is shown in Fig. 1. Here ±θ represent the cross-Kerr nonlinearity materia which add the phase shifts θ ± e i to the coherent probe states α | 〉 1 , α | 〉 3 if the number of photon in the signal state coupled with the coherent probe beam is 1 [68][69][70][71] . Polarizing beam splitters (PBS) can transmit horizontal polarization and reflect vertical polarization. The wave plate R 45 is used to implement Hadamard operation on polarization DOF 72 Beam splitter(BS) can implement Hadamard operation on spatial-mode DOF represent four spatial modes of photon A 2 , a b , 3 3 are two spatial modes of 4 represent four spatial modes of photon B 1 . Beam Splitter (BS) can implement Hadamard operation in spatial-mode DOF. The wave plate R 45 is used to implement Hadamard operation in polarization DOF.
2 are four spatial modes of photon A 2 and ′ ′ a b a b , , , 4 4 4 4 represent four spatial modes of photon B 1 .
To implement Hadamard operation on photons A B , 2 1 in spatial-mode DOF, Alice and Bob let photons A B , 2 1 in spatial modes ′ ′ a b a b , , , 4 pass through BSs, interact with probe coherent beams α | 〉 1 , α | 〉 3 with the cross-Kerr linearities. The interaction between photon A 2 and α | 〉 1 adds a phase shift θ e i to probe coherent www.nature.com/scientificreports www.nature.com/scientificreports/ beam if the number of photon in spatial mode ′ a 2 or ′ b 2 is 1. The interaction between photon B 2 and α | 〉 3 adds a phase shift θ e i to probe coherent beam if the number of photon in spatial mode ′ a 4 or ′ b 4 is 1. After the cross-Kerr linearities, the state of four photons becomes (without normalization)  Since the X quadrature measurements performed on probe coherent beams α 1 evolves to   To implement the nonlocal CNOT operation in polarization DOF, Alice and Bob let photons A B , 2 1 interact with probe coherent beams α | 〉 2 , α | 〉 4 via cross-Kerr nonlinearities and perform X quadrature measurements on probe coherent beams. The interaction between photon A 2 and α | 〉 2 adds a phase shift θ e i to probe coherent beam if the number of photon in spatial mode ′ a 2 or ′ b 2 is 1. The relation between the X quadrature measurements results of probe coherent beams α | 〉 2 , α | 〉 4 , the state of four photons after the X quadrature measurements and the recovery operation performed by Alice and Bob according to the measurement results is shown in Table 1. Here

4 4 represent the bit-flip operations in spatial modes
2 represents σ z p operation in the polarization DOF of photon i 72 . For example, we assume the outcome of the X quadrature measurements is α α . In this situation, the state of photons becomes  Nonlocal CNOT operation in spatial-mode DOF. Now, let us consider the implementation of nonlocal CNOT operation in spatial-mode DOF. Assisted by spatial-mode entanglement of the hyperentangled state, cross-Kerr nonlinearity and linear-optics elements, CNOT operation in spatial-mode DOF can be implemented nonlocally.
To nonlocally implement CNOT operation in polarization and spatial-mode DOFs simultaneously, the agents implement nonlocal CNOT operation in spatial-mode DOF via spatial-mode entanglement of the hyperentangled state by using the spatial-mode state of photon A 1 as the control qubit after transform the state of photons A A B B , , , 1 to ψ | 〉 3 . The quantum circuit for nonlocal implement of CNOT operation in spatial-mode DOF is shown in Fig. 2. Similar to ref. 67   Here ′ ′ a b a b , , , 2 represent four spatial modes of photon A 2 , a b , 4 4 are two spatial modes of photon B 1 .

The state of photons A A B B
, , ,

represents the polarization state of photons A A B B
, , , 2 represent four spatial modes of photon A 2 , a b , 3 3 are two spatial modes of photon B 2 , ′ ′ a b a b , , , 4 4 4 Scientific RepoRtS |  The state of composite system composed photons A A B B , , , 1 and two probe coherent beams α after the cross-Kerr interactions. Here Alice and Bob perform X quadrature measurements on the probe coherent beams α | 〉 1 , α | 〉 2 . The state of pho- if the X quadrature measurements result is α α

. The agents can transform the state of photons A A B B
, , , 1 to φ | 〉 3 by performing corresponding unitary operation I, if the X quadrature measurements result of the probe coherent beams α  The state of composite system composed of photons A A , 1 2 , B B , 2 1 and probe coherent beam evolves to after the corss-Kerr interaction between photon A 2 and the probe coherent beam α | 〉 3 . The state of photons A A , Suppose the X quadrature measurement result is α | 〉 θ e i 3 , the state of photons becomes φ | 〉 4 . To implement the CNOT operation nonlocally, Bob applies -I operation in spatial mode ′ b 4 , let photon B 1 pass through BS k = k ( 6,7,8,9) and interact with the probe coherent beam α | 〉 4 . After Bob applies -I operation in spatial mode ′ b 4 , the state of four photons evolves to After the -I operation, Bob let photon B 1 pass through BS BS , 6 7 which implement Hadamard operation in spatial modes ′ ′ a b a b , , , The state of four photons becomes For implementation of the nonlocal CNOT operation in spatial-mode DOF, Bob let photon B 1 pass through BS BS , 6 7 which implement Hadamard operation in spatial mode After photon B 1 pass through BS BS , 6 7 , the state of photons becomes   To implement CNOT in polarization and spatial-mode DOFs, Alice and Bob perform single-qubit measurements on photons A 2 , B 2 and apply corresponding operations on photons A 1 , B 1 . That is, Alice lets photons A 2 pass through BS 10 which implement Hadamard operation in saptial modes ′ a b , 2 2 . Bob first lets photon B 2 pass through BS 11 which implement Hadamard operation in saptial modes a b , 3 3 , then lets photons B 2 pass through wave plates R 45 in spatial modes a b , 3 3 2  2  2  2  2  2   3  3  3  3  3  3 After photons A B , 2 2 pass through BS 10 , BS 11 and wave plates R 45 , the state of photons A A B B , , , Here The nonlocal CNOT operation in spatial-mode DOF can be realized remotely by applying -I operation in spatial mode ′ b 4 , implementing Hadamard operation in spatial-mode DOF via BSs, introducing probe coherent beams and performing corresponding unitary operations on photons A 1 when the X quadrature measurement result is α | 〉  Table 3.

Discussion
Up to now, the utilization of cross-Kerr nonlinearities has been widely considered in the implementation of quantum information processing both theoretically and experimentally [73][74][75][76][77][78][79] . In 2003, Hofmann et al. showed than phase shift π can be obtained via single two-level atom in one-side cavity 73 . In 2013, Hoi et al. demonstrated cross-Kerr nonlinearity up to 20 degrees per photon at the single-photon level via superconducting qubit 74 . In 2016, Brod and Combes showed that cross-Kerr nonlinearity can be used to construct controlled-phase gate 75 . In addition, our protocol for hyper-parllel nonlocal CNOT operation requires only small phase shift as long as it can be distinguished from zero which makes our protocol for nonlocal CNOT operation in two DOFs more convenient in application than others.
In summary, we have proposed a protocol for parallel nonlocal implementation of CNOT operation in polarization and spatial-mode DOFs simultaneously. Assisted by cross-Kerr nonlinearity, hyperentangled state, the CNOT operation is teleported from implementing on local qubits in 2 DOFs to implementing on remote qubits. The agents first implement nonlocal CNOT operation in polarization via polarization entanglement of the hyperentangled state, then apply nonlocal CNOT operation in spatial-mode DOF via spatial-mode entanglement of the hyperentangled state. It is shown that the protocol for hyper-parallel nonlocal CNOT operation can enhance the communication for distributed quantum computation and large-scale quantum network. If large number of qubits are stored and manipulated in distributed quantum systems connected by entangled channel, nonlocal CNOT operation is required to implement distributed quantum computation. In hyper-parallel nonlocal CNOT operation, nonlocal CNOT operations are nonlocal implemented simultaneously in two DOFs which can enhance the channel capacity for large-scale quantum communication. Therefore, our protocol may be useful for large-scale quantum computation assisted by hyperentangled states.