复杂系统的量子隐形传态

量子隐形传态是量子信息技术中重要的信息传输手段。近三十年来，人们针对量子隐形传态进行了深入地理论和实验研究，各种重要研究成果层出不穷。如今，量子隐形传态研究已从单粒子到多粒子、单自由度到多自由度、二维系统到高维系统；同时，传输距离也从实验室距离到百公里、再到千公里级别。量子隐形传态研究朝着日趋复杂的方向发展，产生了更多有趣且重要的科学问题等待人们研究。量子门隐形传输是量子隐形传态研究的重要分支，是实现分布式量子计算、分布式量子编码解码等复杂量子任务中一项不可或缺的技术，在线性光学、离子阱、中性原子、超导量子比特等诸多实验系统中得到了实现。但目前对量子门隐形传输的研究仍然主要关注两方量子门隐形传输问题，多方量子门隐形传输问题鲜有涉及。
在本文中，我们首先梳理量子隐形传态的发展历程，介绍多粒子系统、单粒子多自由度系统以及高维系统等多种复杂量子系统隐形传态理论与实验实现方法。然后，我们重点关注多方间量子门隐形传输问题，提出基于多比特 GHZ 态三方间和多方间量子受控非门隐形传输理论方案，并设计基于光子偏振和路径自由度的线性光学系统实验光路，进行三方间受控非门量子隐形传输实验验证。最终，我们梳理量子系统表征方法，对实验中三方量子门隐形传输保真度进行分析，估计保真度在0.839 到 0.866 之间，并通过量子过程层析证实保真度为 0.861±0.004。另外，为进一步证明该方案的有效性，我们基于三方受控非门量子隐形传输协议，完成分布式量子编码方案演示，成功对量子态进行比特编码，将 |+〉 编码为 (|000〉+|111〉)/√2，获得 0.866 ± 0.019 的保真度。

Quantum teleportation (QT) is an important tool of information transmission in quantum information technology. For nearly three decades, QT has been studied both theoretically and experimentally, and various important research results emerge one after another. Nowadays, research on quantum state teleportation (QST) has been developed from single particle to many particles, one degree/variable to multiple degrees/variables, two-dimensional to higher-dimensional systems, and the transmission distance has also been increased from laboratory platforms to thousands-of-kilometer. The research on QT has become more and more complex, bringing out lots of unsolved interesting and important scientific questions. Quantum gate teleportation (QGT) is an important branch of QT. It is an indispensable technology for realizing complex quantum tasks such as distributed quantum computing, distributed quantum encoding and decoding. It has been realized in many experimental systems such as linear optics, ion traps, neutral atoms, and superconducting qubits. However, the research on QGT mainly focuses on teleportation between two parties, rarely among multi-parties. 

In this paper, we first review the development of QT, and introduce QT theory and corresponding experimental methods for various complex quantum systems. Then, we focus on the multi-party QGT problem, and propose a theoretical scheme for multi-party quantum CNOT gate teleportation based on multipartite GHZ states. We design the experiment setup using photonic polarization and path degrees of freedom, and realize the multipartite QGT in experiments. Finally, we analyze and obtain the fidelity of the tripartite QGT 0.839≤F≤0.866, and verify that the fidelity is 0.861±0.004 by quantum process tomography. In addition, in order to further prove the effectiveness of the scheme, we experimentally demonstrated the distributed quantum encoding scheme based on the QT protocol of the controlled NOT gate among the three parties, and successfully encoded the single-qubit state |+〉 to a three-qubit state as (|000〉+|111〉)/√2 shared by three parties with fidelity 0.866±0.019.  

[1] KNILL E, LAFLAMME R, MILBURN G J. A scheme for efficient quantum computation with linear optics[J]. Nature, 2001, 409(6816): 46-52.
[2] WALTHER P, RESCH K J, RUDOLPH T, et al. Experimental one-way quantum computing[J]. Nature, 2005, 434(7030): 169-176.
[3] LU C Y, BROWNE D E, YANG T, et al. Demonstration of a compiled version of shor’s quantum factoring algorithm using photonic qubits[J]. Phys. Rev. Lett., 2007, 99: 250504.
[4] ARUTE F, ARYA K, BABBUSH R, et al. Quantum supremacy using a programmable superconducting processor[J]. Nature, 2019, 574(7779): 505-510.
[5] ZHONG H S, WANG H, DENG Y H, et al. Quantum computational advantage using photons[J]. Science, 2020, 370(6523): 1460-1463.
[6] POLITI A, MATTHEWS J C F, O’BRIEN J L. Shor’s quantum factoring algorithm on a photonic chip[J]. Science, 2009, 325(5945): 1221-1221.
[7] HWANG W Y. Quantum key distribution with high loss: Toward global secure communication[J]. Phys. Rev. Lett., 2003, 91: 057901.
[8] LO H K, CURTY M, QI B. Measurement-device-independent quantum key distribution[J]. Phys. Rev. Lett., 2012, 108: 130503.
[9] LIAO S K, CAI W Q, LIU W Y, et al. Satellite-to-ground quantum key distribution[J]. Nature, 2017, 549(7670): 43-47.
[10] BENNETT C H, BRASSARD G, CRÉPEAU C, et al. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels[J]. Phys. Rev. Lett., 1993, 70: 1895-1899.
[11] BOUWMEESTER D, PAN J W, MATTLE K, et al. Experimental quantum teleportation[J]. Nature, 1997, 390(6660): 575-579.
[12] PORRAS D, CIRAC J I. Effective quantum spin systems with trapped ions[J]. Phys. Rev. Lett.,2004, 92: 207901.
[13] FRIEDENAUER A, SCHMITZ H, GLUECKERT J T, et al. Simulating a quantum magnet withtrapped ions[J]. Nature Physics, 2008, 4(10): 757-761.
[14] ZHANG J, PAGANO G, HESS P W, et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator[J]. Nature, 2017, 551(7682): 601-604.
[15] KASEVICH M, CHU S. Atomic interferometry using stimulated raman transitions[J]. Phys. Rev. Lett., 1991, 67: 181-184.
[16] WEISS D S, YOUNG B C, CHU S. Precision measurement of ℏ/mcs based on photon recoil using laser-cooled atoms and atomic interferometry[J]. Applied Physics B, 1994, 59(3): 217- 256.
[17] ERHARD M, KRENN M, ZEILINGER A. Advances in high-dimensional quantum entanglement[J]. Nature Reviews Physics, 2020, 2(7): 365-381.
[18] BOSCHI D, BRANCA S, DE MARTINI F, et al. Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels[J]. Phys. Rev. Lett., 1998, 80: 1121-1125.
[19] ZHAO Z, CHEN Y A, ZHANG A N, et al. Experimental demonstration of five-photon entanglement and open-destination teleportation[J]. Nature, 2004, 430(6995): 54-58.
[20] HILLERY M, BUZEK V, BERTHIAUME A. Quantum secret sharing[J]. Phys. Rev. A, 1999, 59: 1829-1834.
[21] ZHANG Q, GOEBEL A, WAGENKNECHT C, et al. Experimental quantum teleportation of a two-qubit composite system[J]. Nature Physics, 2006, 2(10): 678-682.
[22] WANG X L, CAI X D, SU Z E, et al. Quantum teleportation of multiple degrees of freedom of a single photon[J]. Nature, 2015, 518(7540): 516-519.
[23] LUO Y H, ZHONG H S, ERHARD M, et al. Quantum teleportation in high dimensions[J]. Phys. Rev. Lett., 2019, 123: 070505.
[24] ZHANG H, ZHANG C, HU X M, et al. Arbitrary two-particle high-dimensional bell-state measurement by auxiliary entanglement[J]. Phys. Rev. A, 2019, 99: 052301.
[25] URSIN R, JENNEWEIN T, ASPELMEYER M, et al. Quantum teleportation across the danube[J]. Nature, 2004, 430(7002): 849-849.
[26] JIN X M, REN J G, YANG B, et al. Experimental free-space quantum teleportation[J]. Nature Photonics, 2010, 4(6): 376-381.
[27] YIN J, REN J G, LU H, et al. Quantum teleportation and entanglement distribution over 100- kilometre free-space channels[J]. Nature, 2012, 488(7410): 185-188.
[28] MA X S, HERBST T, SCHEIDL T, et al. Quantum teleportation over 143 kilometres using active feed-forward[J]. Nature, 2012, 489(7415): 269-273.
[29] REN J G, XU P, YONG H L, et al. Ground-to-satellite quantum teleportation[J]. Nature, 2017, 549(7670): 70-73.
[30] SENKO C, RICHERME P, SMITH J, et al. Realization of a quantum integer-spin chain with controllable interactions[J]. Phys. Rev. X, 2015, 5: 021026.
[31] YAN B, MOSES S A, GADWAY B, et al. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules[J]. Nature, 2013, 501(7468): 521-525.
[32] PARIGI V, D’AMBROSIO V, ARNOLD C, et al. Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory[J]. Nature Communications, 2015, 6 (1): 7706.
[33] DING D S, ZHANG W, SHI S, et al. High-dimensional entanglement between distant atomicensemble memories[J]. Light: Science & Applications, 2016, 5(10): e16157-e16157.
[34] NEELEY M, ANSMANN M, BIALCZAK R C, et al. Emulation of a quantum spin with a superconducting phase qudit[J]. 2009, 325(5941): 722-725.
[35] CHOI S, CHOI J, LANDIG R, et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system[J]. Nature, 2017, 543(7644): 221-225.
[36] GOTTESMAN D, CHUANG I L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations[J]. Nature, 1999, 402(6760): 390-393.
[37] CIRAC J I, ZOLLER P, KIMBLE H J, et al. Quantum state transfer and entanglement distribution among distant nodes in a quantum network[J]. Phys. Rev. Lett., 1997, 78: 3221-3224.
[38] DUAN L, BLINOV B B, MOEHRING D L, et al. Scalable trapped ion quantum computation with a probabilistic ion-photon mapping[J]. Quantum Inf. Comput., 2004, 4(3): 165-173.
[39] CIRAC J I, EKERT A K, HUELGA S F, et al. Distributed quantum computation over noisy channels[J]. Phys. Rev. A, 1999, 59: 4249-4254.
[40] NIELSEN M A, CHUANG I L. Programmable quantum gate arrays[J]. Phys. Rev. Lett., 1997, 79: 321-324.
[41] SØRENSEN A, MØLMER K. Error-free quantum communication through noisy channels[J]. Phys. Rev. A, 1998, 58: 2745-2749.
[42] COLLINS D, LINDEN N, POPESCU S. Nonlocal content of quantum operations[J]. Phys. Rev. A, 2001, 64: 032302.
[43] EISERT J, JACOBS K, PAPADOPOULOS P, et al. Optimal local implementation of nonlocal quantum gates[J]. Phys. Rev. A, 2000, 62: 052317.
[44] HUANG Y F, REN X F, ZHANG Y S, et al. Experimental teleportation of a quantum controllednot gate[J]. Phys. Rev. Lett., 2004, 93: 240501.
[45] GAO W B, GOEBEL A M, LU C Y, et al. Teleportation-based realization of an optical quantum two-qubit entangling gate[J]. Proceedings of the National Academy of Sciences, 2010, 107(49): 20869-20874.
[46] WAN Y, KIENZLER D, ERICKSON S D, et al. Quantum gate teleportation between separated qubits in a trapped-ion processor[J]. Science, 2019, 364(6443): 875-878.
[47] CHOU K S, BLUMOFF J Z, WANG C S, et al. Deterministic teleportation of a quantum gate between two logical qubits[J]. Nature, 2018, 561(7723): 368-373.
[48] HU X M, ZHANG C, LIU B H, et al. Experimental high-dimensional quantum teleportation[J]. Phys. Rev. Lett., 2020, 125: 230501.
[49] WANG C, DENG F G, LI Y S, et al. Quantum secure direct communication with high-dimension quantum superdense coding[J]. Phys. Rev. A, 2005, 71: 044305.
[50] BARBIERI M, VALLONE G, MATALONI P, et al. Complete and deterministic discrimination of polarization bell states assisted by momentum entanglement[J]. Phys. Rev. A, 2007, 75: 042317.
[51] BARBIERI M, CINELLI C, MATALONI P, et al. Polarization-momentum hyperentangled states: Realization and characterization[J]. Phys. Rev. A, 2005, 72: 052110.
[52] VALLONE G, CECCARELLI R, DE MARTINI F, et al. Hyperentanglement of two photons in three degrees of freedom[J]. Phys. Rev. A, 2009, 79: 030301.
[53] SCHUCK C, HUBER G, KURTSIEFER C, et al. Complete deterministic linear optics bell state analysis[J]. Phys. Rev. Lett., 2006, 96: 190501.
[54] SCHRÖDINGER E. The present status of quantum mechanics[J]. Die Naturwissenschaften, 1935, 23(48): 1-26.
[55] EINSTEIN A, PODOLSKY B, ROSEN N. Can quantum-mechanical description of physical reality be considered complete?[J]. Phys. Rev., 1935, 47: 777-780.
[56] BOHM D. A suggested interpretation of the quantum theory in terms of ”hidden” variables[J]. Phys. Rev., 1952, 85: 166-179.
[57] BELL J S. On the einstein podolsky rosen paradox[J]. Physics Physique Fizika, 1964, 1: 195- 200.
[58] FREEDMAN S J, CLAUSER J F. Experimental test of local hidden-variable theories[J]. Phys. Rev. Lett., 1972, 28: 938-941.
[59] ASPECT A, DALIBARD J, ROGER G. Experimental test of bell’s inequalities using timevarying analyzers[J]. Phys. Rev. Lett., 1982, 49: 1804-1807.
[60] WEIHS G, JENNEWEIN T, SIMON C, et al. Violation of bell’s inequality under strict einstein locality conditions[J]. Phys. Rev. Lett., 1998, 81: 5039-5043.
[61] ROWE M A, KIELPINSKI D, MEYER V, et al. Experimental violation of a bell’s inequality with efficient detection[J]. Nature, 2001, 409(6822): 791-794.
[62] MATSUKEVICH D N, MAUNZ P, MOEHRING D L, et al. Bell inequality violation with two remote atomic qubits[J]. Phys. Rev. Lett., 2008, 100: 150404.
[63] ANSMANN M, WANG H, BIALCZAK R C, et al. Violation of bell’s inequality in josephson phase qubits[J]. Nature, 2009, 461(7263): 504-506.
[64] HENSEN B, BERNIEN H, DRÉAU A E, et al. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres[J]. Nature, 2015, 526(7575): 682-686.
[65] WERNER R F. Quantum states with einstein-podolsky-rosen correlations admitting a hiddenvariable model[J]. Phys. Rev. A, 1989, 40: 4277-4281.
[66] KWIAT P G, MATTLE K, WEINFURTER H, et al. New high-intensity source of polarizationentangled photon pairs[J]. Phys. Rev. Lett., 1995, 75: 4337-4341.
[67] GREENBERGER D M, HORNE M, ZEILINGER A. Similarities and differences between two-particle and three-particle interference[J]. Fortschritte der Physik, 2000, 48(4): 243-252.
[68] DÜR W, VIDAL G, CIRAC J I. Three qubits can be entangled in two inequivalent ways[J]. Phys. Rev. A, 2000, 62: 062314.
[69] LU C Y, ZHOU X Q, GÜHNE O, et al. Experimental entanglement of six photons in graph states[J]. Nature Physics, 2007, 3(2): 91-95.
[70] BOUWMEESTER D, PAN J W, DANIELL M, et al. Observation of three-photon greenbergerhorne-zeilinger entanglement[J]. Phys. Rev. Lett., 1999, 82: 1345-1349.
[71] PAN J W, DANIELL M, GASPARONI S, et al. Experimental demonstration of four-photon entanglement and high-fidelity teleportation[J]. Phys. Rev. Lett., 2001, 86: 4435-4438.
[72] YAO X C, WANG T X, XU P, et al. Observation of eight-photon entanglement[J]. Nature Photonics, 2012, 6(4): 225-228.
[73] YAO X C, WANG T X, CHEN H Z, et al. Experimental demonstration of topological error correction[J]. Nature, 2012, 482(7386): 489-494.
[74] WANG X L, CHEN L K, LI W, et al. Experimental ten-photon entanglement[J]. Phys. Rev. Lett., 2016, 117: 210502.
[75] CHEN L K, LI Z D, YAO X C, et al. Observation of ten-photon entanglement using thin bib3o6 crystals[J]. Optica, 2017, 4(1): 77-83.
[76] ZHONG H S, LI Y, LI W, et al. 12-photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairs from parametric down-conversion[J]. Phys. Rev. Lett., 2018, 121: 250505.
[77] BARREIRO J T, LANGFORD N K, PETERS N A, et al. Generation of hyperentangled photon pairs[J]. Phys. Rev. Lett., 2005, 95: 260501.
[78] KWIAT P G, WEINFURTER H. Embedded bell-state analysis[J]. Phys. Rev. A, 1998, 58: R2623-R2626.
[79] SIMON C, PAN J W. Polarization entanglement purification using spatial entanglement[J]. Phys. Rev. Lett., 2002, 89: 257901.
[80] BRUẞ D, MACCHIAVELLO C. Optimal eavesdropping in cryptography with threedimensional quantum states[J]. Phys. Rev. Lett., 2002, 88: 127901.
[81] JOZSA R. Fidelity for mixed quantum states[J]. Journal of Modern Optics, 1994, 41(12): 2315-2323.
[82] NIELSEN M A, CHUANG I L. Quantum computation and quantum information: 10th anniversary edition[M]. [S.l.]: Cambridge University Press, 2010.
[83] NIELSEN M A. A simple formula for the average gate fidelity of a quantum dynamical operation[J]. Physics Letters A, 2002, 303(4): 249-252.
[84] DIRAC P. The principles of quantum mechanics[M]. Oxford: Oxford Science Publications, 1958: 29-48.
[85] WOOTTERS W K, ZUREK W H. A single quantum cannot be cloned[J]. Nature, 1982, 299 (5886): 802-803.
[86] THEW R T, NEMOTO K, WHITE A G, et al. Qudit quantum-state tomography[J]. Phys. Rev. A, 2002, 66: 012303.
[87] SCHAEFER B, COLLETT E, SMYTH R, et al. Measuring the stokes polarization parameters[J]. American Journal of Physics, 2007, 75(2): 163-168.
[88] JAMES D F V, KWIAT P G, MUNRO W J, et al. Measurement of qubits[J]. Phys. Rev. A, 2001, 64: 052312.
[89] HUSZÁR F, HOULSBY N M T. Adaptive bayesian quantum tomography[J]. Phys. Rev. A, 2012, 85: 052120.
[90] OKAMOTO R, IEFUJI M, OYAMA S, et al. Experimental demonstration of adaptive quantum state estimation[J]. Phys. Rev. Lett., 2012, 109: 130404.
[91] CHUANG I L, NIELSEN M A. Prescription for experimental determination of the dynamics of a quantum black box[J]. Journal of Modern Optics, 1997, 44(11-12): 2455-2467.
[92] POYATOS J F, CIRAC J I, ZOLLER P. Complete characterization of a quantum process: The two-bit quantum gate[J]. Phys. Rev. Lett., 1997, 78: 390-393.
[93] CHILDS A M, CHUANG I L, LEUNG D W. Realization of quantum process tomography in nmr[J]. Phys. Rev. A, 2001, 64: 012314.
[94] WEINSTEIN Y S, HAVEL T F, EMERSON J, et al. Quantum process tomography of the quantum fourier transform[J]. The Journal of Chemical Physics, 2004, 121(13): 6117-6133.
[95] DE MARTINI F, MAZZEI A, RICCI M, et al. Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single-qubit device[J]. Phys. Rev. A, 2003, 67: 062307.
[96] ALTEPETER J B, BRANNING D, JEFFREY E, et al. Ancilla-assisted quantum process tomography[J]. Phys. Rev. Lett., 2003, 90: 193601.
[97] O’BRIEN J L, PRYDE G J, GILCHRIST A, et al. Quantum process tomography of a controllednot gate[J]. Phys. Rev. Lett., 2004, 93: 080502.
[98] MITCHELL M W, ELLENOR C W, SCHNEIDER S, et al. Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography[J]. Phys. Rev. Lett., 2003, 91: 120402.
[99] RIEBE M, KIM K, SCHINDLER P, et al. Process tomography of ion trap quantum gates[J]. Phys. Rev. Lett., 2006, 97: 220407.
[100] HANNEKE D, HOME J P, JOST J D, et al. Realization of a programmable two-qubit quantum processor[J]. Nature Physics, 2010, 6(1): 13-16.
[101] NEELEY M, ANSMANN M, BIALCZAK R C, et al. Process tomography of quantum memory in a josephson-phase qubit coupled to a two-level state[J]. Nature Physics, 2008, 4(7): 523-526.
[102] CHOW J M, GAMBETTA J M, TORNBERG L, et al. Randomized benchmarking and process tomography for gate errors in a solid-state qubit[J]. Phys. Rev. Lett., 2009, 102: 090502.
[103] BIALCZAK R C, ANSMANN M, HOFHEINZ M, et al. Quantum process tomography of a universal entangling gate implemented with josephson phase qubits[J]. Nature Physics, 2010, 6(6): 409-413.
[104] YAMAMOTO T, NEELEY M, LUCERO E, et al. Quantum process tomography of two-qubit controlled-z and controlled-not gates using superconducting phase qubits[J]. Phys. Rev. B, 2010, 82: 184515.
[105] CHOW J M, CÓRCOLES A D, GAMBETTA J M, et al. Simple all-microwave entangling gate for fixed-frequency superconducting qubits[J]. Phys. Rev. Lett., 2011, 107: 080502.
[106] DEWES A, ONG F R, SCHMITT V, et al. Characterization of a two-transmon processor with individual single-shot qubit readout[J]. Phys. Rev. Lett., 2012, 108: 057002.
[107] ZHANG J, SOUZA A M, BRANDAO F D, et al. Protected quantum computing: Interleaving gate operations with dynamical decoupling sequences[J]. Phys. Rev. Lett., 2014, 112: 050502.
[108] HOWARD M, TWAMLEY J, WITTMANN C, et al. Quantum process tomography and linblad estimation of a solid-state qubit[J]. New Journal of Physics, 2006, 8(3): 33-33.
[109] CHUANG I L, GERSHENFELD N, KUBINEC M G, et al. Bulk quantum computation with nuclear magnetic resonance: theory and experiment[J]. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 1998, 454(1969): 447- 467.
[110] SENKO C, RICHERME P, SMITH J, et al. Realization of a quantum integer-spin chain with controllable interactions[J]. Phys. Rev. X, 2015, 5: 021026.
[111] YAN B, MOSES S A, GADWAY B, et al. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules[J]. Nature, 2013, 501(7468): 521-525.
[112] PARIGI V, D’AMBROSIO V, ARNOLD C, et al. Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory[J]. Nature Communications, 2015, 6 (1): 7706.
[113] DING D S, ZHANG W, SHI S, et al. High-dimensional entanglement between distant atomicensemble memories[J]. Light: Science & Applications, 2016, 5(10): e16157-e16157.
[114] NEELEY M, ANSMANN M, BIALCZAK R C, et al. Emulation of a quantum spin with a superconducting phase qudit[J]. Science, 2009, 325(5941): 722-725.
[115] CHOI S, CHOI J, LANDIG R, et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system[J]. Nature, 2017, 543(7644): 221-225.
[116] RIGOLIN G. Quantum teleportation of an arbitrary two-qubit state and its relation to multipartite entanglement[J]. Phys. Rev. A, 2005, 71: 032303.
[117] LIU S, LOU Y, JING J. Orbital angular momentum multiplexed deterministic all-optical quantum teleportation[J]. Nature Communications, 2020, 11(1): 3875.
[118] WANG X L, CAI X D, SU Z E, et al. Quantum teleportation of multiple degrees of freedom of a single photon[J]. Nature, 2015, 518(7540): 516-519.
[119] HU X M, ZHANG C, LIU B H, et al. Experimental high-dimensional quantum teleportation[J]. Phys. Rev. Lett., 2020, 125: 230501.
[120] PRESKILL J. Quantum computing in the nisq era and beyond[J]. Quantum, 2018, 2.
[121] AVRON J, CASPER O, ROZEN I. Quantum advantage and noise reduction in distributed quantum computing[J]. Phys. Rev. A, 2021, 104: 052404.
[122] EISERT J, JACOBS K, PAPADOPOULOS P, et al. Optimal local implementation of nonlocal quantum gates[J]. Phys. Rev. A, 2000, 62: 052317.
[123] JIANG L, TAYLOR J M, SØRENSEN A S, et al. Distributed quantum computation based on small quantum registers[J]. Phys. Rev. A, 2007, 76: 062323.
[124] GOTTESMAN D, CHUANG I L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations[J]. Nature, 1999, 402(6760): 390-393.
[125] EISERT J, JACOBS K, PAPADOPOULOS P, et al. Optimal local implementation of nonlocal quantum gates[J]. Phys. Rev. A, 2000, 62: 052317.
[126] COLLINS D, LINDEN N, POPESCU S. Nonlocal content of quantum operations[J]. Phys. Rev. A, 2001, 64: 032302.
[127] HOFMANN H F. Complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations[J]. Phys. Rev. Lett., 2005, 94: 160504.
[128] HOFMANN H F, OKAMOTO R, TAKEUCHI S. Analysis of an experimental quantum logic gate by complementary classical operations[J]. Modern Physics Letters A, 2006, 21(24): 1837- 1850.
[129] HOFMANN H F, OKAMOTO R, TAKEUCHI S. Locally observable conditions for the successful implementation of entangling multi-qubit quantum gates[M]. [S.l.: s.n.]: 68-71.