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Quantum computing has been proven to be powerful, however, there are still great challenges for building real quantum computers due to the requirements of both fault-tolerance and universality. People still lack a systematic way to design fast quantum algorithms and identify the key quantum resources. In this work, we develop a resource-theoretic approach to characterize universal quantum computing models and the universal resources for quantum computing. Our theory combines the framework of universal quantum computing model (UQCM) and the quantum resource theory (QRT). The former has played major roles in quantum computing, while the later was developed mainly for quantum information theory. Putting them together proves to be ‘win-win’: on one hand, using QRT can provide a resource-theoretic characterization of a UQCM, the relation among models and inspire new ones, and on the other hand, using UQCM offers a framework to apply resources, study relation among resources and classify them. In quantum theory, we mainly study states, evolution, observable, and probability from measurements, and this motivates the introduction of different families of UQCMs. A family also includes generations depending on a hierarchical structure of resource theories. We introduce a table of UQCMs by first classifying two categories of models: one referring to the format of information, and one referring to the logical evolution of information requiring quantum error-correction codes. Each category contains a few families of models, leading to more than one hundred of them in total. Such a rich spectrum of models include some well-known ones that people use, such as the circuit model, the adiabatic model, but many of them are relatively new and worthy of more study in the future. Among them are the models of quantum von Neumann architectures established recently. This type of architecture or model circumvents the no-go theorems on both the quantum program storage and quantum control unit, enabling the construction of more complete quantum computer systems and high-level programming. Correspondingly, each model is captured by a unique quantum resource. For instance, in the state family, the universal resource for the circuit model is coherence, for the local quantum Turing machine is bipartite entanglement, and for the cluster-state based, also known as measurement-based model is a specific type of entanglement relevant to symmetry-protected topological order. As program-storage is a central feature of the quantum von Neumann architecture, we find the quantum resources for it are quantum memories, which are dynamical resources closely related to entanglement. In other words, our classification of UQCMs also serves as a computational classification of quantum resources. This can be used to resolve the dispute over the computing power of resources, such as interference, entanglement, or contextuality. In all, we believe our theory lays down a solid framework to study computing models, resources, and design algorithms. -
图 1 经典与量子信息领域的一些发展阶段. 经典(上部): 在世纪之交, 希尔伯特提出了著名的23个问题, 其中一个启发了图灵对于计算的研究, 直接奠定了计算机科学的理论基础. 香农证明了通信的三大定理, 为纠错码理论奠定基础. 同时, 冯诺依曼提出了通用计算机的架构理论. 之后, PN结和三极管的发明奠定了电子计算机的硬件基础, 然后发展到大规模可编程集成电路(IC). 量子(下部): 早期有EPR和Bell关于量子纠缠和非定域性的探讨. 之后, 经Holevo, Kraus等人将量子信道演化、退相干、测量等数学形式发展出来. BB84是首个利用量子不确定性的保密通信方案, 整个领域从此开始起步. 在理论方面, 量子资源理论(QRT)作为描述量子信息的完备理论逐渐发展成熟
Figure 1. Development of classical and quantum information science. Classical (up): From the 23 problems of Hilbert, Turing laid the foundation of computation science. Shannon established the theory of communication, and von Neumann established the architecture of computers. The next breakthrough include PN junction and transistor, forming the building blocks of modern integrated circuits. Quantum (down): With the early study of EPR and Bell, the mathematical formalism of quantum channel, decoherence, and measurement were developed by Holevo, Kraus, etc. The BB84 secure protocol boosted the field. The theoretical achievement is the recent development of quantum resource theory as the theory of quantum information.
图 2 计算机系统的层次化设计原理. 从整体上看, 可以分为硬件层次和软件层次, 也可以区分出软硬件之间的架构层次, 即微体系结构的设计
Figure 2. Hierarchy of computer system. There are layers of hardware and software, and also the layers of system architecture.
图 3 量子线路模型示意及算法设计结构. 基本结构(左上)包括某经典算法A和它设计的量子线路 U以及测量方式(三角符号). 也可以扩展为经典-量子混合的迭代结构(右上), 或等价地表示为线性方式(下)
Figure 3. Structures of quantum circuit model and quantum algorithms. The basic structure (top-left) has a classical algorithm A that designs the quantum circuit U and measurement. It extends to the iterative classical-quantum algorithms (top-right), which can be “stretched” into a linear flow (bottom).
图 4 量子纠错过程示意, 即用量子超信道$ \hat{{\cal{S}}} $将$ \Phi^{\otimes n} $近似地转化为$ {1 1}^{\otimes k} $
Figure 4. Structure of quantum error correction that converts $ \Phi^{\otimes n} $ into $ {1 1}^{\otimes k} $ approximately by a superchannel $ \hat{{\cal{S}}} $.
图 5 量子资源理论框架下的计算模型. 第I类模型主要是针对输入端, 即信息的不同表示形式, 第II类模型主要是针对编码后的逻辑操作的形式
Figure 5. Structure of quantum computing model via quantum resource theory. The Category-I (-II) models are defined for different types of input (logical operations).
图 6 通用量子计算模型分类表. 第I类模型即形式类有12个模型, 第II类模型即演化类有9个模型, 因而一共108个完备的模型(灰色方格). 其中研究最多的是基于线路模型的各种方案. 信道家族的模型统称为量子冯诺依曼模型或架构. 模型之间也可以进行混合搭配
Figure 6. The classification table of universal quantum computing models. There are 12 (9) Category-I (-II) models, hence in total 108 complete models (grey boxes). The most well-studied are those based on circuit model. The channel-family models are all von Neumann architecture or models. Hybridization among models are also allowed.
图 7 矩阵乘积态的等价表示方式. (上) 张量形式: 横线是纠缠空间, 竖线是不同的物理空间, 方框代表张量(或矩阵). (中) VBS或AKLT形式[69]: 张量由圈代表的算子构造, 横向线段代表Bell态, 对应公式(11). (下) 量子线路形式: 每个张量可以由幺正过程(大框)实现
Figure 7. Representations of matrix-product states. (Top) Tensor form: the top register is the entanglement space, the vertical wires are physical sites, the boxes are the tensors or matrices. (Middle) VBS or AKLT form[69]: tensors are defined by local operators (circles) acting on Bell states (Eq.11). (Bottom) Circuit form: each tensor is realized by a unitary circuit (big boxes).
图 8 量子冯诺依曼架构示意图(左)与量子线路模型(右). 对比来看, 在通常的线路模型中不考虑量子的控制单元和量子的程序存储
Figure 8. Schematics of the quantum von Neumann architecture (left) and the circuit model (right). For the later, there is no explicit quantum control unit and storage of quantum programs.
图 9 量子超算法结构示意. 其母算法(阴影部分)将输入的数据(方框)转化为所需的算法即子算法, 完成上端数据系统的输入输出过程(自左向右). 经典-量子混合架构是其特例(图3), 且MPS结构(图7)也可以看作其特例. 输入(方框)之间也可以存在量子关联(未表示)
Figure 9. Schemetics of quantum super-algorithm. The `mother' algorithm (shaded) maps the input data (boxes) into the desired `child' algorithm, which acts on the data system (top register). The classical-quantum hybrid algorithm (Fig. 3) is a special case, and the MPS formula (Fig. 7) is also a special case of it. There can also be quantum correlation or memory (unshown) between the input (boxes).
图 10 在现有的经典计算和未来通用量子计算之间, 还存在其他的研究范式, 比如专用光芯片、忆阻器、适用于人工智能的GPU、以及量子模拟等
Figure 10. In between the current classical computers and the universal quantum computers in the future, there are other research paradigm, such as optical chips, memristors, GPU for AI, and quantum simulators etc.
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[1] Preskill J 2018 Quantum 2 79
Google Scholar
[2] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge U.K.: Cambridge University Press
[3] Pan J W 2024 Acta Phys. Sin. 73 010301
Google Scholar
[4] Bell J S 1966 Rev. Mod. Phys. 38 447
Google Scholar
[5] Kraus K 1983 States, Effects, and Operations: Fundamental Notions of Quantum Theory, vol. 190 of Lecture Notes in Physics (Berlin: Springer-Verlag
[6] Holevo A S 1982 Probabilistic and Statistical Aspect of Quantum Theory (North-Holland, Amsterdam
[7] Feynman R P 1982 Int. J. Theor. Phys. 21 467
Google Scholar
[8] Deutsch D 1985 In Proc. R. Soc. (London), Ser. A, vol. 400 (The Royal Society), pp 97–117
[9] Yao A C C 1993 In Foundations of Computer Science, 1993. Proceedings., 34 th Annual Symposium on (IEEE), pp 352–361
[10] Bernstein E, Vazirani U 1997 SIAM J. Comput. 26 1411
Google Scholar
[11] Shor P W 1994 In Proceedings 35 th annual symposium on foundations of computer science (IEEE), pp 124–134
[12] Harris D M, Harris S L 2013 Digital design and computer architecture (Elsevier
[13] Shannon C 1948 The Bell System Technical Journal 27 379
Google Scholar
[14] von Neumann J 1993 IEEE Annals of the History of Computing 15 27
Google Scholar
[15] Lidar D, Brun T A, editors 2013 Quantum error correction (Cambridge University Press
[16] Ladd T D, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O’ Brien J L 2010 Nature 464 45
Google Scholar
[17] Grover L K 1996 In Proceedings of the twenty-eighth annual ACM Symposium on Theory of Computing
[18] Harrow A W, Hassidim A, Lloyd S 2009 Phys. Rev. Lett. 103 150502
Google Scholar
[19] Long G L 2011 Int. J. Theor. Phys. 50 1305
Google Scholar
[20] Martyn J M, Rossi Z M, Tan A K, Chuang I L 2021 PRX Quantum 2 040203
Google Scholar
[21] Watrous J 2018 The Theory of Quantum Information (Cambridge University Press
[22] Hayashi M 2017 Quantum Information Theory: Mathematical Foundation, 2 nd edition (Springer
[23] Wilde M 2017 Quantum Information Theory (Cambridge University Press
[24] Chitambar E, Gour G 2019 Rev. Mod. Phys. 91 025001
Google Scholar
[25] Wang D S 2023 Commun. Theor. Phys. 75 125101
Google Scholar
[26] Albash T, Lidar D A 2018 Rev. Mod. Phys. 90 015002
Google Scholar
[27] Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Rev. Mod. Phys. 80 1083
Google Scholar
[28] Childs A M, Gosset D, Webb Z 2013 Science 339 791
Google Scholar
[29] Arrighi P 2019 Natural Computing 18 885
Google Scholar
[30] Briegel H J, Browne D E, Dür W, Raussendorf R, Van den Nest M 2009 Nat. Phys. 5 19
Google Scholar
[31] Barenco A, Bennett C H, Cleve R, DiVincenzo D P, Margolus N, Shor P, Sleator T, Smolin J A, Weinfurter H 1995 Phys. Rev. A 52 3457
Google Scholar
[32] DiVincenzo D P 2000. ArXiv: quant-ph/0002077
[33] Nielsen M A, Chuang I L 1997 Phys. Rev. Lett. 79 321
Google Scholar
[34] Yang Y, Renner R, Chiribella G 2020 Phys. Rev. Lett. 125 210501
Google Scholar
[35] Wang D S 2022 Commun. Theor. Phys. 74 095103
Google Scholar
[36] Dawson C M, Nielsen M A 2006 Quantum Inf. Comput. 6 81
[37] Lloyd S 1996 Science 273 1073
Google Scholar
[38] Brassard G, Høyer P, Mosca M, Tapp A 2002 Contem. Mathemat. 305 53
[39] Knill E, Laflamme R 1997 Phys. Rev. A 55 900
Google Scholar
[40] Chiribella G, D’Ariano G M, Perinotti P 2008 Europhys. Lett. 83 30004
Google Scholar
[41] Chiribella G, D’Ariano G M, Perinotti P 2008 Phys. Rev. Lett. 101 060401
Google Scholar
[42] Chiribella G, D’Ariano G M, Perinotti P 2009 Phys. Rev. A 80 022339
Google Scholar
[43] Choi M D 1975 Linear Algebra Appl. 10 285
Google Scholar
[44] Bény C, Oreshkov O 2010 Phys. Rev. Lett. 104 120501
Google Scholar
[45] Gottesman D 1998 Phys. Rev. A 57 127
Google Scholar
[46] Wang D S, Zhu G, Okay C, Laflamme R 2020 Phys. Rev. Res. 2 033116
Google Scholar
[47] Wang D S, Wang Y J, Cao N, Zeng B, Laflamme R 2022 New J. Phys. 24 023019
Google Scholar
[48] Zhou S, Liu Z W, Jiang L 2021 Quantum 5 521
Google Scholar
[49] Yang Y, Mo Y, Renes J M, Chiribella G, Woods M P 2022 Phys. Rev. Res. 4 023107
Google Scholar
[50] Kubica A, Demkowicz-Dobrzański R 2021 Phys. Rev. Lett. 126 150503
Google Scholar
[51] Viola L, Knill E, Lloyd S 1999 Phys. Rev. Lett. 82 2417
Google Scholar
[52] Kitaev A Y 2003 Ann. Phys. 303 2
Google Scholar
[53] Ryan W E, Lin S 2009 Channel Codes: Classical and Modern (Cambridge University Press
[54] Breuckmann N P, Eberhardt J N 2021 PRX Quantum 2 040101
Google Scholar
[55] Wang D S, Liu Y D, Wang Y J, Luo S 2024 Phys. Rev. A 110 032413
Google Scholar
[56] Coecke B, Fritz T, Spekkens R W 2016 Information and Computation 250 59
Google Scholar
[57] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
Google Scholar
[58] Streltsov A, Adesso G, Plenio M B 2017 Rev. Mod. Phys. 89 041003
Google Scholar
[59] Wang D S 2021 Quantum Engineering 2 e85
[60] Wang D S 2020 Quant. Infor. Comput. 20 0213
[61] Raussendorf R, Briegel H J 2001 Phys. Rev. Lett. 86 5188
Google Scholar
[62] Nielsen M A 2006 Reports on Math. Phys. 57 147
Google Scholar
[63] Wang D S, Stephen D T, Raussendorf R 2017 Phys. Rev. A 95 032312
Google Scholar
[64] Stephen D T, Wang D S, Prakash A, Wei T C, Raussendorf R 2017 Phys. Rev. Lett. 119 010504
Google Scholar
[65] Raussendorf R, Okay C, Wang D S, Stephen D T, Nautrup H P 2019 Phys. Rev. Lett. 122 090501
Google Scholar
[66] Molina A, Watrous J 2019 Proc. Royal Soc. A 475 20180767
Google Scholar
[67] Paetznick A, Reichardt B W 2013 Phys. Rev. Lett. 111 090505
Google Scholar
[68] Tóth G, Apellaniz I 2014 J. Phys. A: Math. Theor. 47 424006
Google Scholar
[69] Affleck I, Kennedy T, Lieb E H, Tasaki H 1987 Phys. Rev. Lett. 59 799
Google Scholar
[70] Fannes M, Nachtergaele B, Werner R F 1992 Commun. Math. Phys. 144 443
Google Scholar
[71] Perez-Garcia D, Verstraete F, Wolf M, Cirac J 2007 Quantum Inf. Comput. 7 401
[72] Sarovar M, Proctor T, Rudinger K, Young K, Nielsen E, Blume-Kohout R 2020 Quantum 4 321
Google Scholar
[73] Crépeau C, Gottesman D, Smith A 2002 In STOC ’ 02: Proc. 34 rd Annual ACM Symp. Theory of Computing. p 643
[74] Broadbent A, Fitzsimons J, Kashefi E 2009 In Proceedings of the 50 th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, CA, 2009). pp 517–527
[75] Myers J M 1997 Phys. Rev. Lett. 78 1823
Google Scholar
[76] Cirac J I, Pérez-García D, Schuch N, Verstraete F 2021 Rev. Mod. Phys. 93 045003
Google Scholar
[77] Wehner S, Elkouss D, Hanson R 2018 Science 362 303
[78] Wang D S 2019 Int. J. Mod. Phys. B 33 1930004
Google Scholar
[79] Wang D S 2020 Phys. Rev. A 101 052311
Google Scholar
[80] Van den Nest M, Dür W, Vidal G, Briegel H J 2007 Phys. Rev. A 75 012337
Google Scholar
[81] Van den Nest M, Dür W, Miyake A, Briegel H J 2007 New J. Phys. 9 204
Google Scholar
[82] Gross D, Flammia S T, Eisert J 2009 Phys. Rev. Lett. 102 190501
Google Scholar
[83] Bremner M J, Mora C, Winter A 2009 Phys. Rev. Lett. 102 190502
Google Scholar
[84] Gu Z C, Wen X G 2009 Phys. Rev. B 80 155131
Google Scholar
[85] Chen X, Gu Z C, Wen X G 2011 Phys. Rev. B 83 035107
Google Scholar
[86] Schuch N, Pérez-García D, Cirac I 2011 Phys. Rev. B 84 165139
Google Scholar
[87] Bartolucci S, Birchall P, Bombin H, Cable H, Dawson C, Gimeno-Segovia M, Johnston E, Kieling K, Nickerson N, Pant M, Pastawski F, Rudolph T, Sparrow C 2023 Nat. Commun. 14 912
Google Scholar
[88] Kitaev A, Shen A H, Vyalyi M N 2002 Classical and Quantum Computation, vol. 47 of Graduate Studies in Mathematics (Providence: American Mathematical Society
[89] Wocjan P, Roetteler M, Janzing D, Beth T 2002 Quant. Infor. Comput. 2 133
[90] Dodd J L, Nielsen M A, Bremner M J, Thew R T 2002 Phys. Rev. A 65 040301
Google Scholar
[91] Cubitt T S, Montanaro A, Piddock S 2018 Proceedings of the National Academy of Sciences 115 9497
Google Scholar
[92] Kohler T, Piddock S, Bausch J, Cubitt T 2021 Henri Poincaré 23 223
[93] Kohler T, Piddock S, Bausch J, Cubitt T 2022 PRX Quantum 3 010308
Google Scholar
[94] Berry D W, Ahokas G, Cleve R, Sanders B C 2007 Commun. Math. Phys. 270 359
Google Scholar
[95] Cirac J I, Zoller P 2012 Nat. Phys. 8 264
Google Scholar
[96] Shepherd D J, Franz T, Werner R F 2006 Phys. Rev. Lett. 97 020502
Google Scholar
[97] Janzing D 2007 Phys. Rev. A 75 012307
Google Scholar
[98] Nagaj D, Wocjan P 2008 Phys. Rev. A 78 032311
Google Scholar
[99] Nagaj D 2012 Phys. Rev. A 85 032330
Google Scholar
[100] Lloyd S, Terhal B 2016 New J. Phys. 18 023042
Google Scholar
[101] Toffoli T, Margolus N 1987 Cellular Automata Machines: A new environment for modeling (MIT Press
[102] Bisio A, D’ Ariano G M, Tosini A 2015 Ann. Phys. 354 244
Google Scholar
[103] Heim B, Rønnow T F, Isakov S V, Troyer M 2015 Science 348 215
Google Scholar
[104] Villanueva A, Najafi P, Kappen H J 2023 J. Phys. A: Math. Theor. 56 465304
Google Scholar
[105] Bravyi S, DiVincenzo D P, Oliveira R I, Terhal B M 2008 Quantum Infor. Comput. 8 0361
[106] Zhang J, Kyaw T H, Filipp S, Kwek L C, Sjöqvist E, Tong D 2023 Phys. Reports 1027 1
Google Scholar
[107] Wootters W K 1987 Ann. Phys. 176 1
Google Scholar
[108] Gross D 2006 J. Math. Phys. 47 122107
Google Scholar
[109] Bravyi S, Kitaev A 2005 Phys. Rev. A 71 022316
Google Scholar
[110] Popescu S, Rohrlich D 1994 Found. Phys. 24 379
Google Scholar
[111] Spekkens R W 2008 Phys. Rev. Lett. 101 020401
Google Scholar
[112] Spekkens R W 2005 Phys. Rev. A 71 052108
Google Scholar
[113] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895
Google Scholar
[114] Gottesman D, Chuang I L 1999 Nature 402 390
Google Scholar
[115] Long G L 2006 Commun. Theor. Phys. 45 825
Google Scholar
[116] Childs A M, Wiebe N 2012 Quant. Infor. Comput. 12 901
[117] Berry D W, Childs A M, Cleve R, Kothari R, Somma R D 2015 Phys. Rev. Lett. 114 090502
Google Scholar
[118] Wei S, Long G L 2016 Quantum Infor. Processing 15 1189
Google Scholar
[119] Zhou X, Leung D W, Chuang I L 2000 Phys. Rev. A 62 052316
Google Scholar
[120] Broadbent A 2016 Phys. Rev. A 94 022318
Google Scholar
[121] Clauser J F, Horne M A, Shimony A, Holt R A 1969 Phys. Rev. Lett. 23 880
Google Scholar
[122] Vaidman L 2003 Phys. Rev. Lett. 90 010402
Google Scholar
[123] Brassard G, Buhrman H, Linden N, Méthot A A, Tapp A, Unger F 2006 Phys. Rev. Lett. 96 250401
Google Scholar
[124] Chitambar E, Leung D, Mančinska L, Ozols M, Winter A 2014 Commun. Math. Phys. 328 303
Google Scholar
[125] Bennett C H, DiVincenzo D P, Smolin J A 1997 Phys. Rev. Lett. 78 3217
Google Scholar
[126] Gheorghiu A, Kapourniotis T, Kashefi E 2019 Theory of Computing Systems 63 715
Google Scholar
[127] Wang D S 2024 Chin. Phys. B 33 080302
Google Scholar
[128] Horodecki M, Shor P, Ruskai M B 2003 Rev. Math. Phys. 15 629
Google Scholar
[129] Rosset D, Buscemi F, Liang Y C 2018 Phys. Rev. X 8 021033
[130] Li L, Hall M J W, Wiseman H M 2018 Phys. Reports 759 1
Google Scholar
[131] Eastin B, Knill E 2009 Phys. Rev. Lett. 102 110502
Google Scholar
[132] Degen C L, Reinhard F, Cappellaro P 2017 Rev. Mod. Phys. 89 035002
Google Scholar
[133] Yoder T J, Takagi R, Chuang I L 2016 Phys. Rev. X 6 031039
[134] Zeng B, Chen X, Zhou D L, Wen X G 2019 Quantum Information Meets Quantum Matter (Springer-Verlag New York
[135] Koenig R, Kuperberg G, Reichardt B W 2010 Ann. Phys. 325 2707
Google Scholar
[136] Brown B J, Loss D, Pachos J K, Self C N, Wootton J R 2016 Rev. Mod. Phys. 88 045005
Google Scholar
[137] Sarma S D, Freedman M, Nayak C 2015 npj Quantum Information 1 15001
Google Scholar
[138] Harper F, Roy R, Rudner M S, Sondhi S L 2020 Ann. Rev. Condensed Matter Phys. 11 345
Google Scholar
[139] Khodjasteh K, Lidar D A 2008 Phys. Rev. A 78 012355
Google Scholar
[140] Verstraete F, Wolf M M, Cirac J I 2009 Nat. Phys. 5 633
Google Scholar
[141] Anderson J T, Duclos-Cianci G, Poulin D 2014 Phys. Rev. Lett. 113 080501
Google Scholar
[142] Liu Y T, Wang K, Liu Y D, Wang D S 2023 Entropy 25 1187
Google Scholar
[143] Araujo M, Feix A, Costa F, Brukner C 2014 New J. Phys. 16 093026
Google Scholar
[144] Bengtsson I, Życzkowski K 2006 Geometry of Quantum States (Cambridge U.K.: Cambridge University Press
[145] Wang K, Wang D S 2023 New J. Phys. 25 043013
Google Scholar
[146] Bennett C H, Brassard G 1984 In Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (Bangalore, India, (IEEE, New York)), p 175–179
[147] Morris J, Saggio V, Gocanin A, Dakic B 2022 Adv. Quantum Technol. 5 2100118
Google Scholar
[148] Huang H L, Wu D, Fan D, Zhu X 2020 Sci. China Inf. Sci. 63 180501
Google Scholar
[149] Wang J, Sciarrino F, Laing A, Thompson M G 2020 Nat. Photonics 14 273
Google Scholar
[150] Editorial 2022 Nat. Rev. Phys. 4 1
Google Scholar
[151] Ma Y, Ma Y Z, Zhou Z Q, Li C F, Guo G C 2021 Nat. Commun. 12 2381
Google Scholar
[152] Chiribella G, D’Ariano G M, Perinotti P 2008 Phys. Rev. Lett. 101 180501
Google Scholar
[153] Gutoski G, Watrous J 2007 In Proceedings of the 39 th ACM Symposium on Theory of Computing. p 565574
[154] Mehta P, Bukov M, Wang C H, Day A G R, Richardson C, Fisher C K, Schwab D J 2019 Phys. Reports 810 1
Google Scholar
[155] Lim D, Doriguello J F, Rebentrost P 2023. ArXiv: quantph/2304.02262
[156] Dunjko V, Briegel H J 2018 Rep. Prog. Phys. 81 074001
Google Scholar
[157] Verdon G, Pye J, Broughton M 2018. ArXiv: quantph/1806.09729
[158] Benedetti M, Lloyd E, Sack S, Fiorentini M 2019 Quantum Science and Technology 4 043001
Google Scholar
[159] Huang H Y, Kueng R, Preskill J 2021 Phys. Rev. Lett. 126 190505
Google Scholar
[160] Caro M C 2024 ACM Trans. Quantum Comput. 5 2
[161] Cleve R, Ekert A, Macchiavello C, Mosca M 1998 Proc. R. Soc. (London), Ser. A 454 339
Google Scholar
[162] Jozsa R, Linden N 2003 Proc. R. Soc. (London), Ser. A 459 2011
Google Scholar
[163] Steane A M 2003 Studies in History and Philosophy of Modern Physics 34 469
Google Scholar
[164] Van den Nest M 2013 Phys. Rev. Lett. 110 060504
Google Scholar
[165] Howard M, Wallman J, Veitch V, Emerson J 2014 Nature 510 351
Google Scholar
[166] Giovannetti V, Maccone L, Morimae T, Rudolph T G 2013 Phys. Rev. Lett. 111 230501
Google Scholar
[167] Holevo A S 1977 Rep. Math. Phys. 12 273
Google Scholar
[168] Tajima H, Shiraishi N, Saito K 2018 Phys. Rev. Lett. 121 110403
Google Scholar
[169] Chiribella G, Yang Y, Renner R 2021 Phys. Rev. X 11 021014
[170] Weedbrook C, Pirandola S, García-Patrón R, Cerf N J, Ralph T C, Shapiro J H, Lloyd S 2012 Rev. Mod. Phys. 84 621
Google Scholar
[171] Xu K, Fan H 2022 Chin. Phys. B 31 100304
Google Scholar
[172] Emerson J, Weinstein Y S, Saraceno M, Lloyd S, Cory D G 2003 Science 302 2098
Google Scholar
[173] Qin D, Xu X, Li Y 2022 Chin. Phys. B 31 090306
Google Scholar
[174] Smith G, Yard J 2008 Science 321 1812
Google Scholar
[175] Sauerwein D, Wallach N R, Gour G, Kraus B 2018 Phys. Rev. X 8 031020
[176] Google Quantum AI and Collaborators 2024. ArXiv preprint arXiv: 2408.13687
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