1Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, E-28049 Madrid, Spain
2Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
The duration of bidirectional transfer protocols in 1D topological models usually scales exponentially with distance. In this work, we propose transfer protocols in multidomain SSH chains and Creutz ladders that lose the exponential dependence, greatly speeding up the process with respect to their single-domain counterparts, reducing the accumulation of errors and drastically increasing their performance, even in the presence of symmetry-breaking disorder. We also investigate how to harness the localization properties of the Creutz ladder-with two localized modes per domain wall-to choose the two states along the ladder that will be swapped during the transfer protocol, without disturbing the states located in the intermediate walls between them. This provides a 1D network with all-to-all connectivity that can be helpful for quantum information purposes.
Featured image: (a) Four-domain SSH chain, with its protected states. (b) Occupation of the protected states in a left-to-right transfer in the chain above, both for the numerical simulation (solid line) and the analytical effective model (dotted line). (c) Transfer time (on a logarithmic scale) as a function of the total length for the single- and two-domain cases, and for the case with an increasing number of equal domains. The inset shows the same data on a linear scale.
Popular summary
However, the duration of this process depends exponentially on transfer distance, making it extremely slow in large chains, like those needed for long-range quantum information transmission. Additionally, this also causes the fidelity of the protocols to drop dramatically when disorder breaks the protecting symmetry.
In this work, we show that, if we instead consider a chain composed of many small pieces with different topological properties, the boundaries between these regions act as amplifiers and greatly speed up the transfer. We show this process is much more robust than the single-region alternative in the case with broken symmetry.
Finally, we also consider another topological model made with two cross-linked chains threaded by a magnetic field. In it, the transfers can take place between any two region boundaries, not only between the ends, without disturbing the states located between them. This can be used to obtain complete connectivity between the nodes in a quantum network, which is fundamental for quantum communication and computing.
► BibTeX data
► References
[1] John Preskill. “Quantum computing in the NISQ era and beyond”. Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79
[2] N. Y. Yao, C. R. Laumann, A. V. Gorshkov, H. Weimer, L. Jiang, J. I. Cirac, P. Zoller, and M. D. Lukin. “Topologically protected quantum state transfer in a chiral spin liquid”. Nature Communications 4, 1585 (2013).
https://doi.org/10.1038/ncomms2531
[3] C. Dlaska, B. Vermersch, and P. Zoller. “Robust quantum state transfer via topologically protected edge channels in dipolar arrays”. Quantum Science and Technology 2, 015001 (2017).
https://doi.org/10.1088/2058-9565/2/1/015001
[4] Marc Antoine Lemonde, Vittorio Peano, Peter Rabl, and Dimitris G. Angelakis. “Quantum state transfer via acoustic edge states in a 2D optomechanical array”. New Journal of Physics 21, 113030 (2019).
https://doi.org/10.1088/1367-2630/AB51F5
[5] M. Bello, C. E. Creffield, and G. Platero. “Sublattice dynamics and quantum state transfer of doublons in two-dimensional lattices”. Physical Review B 95, 094303 (2017).
https://doi.org/10.1103/PhysRevB.95.094303
[6] Martin Leijnse and Karsten Flensberg. “Quantum information transfer between topological and spin qubit systems”. Physical Review Letters 107, 210502 (2011).
https://doi.org/10.1103/PHYSREVLETT.107.210502
[7] Guilherme M. A. Almeida, Francesco Ciccarello, Tony J. G. Apollaro, and Andre M. C. Souza. “Quantum-state transfer in staggered coupled-cavity arrays”. Physical Review A 93, 032310 (2016).
https://doi.org/10.1103/PhysRevA.93.032310
[8] Nicolai Lang and Hans Peter Büchler. “Topological networks for quantum communication between distant qubits”. npj Quantum Information 3, 47 (2017).
https://doi.org/10.1038/s41534-017-0047-x
[9] L. Jin, P. Wang, and Z. Song. “Su-Schrieffer-Heeger chain with one pair of PT-symmetric defects”. Scientific Reports 7, 5903 (2017).
https://doi.org/10.1038/s41598-017-06198-9
[10] Marta P. Estarellas, Irene D’Amico, and Timothy P. Spiller. “Topologically protected localised states in spin chains”. Scientific Reports 7, 42904 (2017).
https://doi.org/10.1038/srep42904
[11] Feng Mei, Gang Chen, Lin Tian, Shi Liang Zhu, and Suotang Jia. “Robust quantum state transfer via topological edge states in superconducting qubit chains”. Physical Review A 98, 012331 (2018).
https://doi.org/10.1103/PhysRevA.98.012331
[12] Péter Boross, János K. Asbóth, Gábor Széchenyi, László Oroszlány, and András Pályi. “Poor man’s topological quantum gate based on the Su-Schrieffer-Heeger model”. Physical Review B 100, 045414 (2019).
https://doi.org/10.1103/PhysRevB.100.045414
[13] C. Yuce. “Spontaneous topological pumping in non-Hermitian systems”. Physical Review A 99, 032109 (2019).
https://doi.org/10.1103/PHYSREVA.99.032109
[14] Felippo M D’Angelis, Felipe A Pinheiro, David Guéry-Odelin, Stefano Longhi, and François Impens. “Fast and robust quantum state transfer in a topological Su-Schrieffer-Heeger chain with next-to-nearest-neighbor interactions”. Physical Review Research 2, 033475 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033475
[15] Senmao Tan, Raditya Weda Bomantara, and Jiangbin Gong. “High-fidelity and long-distance entangled-state transfer with Floquet topological edge modes”. Physical Review A 102, 022608 (2020).
https://doi.org/10.1103/PHYSREVA.102.022608
[16] P. Comaron, V. Shahnazaryan, and M. Matuszewski. “Coherent transfer of topological interface states”. Optics Express 28, 38698–38709 (2020).
https://doi.org/10.1364/OE.409715
[17] Shi Hu, Yongguan Ke, and Chaohong Lee. “Topological quantum transport and spatial entanglement distribution via a disordered bulk channel”. Physical Review A 101, 052323 (2020).
https://doi.org/10.1103/PHYSREVA.101.052323
[18] Ji Cao, Wen Xue Cui, Xuexi Yi, and Hong Fu Wang. “Topological Phase Transition and Topological Quantum State Transfer in Periodically Modulated Circuit-QED Lattice”. Annalen der Physik 533, 2100120 (2021).
https://doi.org/10.1002/ANDP.202100120
[19] Ze Guo Chen, Weiyuan Tang, Ruo Yang Zhang, Zhaoxian Chen, and Guancong Ma. “Landau-Zener Transition in the Dynamic Transfer of Acoustic Topological States”. Physical Review Letters 126, 054301 (2021).
https://doi.org/10.1103/PHYSREVLETT.126.054301
[20] N. E. Palaiodimopoulos, I. Brouzos, F. K. Diakonos, and G. Theocharis. “Fast and robust quantum state transfer via a topological chain”. Physical Review A 103, 052409 (2021).
https://doi.org/10.1103/PHYSREVA.103.052409
[21] Jin Xuan Han, Jin Lei Wu, Yan Wang, Yan Xia, Yong Yuan Jiang, and Jie Song. “Large-scale Greenberger-Horne-Zeilinger states through a topologically protected zero-energy mode in a superconducting qutrit-resonator chain”. Physical Review A 103, 032402 (2021).
https://doi.org/10.1103/PHYSREVA.103.032402
[22] Jiale Yuan, Chenran Xu, Han Cai, and Da Wei Wang. “Gap-protected transfer of topological defect states in photonic lattices”. APL Photonics 6, 030803 (2021).
https://doi.org/10.1063/5.0037394
[23] W. P. Su, J. R. Schrieffer, and A. J. Heeger. “Soliton excitations in polyacetylene”. Physical Review B 22, 2099 (1980).
https://doi.org/10.1103/PhysRevB.22.2099
[24] F. Munoz, Fernanda Pinilla, J. Mella, and Mario I. Molina. “Topological properties of a bipartite lattice of domain wall states”. Scientific Reports 8, 1–9 (2018).
https://doi.org/10.1038/s41598-018-35651-6
[25] Stefano Longhi. “Topological pumping of edge states via adiabatic passage”. Physical Review B 99, 155150 (2019).
https://doi.org/10.1103/PhysRevB.99.155150
[26] Lu Qi, Yu Yan, Yan Xing, Xue Dong Zhao, Shutian Liu, Wen Xue Cui, Xue Han, Shou Zhang, and Hong Fu Wang. “Topological router induced via long-range hopping in a Su-Schrieffer-Heeger chain”. Physical Review Research 3, 023037 (2021).
https://doi.org/10.1103/PHYSREVRESEARCH.3.023037
[27] Michael Creutz. “End States, Ladder Compounds, and Domain-Wall Fermions”. Physical Review Letters 83, 2636–2639 (1999).
https://doi.org/10.1103/PhysRevLett.83.2636
[28] Bo Song, Long Zhang, Chengdong He, Ting Fung Jeffrey Poon, Elnur Hajiyev, Shanchao Zhang, Xiong Jun Liu, and Gyu Boong Jo. “Observation of symmetry-protected topological band with ultracold fermions”. Science Advances 4, eaao4748 (2018).
https://doi.org/10.1126/sciadv.aao4748
[29] Jin Hyoun Kang, Jeong Ho Han, and Y Shin. “Creutz ladder in a resonantly shaken 1D optical lattice”. New Journal of Physics 22, 013023 (2020).
https://doi.org/10.1088/1367-2630/ab61d7
[30] Yanyan He, Ruosong Mao, Han Cai, Jun Xiang Zhang, Yongqiang Li, Luqi Yuan, Shi Yao Zhu, and Da Wei Wang. “Flat-Band Localization in Creutz Superradiance Lattices”. Physical Review Letters 126, 103601 (2021).
https://doi.org/10.1103/PHYSREVLETT.126.103601
[31] Hadiseh Alaeian, Chung Wai Sandbo Chang, Mehran Vahdani Moghaddam, Christopher M. Wilson, Enrique Solano, and Enrique Rico. “Creating lattice gauge potentials in circuit QED: The bosonic Creutz ladder”. Physical Review A 99, 053834 (2019).
https://doi.org/10.1103/PhysRevA.99.053834
[32] Jimmy S. C. Hung, J. H. Busnaina, C. W. Sandbo Chang, A. M. Vadiraj, I. Nsanzineza, E. Solano, H. Alaeian, E. Rico, and C. M. Wilson. “Quantum Simulation of the Bosonic Creutz Ladder with a Parametric Cavity”. Physical Review Letters 127, 100503 (2021).
https://doi.org/10.1103/PhysRevLett.127.100503
[33] Sebabrata Mukherjee, Marco Di Liberto, Patrik Öhberg, Robert R. Thomson, and Nathan Goldman. “Experimental Observation of Aharonov-Bohm Cages in Photonic Lattices”. Physical Review Letters 121, 075502 (2018).
https://doi.org/10.1103/PhysRevLett.121.075502
[34] Julien Vidal, Rémy Mosseri, and Benoit Douçot. “Aharonov-Bohm cages in two-dimensional structures”. Physical Review Letters 81, 5888–5891 (1998).
https://doi.org/10.1103/PhysRevLett.81.5888
[35] C. E. Creffield and G. Platero. “Coherent Control of Interacting Particles Using Dynamical and Aharonov-Bohm Phases”. Physical Review Letters 105, 086804 (2010).
https://doi.org/10.1103/PhysRevLett.105.086804
[36] M. Bello, C. E. Creffield, and G. Platero. “Long-range doublon transfer in a dimer chain induced by topology and ac fields”. Scientific Reports 6, 22562 (2016).
https://doi.org/10.1038/srep22562
[37] J. Jünemann, A. Piga, S. J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez. “Exploring interacting topological insulators with ultracold atoms: The synthetic Creutz-Hubbard model”. Physical Review X 7, 031057 (2017).
https://doi.org/10.1103/PhysRevX.7.031057
[38] Murad Tovmasyan, Evert P.L. Van Nieuwenburg, and Sebastian D. Huber. “Geometry-induced pair condensation”. Physical Review B 88, 220510 (2013).
https://doi.org/10.1103/PHYSREVB.88.220510
[39] Shintaro Takayoshi, Hosho Katsura, Noriaki Watanabe, and Hideo Aoki. “Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands”. Physical Review A 88, 063613 (2013).
https://doi.org/10.1103/PHYSREVA.88.063613
[40] Oleg Derzhko, Johannes Richter, and Mykola Maksymenko. “Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons”. International Journal of Modern Physics B 29, 1530007 (2015).
https://doi.org/10.1142/S0217979215300078
[41] Yoshihito Kuno, Tomonari Mizoguchi, and Yasuhiro Hatsugai. “Interaction-induced doublons and embedded topological subspace in a complete flat-band system”. Physical Review A 102, 063325 (2020).
https://doi.org/10.1103/PHYSREVA.102.063325
[42] G. Pelegrí, A. M. Marques, V. Ahufinger, J. Mompart, and R. G. Dias. “Interaction-induced topological properties of two bosons in flat-band systems”. Physical Review Research 2, 033267 (2020).
https://doi.org/10.1103/PHYSREVRESEARCH.2.033267
[43] Nilanjan Roy, Ajith Ramachandran, and Auditya Sharma. “Interplay of disorder and interactions in a flat-band supporting diamond chain”. Physical Review Research 2, 043395 (2020).
https://doi.org/10.1103/PHYSREVRESEARCH.2.043395
[44] Tilen Čadež, Yeongjun Kim, Alexei Andreanov, and Sergej Flach. “Metal-insulator transition in infinitesimally weakly disordered flat bands”. Physical Review B 104, L180201 (2021).
https://doi.org/10.1103/PHYSREVB.104.L180201
[45] Hiroki Nakai and Chisa Hotta. “Perfect flat band with chirality and charge ordering out of strong spin-orbit interaction”. Nature Communications 13, 579 (2022).
https://doi.org/10.1038/s41467-022-28132-y
[46] Yeongjun Kim, Tileň Cadež, Alexei Andreanov, and Sergej Flach. “Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder” (2022). arXiv:2211.09410v1.
https://doi.org/10.1103/PhysRevB.107.174202
arXiv:2211.09410v1
[47] M. Röntgen, C. V. Morfonios, I. Brouzos, F. K. Diakonos, and P. Schmelcher. “Quantum Network Transfer and Storage with Compact Localized States Induced by Local Symmetries”. Physical Review Letters 123, 080504 (2019).
https://doi.org/10.1103/PHYSREVLETT.123.080504
[48] Yoshihito Kuno, Takahiro Orito, and Ikuo Ichinose. “Flat-band many-body localization and ergodicity breaking in the Creutz ladder”. New Journal of Physics 22, 013032 (2020).
https://doi.org/10.1088/1367-2630/AB6352
[49] Linhu Li and Shu Chen. “Characterization of topological phase transitions via topological properties of transition points”. Physical Review B 92, 085118 (2015).
https://doi.org/10.1103/PhysRevB.92.085118
[50] Juan Zurita, Charles E. Creffield, and Gloria Platero. “Topology and Interactions in the Photonic Creutz and Creutz‐Hubbard Ladders”. Advanced Quantum Technologies 3, 1900105 (2019).
https://doi.org/10.1002/qute.201900105
[51] Sriram Ganeshan, Kai Sun, and S. Das Sarma. “Topological zero-energy modes in gapless commensurate aubry-andré-harper models”. Phys. Rev. Lett. 110, 180403 (2013).
https://doi.org/10.1103/PhysRevLett.110.180403
[52] A. Flesch, M. Cramer, I. P. McCulloch, U. Schollwöck, and J. Eisert. “Probing local relaxation of cold atoms in optical superlattices”. Phys. Rev. A 78, 033608 (2008).
https://doi.org/10.1103/PhysRevA.78.033608
[53] Peter Stano and Daniel Loss. “Review of performance metrics of spin qubits in gated semiconducting nanostructures”. Nature Reviews Physics 4, 672–688 (2022).
https://doi.org/10.1038/s42254-022-00484-w
[54] Stefano Longhi, Gian Luca Giorgi, and Roberta Zambrini. “Landau–Zener Topological Quantum State Transfer”. Advanced Quantum Technologies 2, 1800090 (2019).
https://doi.org/10.1002/qute.201800090
[55] E. Knill. “Quantum computing with realistically noisy devices”. Nature 434, 39–44 (2005).
https://doi.org/10.1038/nature03350
[56] David S. Wang, Austin G. Fowler, and Lloyd C.L. Hollenberg. “Surface code quantum computing with error rates over 1 percent”. Physical Review A 83, 020302(R) (2011).
https://doi.org/10.1103/PHYSREVA.83.020302
[57] Austin G. Fowler, Adam C. Whiteside, and Lloyd C.L. Hollenberg. “Towards practical classical processing for the surface code”. Physical Review Letters 108, 180501 (2012).
https://doi.org/10.1103/PHYSREVLETT.108.180501
[58] Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R. Brown, Marko Cetina, and Christopher Monroe. “Fault-Tolerant Operation of a Quantum Error-Correction Code” (2020). arXiv:2009.11482.
arXiv:2009.11482
[59] T. J.G. Apollaro, L. Banchi, A. Cuccoli, R. Vaia, and P. Verrucchi. “99%-fidelity ballistic quantum-state transfer through long uniform channels”. Physical Review A 85, 052319 (2012).
https://doi.org/10.1103/PHYSREVA.85.052319
[60] Leonardo Banchi and Ruggero Vaia. “Spectral problem for quasi-uniform nearest-neighbor chains”. Journal of Mathematical Physics 54, 043501 (2013).
https://doi.org/10.1063/1.4797477
[61] Janos K. Asboth, Laszlo Oroszlany, and Andras Palyi. “A Short Course on Topological Insulators”. Springer. (2015).
https://doi.org/10.1007/978-3-319-25607-8
[62] Leonardo Banchi, Enrico Compagno, and Sougato Bose. “Perfect wave-packet splitting and reconstruction in a one-dimensional lattice”. Physical Review A 91, 052323 (2015).
https://doi.org/10.1103/PhysRevA.91.052323
[63] Claudio Albanese, Matthias Christandl, Nilanjana Datta, and Artur Ekert. “Mirror Inversion of Quantum States in Linear Registers”. Physical Review Letters 93, 230502 (2004).
https://doi.org/10.1103/PhysRevLett.93.230502
[64] Matthias Christandl, Nilanjana Datta, Artur Ekert, and Andrew J. Landahl. “Perfect state transfer in quantum spin networks”. Physical Review Letters 92, 187902 (2004).
https://doi.org/10.1103/PhysRevLett.92.187902
[65] Diego Porras and Samuel Fernández-Lorenzo. “Topological amplification in photonic lattices”. Physical Review Letters 122, 143901 (2019).
https://doi.org/10.1103/PhysRevLett.122.143901
[66] Clara C. Wanjura, Matteo Brunelli, and Andreas Nunnenkamp. “Topological framework for directional amplification in driven-dissipative cavity arrays”. Nature Communications 11, 3149 (2020).
https://doi.org/10.1038/s41467-020-16863-9
[67] Tomás Ramos, Juan José García-Ripoll, and Diego Porras. “Topological input-output theory for directional amplification”. Physical Review A 103, 033513 (2021).
https://doi.org/10.1103/PHYSREVA.103.033513
[68] Clara C. Wanjura, Matteo Brunelli, and Andreas Nunnenkamp. “Correspondence between Non-Hermitian Topology and Directional Amplification in the Presence of Disorder”. Physical Review Letters 127, 213601 (2021).
https://doi.org/10.1103/PHYSREVLETT.127.213601
[69] Tomás Ramos, Álvaro Gómez-León, Juan José García-Ripoll, Alejandro González-Tudela, and Diego Porras. “Directional Josephson traveling-wave parametric amplifier via non-Hermitian topology” (2022). arXiv:2207.13728.
arXiv:2207.13728
[70] Álvaro Gómez-León, Tomás Ramos, Alejandro González-Tudela, and Diego Porras. “Driven-dissipative topological phases in parametric resonator arrays”. Quantum 7, 1016 (2023).
https://doi.org/10.22331/q-2023-05-23-1016
[71] Ning Sun and Lih-King King Lim. “Quantum charge pumps with topological phases in a Creutz ladder”. Physical Review B 96, 035139 (2017).
https://doi.org/10.1103/PhysRevB.96.035139
[72] Alexander Altland and Martin R. Zirnbauer. “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures”. Physical Review B 55, 1142–1161 (1997).
https://doi.org/10.1103/PhysRevB.55.1142
[73] Abhijeet Alase, Emilio Cobanera, Gerardo Ortiz, and Lorenza Viola. “Generalization of Bloch’s theorem for arbitrary boundary conditions: Theory”. Physical Review B 96, 195133 (2017).
https://doi.org/10.1103/PhysRevB.96.195133
[74] Emilio Cobanera, Abhijeet Alase, Gerardo Ortiz, and Lorenza Viola. “Generalization of Bloch’s theorem for arbitrary boundary conditions: Interfaces and topological surface band structure”. Physical Review B 98, 245423 (2018).
https://doi.org/10.1103/PhysRevB.98.245423
[75] Motohiko Ezawa. “Non-Abelian braiding of Majorana-like edge states and topological quantum computations in electric circuits”. Physical Review B 102, 075424 (2020).
https://doi.org/10.1103/PhysRevB.102.075424
[76] Frank Schindler, Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, B. Andrei Bernevig, and Titus Neupert. “Higher-order topological insulators”. Science Advances 4, eaat0346 (2018).
https://doi.org/10.1126/sciadv.aat0346
[77] Shuo Liu, Wenlong Gao, Qian Zhang, Shaojie Ma, Lei Zhang, Changxu Liu, Yuan Jiang Xiang, Tie Jun Cui, and Shuang Zhang. “Topologically Protected Edge State in Two-Dimensional Su–Schrieffer–Heeger Circuit”. Research 2019, 8609875 (2019).
https://doi.org/10.34133/2019/8609875
[78] Avik Dutt, Momchil Minkov, Ian A. D. Williamson, and Shanhui Fan. “Higher-order topological insulators in synthetic dimensions”. Light: Science & Applications 9, 131 (2020).
https://doi.org/10.1038/s41377-020-0334-8
[79] Shao-Liang Zhang and Qi Zhou. “Shaping topological properties of the band structures in a shaken optical lattice”. Phys. Rev. A 90, 051601 (2014).
https://doi.org/10.1103/PhysRevA.90.051601
[80] Mark Kremer, Ioannis Petrides, Eric Meyer, Matthias Heinrich, Oded Zilberberg, and Alexander Szameit. “A square-root topological insulator with non-quantized indices realized with photonic Aharonov-Bohm cages”. Nature Communications 11, 907 (2020).
https://doi.org/10.1038/s41467-020-14692-4
[81] Juan Zurita, Charles Creffield, and Gloria Platero. “Tunable zero modes and symmetries in flat-band topological insulators”. Quantum 5, 591 (2021).
https://doi.org/10.22331/q-2021-11-25-591
Cited by
[1] Gabriel Cáceres-Aravena, Bastián Real, Diego Guzmán-Silva, Paloma Vildoso, Ignacio Salinas, Alberto Amo, Tomoki Ozawa, and Rodrigo A. Vicencio, “Spectral edge-to-edge topological state transfer in diamond photonic lattices”, arXiv:2301.04189, (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-06-22 15:27:16). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2023-06-22 15:27:14: Could not fetch cited-by data for 10.22331/q-2023-06-22-1043 from Crossref. This is normal if the DOI was registered recently.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- EVM Finance. Unified Interface for Decentralized Finance. Access Here.
- Quantum Media Group. IR/PR Amplified. Access Here.
- PlatoAiStream. Web3 Data Intelligence. Knowledge Amplified. Access Here.
- Source: https://quantum-journal.org/papers/q-2023-06-22-1043/
