Continuous-Variable Quantum Computation in Circuit QED
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Lu–H–N compounds have drawn increasing attention recently as it was reported that certain Lu–H–N phase exhibits appealing observable macroscopic quantum phenomenon at ambient condition, which may revolutionarily revolt the field of condensed matter physics. The existing literature[1] has performed characterization and pointed out that the compound is the LuH3–δNε in space group of
backed by the x-ray diffraction (XRD) pattern, and doped with N. The synthesis process as they stated in the article is fairly accessible as the precursors are commercially available and experimental condition, such as pressure and temperature, is not critical by current technology. However, debate soars within the community as a group of scientists are skeptical about the results, hence extra experimental confirmations[2] from multiple sources are demanded to settle down the arguments.[3] This has set off a heat wave in the study of lutetium hydrogen systems recently.[4–6] Fortunately, recent advances in high-throughput first-principles calculation along with large-scale database[7–9] make it a feasible method to thoroughly search possible compounds within a certain chemical system, and accurately evaluate their likelihood of existence, way quicker than experimental approaches. In this Letter, we tackle the stability issue of possible phases in Lu–H–N chemical systems harnessing the high-throughput calculation and a large in-house database,[10] which would hopefully provide useful knowledge for the community to understand the energy landscape of the Lu–H–N compounds. From a scientific point of view, lanthanides hydrides are worth investigating as there are several high-temperature superconductors in this phase space under high pressure. For example, lanthanum decahydride (LaH10) exhibits superconductivity under 252 K at the pressure of 170 GPa;[11] yttrium hydride (YH9) exhibits superconductivity under 243 K at the pressure of 201 GPa.[12] For Lu–H systems, recent experiments demonstrated that LuH3 undergoes a phase transition into the superconducting states at 12.4 K under 122 GPa.[13] Meanwhile, it was found that the Lu4H23 exhibits superconductivity at 71 K under 218 GPa.[14] However, the lanthanides hydrides are less explored compared to common metals, let alone the ternary compounds within the Lu–H–N chemical space. Given that it is alleged that a certain compound within the Lu–H–N chemical space possesses superconductivity at the very much near ambient condition,[1] it is necessary to have a systematic search for viable structures within the Lu–H–N system.
Leveraging the recent advance in high-throughput first-principles calculation[9,15–18] as well as the large trove of materials data,[10] in this Letter, we closely investigate the possible compounds in the Lu–H–N chemical system. Such a method is fairly effective and has been demonstrated several times in searching for ternary nitrides,[19] Kagome materials,[20] superconductor,[21] and many others, and more importantly, the theoretical predictions have been successfully confirmed by experiments.[22,23] Essentially, we can use the existing compounds in a database as the structure templates and perform element substitution to create new structures within the targeting chemical systems, which is Lu–H–N in this case. Then the high-throughput first-principles calculations can optimize those structures and compute the energy of each compound. By comparing the formation energies of those compounds, the formation energy landscape can be obtained, and the thermodynamic stability of each compound can be derived to justify the likelihood of the existence of those compounds quantitatively.[24]
Methodology. In thorough explanation, those compounds which boast the least formation energy, that is, the compounds that release the largest amount of enthalpy in their synthesis processes, are the ones that comprise the energy hull.[24] This energy hull represents the boundary line of the maximum formation energy of a chemical structure. Therefore, the hull energy designates the lowest energy state potentially attainable by any feasible combination of elements capable of forming a compound with a given chemical composition. Upon this foundation, the energy disparity between a compound and the energy hull may furnish insight into the compound’s instability, or the “energy above the hull” (Ehull). Ehull conveys the minimum amount of energy required to break down the compound into the stable compounds present on the energy convex hull.
In order to methodically assess the feasible compounds present in the Lu–H–N systems, 4478 initial structures are constructed by employing structural templates such as X–H–N, X–H–P, X–H–As, X–Li–N, X–Na–N, and X–K–N for ternary phases, and X–H, X–N, X–Li, and X–P for binary phases, resulting in 1561 newly created compounds after duplication removal. The computation is performed utilizing the Vienna ab initio simulation package (VASP)[25–29] along with unified input parameters for Atomly.net. Further information with regard to the detailed input parameters could be found in Refs. [10,20,21,30]. All the structures and calculation results presented here can be found in Atomly.net materials database.
After conducting high-throughput calculations, totally 1561 structures are computed, including 712 Lu–H–N phases, 432 N–H phases, 116 Lu–H phases, and 301 Lu–N phases. The purpose of these calculations is to create a comprehensive “phase diagram” for Lu–H–N. Subsequently, we select all the compounds with relative stability and Ehull values less than 150 meV/atom. Those compounds are then calculated subjecting to hydrostatic pressure of up to 10 GPa with an interval of 1 GPa. Hence, the phase evolution of those compounds under pressure can be clearly captured and the phase diagrams under different pressured states are constructed and presented in this study.
Results and Discussions. Figure 1 presents the formation energy landscape of the Lu–H–N and its evolution versus external pressure. The phase diagram essentially presents that, at 0 GPa, there are six binary stable compounds, which are LuN
, Lu12N11 (C2/m), NH3 , N2H3 (C2/c), LuH2 , and LuH3 , and there is no stable Lu–H–N ternary phase. Among all the structures we have calculated, Lu20H2N17 (C2/m), Lu2H2N , LuH5N2 (P1), Lu3H6N (P21), Lu10HN8 , Lu(H15N8)2 (P21/c), Lu6HN6 (Cm), Lu2H5N (Pc), LuH3N2 (P21/c), LuH5N (P212121) are ternary metastable phases with small Ehull (< 100 meV/atom), and the Ehull values are listed in Table 1. When pressure is applied, we spot some subtle changes in the phase diagram, for example, under 5 GPa, LuN9 emerges as a stable phase, and under 10 GPa, Lu10H21 is stabilized thermodynamically. There is no thermodynamic stable Lu–H–N ternary phase according to our evaluation under external pressure up to 10 GPa. ![]()
Fig. Fig. 1. Phase diagrams of Lu–H–N chemical system in both the 3D and 2D modes. The phase diagrams under 0 GPa, 5 GPa, and 10 GPa are calculated and presented to showcase the change of energy landscape as a function of hydrostatic pressure. On those plots, each data point shows a compound we calculate. Stable compounds are shown in green square dots, and unstable ones are shown in red circular dots. The energy convex hull is colored based on the formation energy.Table 1. List of some compounds with Ehull < 100 meV/atom at 0 GPa.[10]Composition Space group Atomly Id Ehull (eV/atom) Stability Lu20H2N17 C2/m 1000313230 0.036 × Lu10HN8 1000313257 0.058 × Lu(H15N8)2 P21/c 1000313234 0.064 × Lu6HN6 Cm 1000313346 0.077 × Lu2H5N Pc 1000313150 0.081 × Lu2H2N 1000313854 0.082 × LuH3N2 P21/c 1000313450 0.084 × LuH5N2 P1 1000313474 0.090 × Lu3H6N P21 1000313607 0.095 × LuH5N P212121 1000313153 0.099 × NH3 P213 1000314459 0.000 √ N2H3 C2/c 1000314332 0.000 √ LuH3 0000079762 0.000 √ LuH2 1000314722 0.000 √ LuN 3001350567 0.000 √ Lu12N11 C2/m 1000314893 0.000 √ | Show Table
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Figure 2 presents the phase evolution of binary compounds with or without external pressure. Clearly, the binary phase diagrams imply that the landscape of the Lu–H–N system is sensitive to pressure. For example, NH becomes stable when the pressure reaches 10 GPa. Many of the phases are not able to be found in other databases, evidencing that such a system has not been thoroughly investigated before. The existing databases including Materials Project,[24,31,32] Aflow,[16] and many others may miss the stable phases and thus are unable to extrapolate the correct energy convex hull for this system. Except for the stable phases, the Lu–H phase diagram shows a group of compounds that have fairly low Ehull, such as Lu10H21, Lu4H7, Lu3H8, Lu2H3, Lu3H2, Lu4H, and Lu3H, implying that Lu and H are inter-mixable and can form many likely phases at different compositions. When 10 GPa pressure is applied, a new stable phase of Lu10H21 is observed to form. However, there is no new phase that contains a higher proportion of hydrogen than the original stable phase under 10 GPa.
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Fig. Fig. 2. N–H, Lu–H, and Lu–N binary phase diagrams under 0, 5, and 10 GPa separately. The compounds with Ehull < 150 meV/atom are selected to compute the formation energy under pressure > 0 GPa. (a) N–H phase diagram. NH becomes a stable phase under 10 GPa, while N2H3 becomes an unstable phase. (b) Lu–H phase diagram. Lu10H21 appears to become a new stable phase under the pressure of 10 GPa. (c) Lu–N phase diagram. With the increase of pressure, Lu could combine with more N. As there appears a new phase LuN9 under the pressure of 5 GPa.The Lu–N phase diagram [Fig. 2(c)] suggests that LuN and Lu12N11 are the thermodynamic stable phases, and there are also several low Ehull compounds near Lu : N = 1 : 1 composition, such as Lu20N17, Lu16N13, and Lu3N2. Under 10 GPa, it is found that as the pressure increases, the formation energy of LuN decreases in comparison to its value at 0 GPa, indicating that the system becomes more stable. In addition, higher pressure can also result in the appearance of a new stable phase, LuN9, suggesting that high pressure could facilitate forming of nitrogen-rich Lu–N compound.
Based on our calculation, it is generally difficult to form a Lu–H–N ternary compound as the LuH and LuN do not react with each other as there is no intermediate stable phase between LuN and LuH, and the lowest Ehull can be found from Lu2HN and Lu4HN3, which are Ehull = 168 and 525 meV/atom, respectively. Both of them are too unstable to warrant the existence of those compounds. The reaction calculations give that when LuH and LuN are mixed, the most likely reaction is to form Lu12N11 and LuH2 to reach the more stable phases. There are also no stable or relatively stable compounds within the Immm space group in Lu–H–N phases based on our evaluation with or without pressure.
Figure 3 plots the compounds that share the same diffraction patterns as experimental observations among all the stable phases. Based on these patterns, it appears that LuH2, Lu10H21, LuN, and Lu12N11 are likely to exist. Further investigation reveals that Lu10H21 is actually the same structure as LuH2 but with H interstitials, and Lu12N11 is the same structure as LuN but with N vacancies. This suggests that increasing hydrostatic pressure may introduce H interstitials into LuH2. At low doping concentrations, the H interstitials do not significantly change the volume of LuH2 and may even contract the lattice parameter slightly. However, it is difficult to observe the Lu10H21 phase directly from the XRD as the volume of Lu10H21 is only 0.4% less than that of LuH2. Similarly, the results suggest that N vacancies can be easily formed in LuN, as they are energetically favored and do not add too much strain energy. This may help to explain the experimental formation of LuH3–δNε with additional dopant, as the LuH2 favors interstitial sites and can accommodate extra intercalants at low dopant concentration. When pressure is applied, the XRD peaks of LuH2 shift drastically more than those in LuN, meaning that the LuN is elastically stiffer than LuH2. Young’s moduli of LuH2 and LuN from our calculations are 148.5 GPa and 338.1 GPa, respectively.
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Fig. Fig. 3. The diffraction patterns of stable structures that matches the experimental XRD.[1] Those compounds are LuH2, Lu10H21, LuN, Lu12N11, in which Lu10H21 is basically the LuH2 with addition interstitial H, while Lu12N11 is LuN with N vacancy.Figure 4 exhibits the calculated energy bands of LuH2
and LuH3 . Based on the energy band structures, LuH2 is a metallic phase and the trigonal LuH3 can be classified into a node-line semimetal when spin-orbit coupling is not considered. As shown in Fig. 4, trigonal LuH3 exhibits a clean electronic structure with an electron and hole pocket intersecting right at the Fermi level. Band inversion happens between |Lu+, dxz/dyz〉 and |H1–, s〉 and a nodal line is formed and protected by the glide-plane symmetry. The fact that the density of states is vanishing at the Fermi level makes LuH3 an ideal system to study fundamental physics of topological node-line semimetals. With eliminated contributions of the topologically trivial bands, it is a unique platform to investigate manifestations of electron-hole interactions such as the linear frequency dependent optical conductivity[33] and the hot carrier dynamics.[34] Moreover, pressure-dependent band structure indicates that the nodal line is robust under hydrostatic pressure, yet it can serve as a manipulation of the Fermi velocity of the linearly dispersed bands. ![]()
Fig. Fig. 4. Energy bands of LuH2and LuH3 under equilibrium and external pressure of 5 GPa and 10 GPa from first-principles calculation at the GGA-PBE level. In summary, the recent advancement of high-throughput calculations has greatly improved the exploration and analysis of compound spaces with efficiency and accuracy. The focus of the present particular study focuses on investigating the different phases within the chemical space of Lu–H–N, resulting in the formation energy landscape. Hydrostatic pressure is also taken into consideration to simulate experimental conditions. It is found that there is no thermodynamic stable Lu–H–N ternary phase with or without external pressure up to 10 GPa. Lu20H2N17 (C2/m), Lu2H2N
, LuH5N2 (P1), Lu3H6N (P21), and Lu10HN8 are ternary phases with small Ehull (< 100 meV/atom), hence they are the potential metastable ternary phases. Additionally, the binary phase diagram shows that Lu–H compounds are inter-mixable and may have various structures due to numerously low Ehull Lu–H compounds calculated. The LuH2 has interstitial sites, and thus can accommodate lightly doped intercalants, leading to the formation of Lu10H21 and N-doped LuH2. The XRD compassion reveals that the experimental sample is likely to be composed of LuH2 and LuN, and doped phases of them. Electronic structure analysis indicates that the LuH2 is a metallic phase and LuH3 is a nodal line semimetal. Overall, this work aims to provide valuable insights into the synthesizability of Lu–H–N compounds, and the developed algorithm can be further utilized to explore the vast territory of inorganic compounds and to accelerate the discovery of new materials. -
References
[1] Poulin D, Hastings M B, Wecker D, Wiebe N, Doherty A C, Troyer M 2015 Quantum Inf. Comput. 151 361[2] Wecker D, Bauer B, Clark B K, Hastings M B, Troyer M 2014 Phys. Rev. A 90 022305 doi: 10.1103/PhysRevA.90.022305[3] Shor P W 1999 SIAM Rev. 41 303 doi: 10.1137/S0036144598347011[4] Bruzewicz C D, Chiaverini J, McConnell R, Sage J M 2019 Appl. Phys. Rev. 6 021314 doi: 10.1063/1.5088164[5] Hill C D, Peretz E, Hile S J, House M G, Fuechsle M, Rogge S, Simmons M Y, Hollenberg L C 2015 Sci. Adv. 1 e1500707 doi: 10.1126/sciadv.1500707[6] Gidwani B, Sahu V, Shukla S S, Pandey R, Joshi V, Jain V K, Vyas A 2021 J. Drug Deliv. Sci. Technol. 61 102308 doi: 10.1016/j.jddst.2020.102308[7] Atabaki A H, Moazeni S, Pavanello F, et al.. 2018 Nature 556 349 doi: 10.1038/s41586-018-0028-z[8] Devoret M H, Schoelkopf R J 2013 Science 339 1169 doi: 10.1126/science.1231930[9] Blais A, Grimsmo A L, Girvin S M, Wallraff A 2021 Rev. Mod. Phys. 93 025005 doi: 10.1103/RevModPhys.93.025005[10] Paik H, Schuster D I, Bishop L S, et al.. 2011 Phys. Rev. Lett. 107 240501 doi: 10.1103/PhysRevLett.107.240501[11] Barends R, Kelly J, Megrant A, et al.. 2013 Phys. Rev. Lett. 111 080502 doi: 10.1103/PhysRevLett.111.080502[12] Mooij J, Orlando T, Levitov L, Tian L, Van der Wal C H, Lloyd S 1999 Science 285 1036 doi: 10.1126/science.285.5430.1036[13] Bylander J, Gustavsson S, Yan F, Yoshihara F, Harrabi K, Fitch G, Cory D G, Nakamura Y, Tsai J-S, Oliver W D 2011 Nat. Phys. 7 565 doi: 10.1038/nphys1994[14] Yan F, Gustavsson S, Kamal A, et al.. 2016 Nat. Commun. 7 12964 doi: 10.1038/ncomms12964[15] Pop I M, Geerlings K, Catelani G, Schoelkopf R J, Glazman L I, Devoret M H 2014 Nature 508 369 doi: 10.1038/nature13017[16] Johnson B, Reed M, Houck A A, et al.. 2010 Nat. Phys. 6 663 doi: 10.1038/nphys1710[17] Stassi R, Cirio M, Nori F 2020 NPJ Quantum Inf. 6 67 doi: 10.1038/s41534-020-00294-x[18] Yao Y, Xiang L, Guo Z, et al.. 2023 Nat. Phys. 19 1459 doi: 10.1038/s41567-023-02133-0[19] Xu S, Sun Z Z, Wang K, et al.. 2023 Chin. Phys. Lett. 40 060301 doi: 10.1088/0256-307X/40/6/060301[20] Zhang P, Dong H, Gao Y, et al.. 2023 Nat. Phys. 19 120 doi: 10.1038/s41567-022-01784-9[21] Preskill J 2018 Quantum 2 79 doi: 10.22331/q-2018-08-06-79[22] Arute F, Arya K, Babbush R, et al.. 2019 Nature 574 505 doi: 10.1038/s41586-019-1666-5[23] Wu Y, Bao W S, Cao S, et al.. 2021 Phys. Rev. Lett. 127 180501 doi: 10.1103/PhysRevLett.127.180501[24] Campbell E T, Terhal B M, Vuillot C 2017 Nature 549 172 doi: 10.1038/nature23460[25] Nielsen M A, Chuang I 2022 Quantum Computation and Quantum Information American Association of Physics Teachers[26] Zhao Y, Ye Y, Huang H L, Zhang Y, et al.. 2022 Phys. Rev. Lett. 129 030501 doi: 10.1103/PhysRevLett.129.030501[27] Krinner S, Lacroix N, Remm A, et al.. 2022 Nature 605 669 doi: 10.1038/s41586-022-04566-8[28] Cai W, Ma Y, Wang W, Zou C L, Sun L 2021 Fundam. Res. 1 50 doi: 10.1016/j.fmre.2020.12.006[29] Joshi A, Noh K, Gao Y Y 2021 Quantum Sci. Technol. 6 033001 doi: 10.1088/2058-9565/abe989[30] Cochrane P T, Milburn G J, Munro W J 1999 Phys. Rev. A 59 2631 doi: 10.1103/PhysRevA.59.2631[31] Li L, Zou C L, Albert V V, Muralidharan S, Girvin S, Jiang L 2017 Phys. Rev. Lett. 119 030502 doi: 10.1103/PhysRevLett.119.030502[32] Bergmann M, van Loock P 2016 Phys. Rev. A 94 012311 doi: 10.1103/PhysRevA.94.012311[33] Michael M H, Silveri M, Brierley R, Albert V V, Salmilehto J, Jiang L, Girvin S M 2016 Phys. Rev. X 6 031006 doi: 10.1103/PhysRevX.6.031006[34] Royer B, Singh S, Girvin S 2020 Phys. Rev. Lett. 125 260509 doi: 10.1103/PhysRevLett.125.260509[35] Noh K, Albert V V, Jiang L 2018 IEEE Trans. Inf. Theory 65 2563 doi: 10.1109/TIT.2018.2873764[36] Gottesman D, Kitaev A, Preskill J 2001 Phys. Rev. A 64 012310 doi: 10.1103/PhysRevA.64.012310[37] Mirrahimi M, Leghtas Z, Albert V V, Touzard S, Schoelkopf R J, Jiang L, Devoret M H 2014 New J. Phys. 16 045014 doi: 10.1088/1367-2630/16/4/045014[38] Grimsmo A L, Combes J, Baragiola B Q 2020 Phys. Rev. X 10 011058 doi: 10.1103/PhysRevX.10.011058[39] Li L, Young D J, Albert V V, Noh K, Zou C L, Jiang L 2021 Phys. Rev. A 103 062427 doi: 10.1103/PhysRevA.103.062427[40] Ofek N, Petrenko A, Heeres R, et al.. 2016 Nature 536 441 doi: 10.1038/nature18949[41] Grimm A, Frattini N E, Puri S, Mundhada S O, Touzard S, Mirrahimi M, Girvin S M, Shankar S, Devoret M H 2020 Nature 584 205 doi: 10.1038/s41586-020-2587-z[42] Ma W-L, Puri S, Schoelkopf R J, Devoret M H, Girvin S M, Jiang L 2021 Sci. Bull. 66 1789 doi: 10.1016/j.scib.2021.05.024[43] Hu L, Ma Y, Cai W, et al.. 2019 Nat. Phys. 15 503 doi: 10.1038/s41567-018-0414-3[44] Ni Z, Li S, Deng X, et al.. 2023 Nature 616 56 doi: 10.1038/s41586-023-05784-4[45] Campagne-Ibarcq P, Eickbusch A, Touzard S, et al.. 2020 Nature 584 368 doi: 10.1038/s41586-020-2603-3[46] Sivak V, Eickbusch A, Royer B, et al.. 2023 Nature 616 50 doi: 10.1038/s41586-023-05782-6[47] Gertler J M, Baker B, Li J, Shirol S, Koch J, Wang C 2021 Nature 590 243 doi: 10.1038/s41586-021-03257-0[48] Leghtas Z, Kirchmair G, Vlastakis B, Devoret M H, Schoelkopf R J, Mirrahimi M 2013 Phys. Rev. A 87 042315 doi: 10.1103/PhysRevA.87.042315[49] Wang C, Gao Y Y, Reinhold P, et al.. 2016 Science 352 1087 doi: 10.1126/science.aaf2941[50] Krastanov S, Albert V V, Shen C, Zou C L, Heeres R W, Vlastakis B, Schoelkopf R J, Jiang L 2015 Phys. Rev. A 92 040303 doi: 10.1103/PhysRevA.92.040303[51] Heeres R W, Vlastakis B, Holland E, Krastanov S, Albert V V, Frunzio L, Jiang L, Schoelkopf R J 2015 Phys. Rev. Lett. 115 137002 doi: 10.1103/PhysRevLett.115.137002[52] Reinhold P, Rosenblum S, Ma W L, Frunzio L, Jiang L, Schoelkopf R J 2020 Nat. Phys. 16 822 doi: 10.1038/s41567-020-0931-8[53] Xu Y, Ma Y, Cai W, et al.. 2020 Phys. Rev. Lett. 124 120501 doi: 10.1103/PhysRevLett.124.120501[54] Hastrup J, Park K, Filip R, Andersen U L 2021 Phys. Rev. Lett. 126 153602 doi: 10.1103/PhysRevLett.126.153602[55] Pan X, Schwinger J, Huang N-N, Song P, Chua W, Hanamura F, Joshi A, Valadares F, Filip R, Gao Y Y 2023 Phys. Rev. X 13 021004 doi: 10.1103/PhysRevX.13.021004[56] Eickbusch A, Sivak V, Ding A Z, Elder S S, Jha S R, Venkatraman J, Royer B, Girvin S, Schoelkopf R J, Devoret M H 2022 Nat. Phys. 18 1464 doi: 10.1038/s41567-022-01776-9[57] Diringer A A, Blumenthal E, Grinberg A, Jiang L, Hacohen-Gourgy S 2023 arXiv:2301.09831[58] Rosenblum S, Gao Y Y, Reinhold P, et al.. 2018 Nat. Commun. 9 652 doi: 10.1038/s41467-018-03059-5[59] Chou K S, Blumoff J Z, Wang C S, Reinhold P C, Axline C J, Gao Y Y, Frunzio L, Devoret M, Jiang L, Schoelkopf R 2018 Nature 561 368 doi: 10.1038/s41586-018-0470-y[60] Gao Y Y, Lester B J, Chou K S, Frunzio L, Devoret M H, Jiang L, Girvin S, Schoelkopf R J 2019 Nature 566 509 doi: 10.1038/s41586-019-0970-4[61] Lau H K, Plenio M B 2016 Phys. Rev. Lett. 117 100501 doi: 10.1103/PhysRevLett.117.100501[62] Gao Y Y, Lester B J, Zhang Y, Wang C, Rosenblum S, Frunzio L, Jiang L, Girvin S, Schoelkopf R J 2018 Phys. Rev. X 8 021073 doi: 10.1103/PhysRevX.8.021073[63] Khaneja N, Reiss T, Kehlet C, Schulte-Herbrüggen T, Glaser S J 2005 J. Magn. Reson. 172 296 doi: 10.1016/j.jmr.2004.11.004[64] Abdelhafez M, Schuster D I, Koch J 2019 Phys. Rev. A 99 052327 doi: 10.1103/PhysRevA.99.052327[65] Niu M Y, Boixo S, Smelyanskiy V N, Neven H 2019 NPJ Quantum Inf. 5 33 doi: 10.1038/s41534-019-0141-3[66] Zhang X M, Wei Z, Asad R, Yang X C, Wang X 2019 NPJ Quantum Inf. 5 85 doi: 10.1038/s41534-019-0201-8[67] Chakram S, He K, Dixit A V, Oriani A E, Naik R K, Leung N, Kwon H, Ma W L, Jiang L, Schuster D I 2022 Nat. Phys. 18 879 doi: 10.1038/s41567-022-01630-y[68] Porotti R, Peano V, Marquardt F 2023 PRX Quantum 4 030305 doi: 10.1103/PRXQuantum.4.030305[69] Leghtas Z, Touzard S, Pop I M, et al.. 2015 Science 347 853 doi: 10.1126/science.aaa2085[70] Puri S, Boutin S, Blais A 2017 NPJ Quantum Inf. 3 18 doi: 10.1038/s41534-017-0019-1[71] Puri S, St-Jean L, Gross J A, et al.. 2020 Sci. Adv. 6 5901 doi: 10.1126/sciadv.aay5901[72] Goto H 2016 Phys. Rev. A 93 050301 doi: 10.1103/PhysRevA.93.050301[73] Puri S, Grimm A, Campagne-Ibarcq P, Eickbusch A, Noh K, Roberts G, Jiang L, Mirrahimi M, Devoret M H, Girvin S M 2019 Phys. Rev. X 9 041009 doi: 10.1103/PhysRevX.9.041009[74] Ma Y, Xu Y, Mu X, et al.. 2020 Nat. Phys. 16 827 doi: 10.1038/s41567-020-0893-x[75] Place A P, Rodgers L V, Mundada P, et al.. 2021 Nat. Commun. 12 1779 doi: 10.1038/s41467-021-22030-5[76] Wang C, Li X, Xu H, et al.. 2022 NPJ Quantum Inf. 8 3 doi: 10.1038/s41534-021-00510-2[77] Minev Z K, Serniak K, Pop I M, Leghtas Z, Sliwa K, Hatridge M, Frunzio L, Schoelkopf R J, Devoret M H 2016 Phys. Rev. Appl. 5 044021 doi: 10.1103/PhysRevApplied.5.044021[78] Jiang L, Taylor J M, Sørensen A S, Lukin M D 2007 Phys. Rev. A 76 062323 doi: 10.1103/PhysRevA.76.062323[79] Kimble H J 2008 Nature 453 1023 doi: 10.1038/nature07127[80] Monroe C, Raussendorf R, Ruthven A, Brown K R, Maunz P, Duan L M, Kim J 2014 Phys. Rev. A 89 022317 doi: 10.1103/PhysRevA.89.022317[81] Guillaud J, Mirrahimi M 2019 Phys. Rev. X 9 041053 doi: 10.1103/PhysRevX.9.041053[82] Grimsmo A L, Puri S 2021 PRX Quantum 2 020101 doi: 10.1103/PRXQuantum.2.020101[83] Huang H Y, Kueng R, Preskill J 2020 Nat. Phys. 16 1050 doi: 10.1038/s41567-020-0932-7[84] Wilson C, Otterbach J, Tezak N, Smith R, Polloreno A, Karalekas P J, Heidel S, Alam M S, Crooks G, da Silva M 2018 arXiv:1806.08321[85] DiCarlo L, Chow J M, Gambetta J M, et al.. 2009 Nature 460 240 doi: 10.1038/nature08121 -
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