Thermalization and criticality on an analogue–digital quantum simulator

Andersen, T. I.; Astrakhantsev, N.; Karamlou, A. H.; Berndtsson, J.; Motruk, J.; Szasz, A.; Gross, J. A.; Schuckert, A.; Westerhout, T.; Zhang, Y.; Forati, E.; Rossi, D.; Kobrin, B.; Paolo, A. Di; Klots, A. R.; Drozdov, I.; Kurilovich, V.; Petukhov, A.; Ioffe, L. B.; Elben, A.; Rath, A.; Vitale, V.; Vermersch, B.; Acharya, R.; Beni, L. A.; Anderson, K.; Ansmann, M.; Arute, F.; Arya, K.; Asfaw, A.; Atalaya, J.; Ballard, B.; Bardin, J. C.; Bengtsson, A.; Bilmes, A.; Bortoli, G.; Bourassa, A.; Bovaird, J.; Brill, L.; Broughton, M.; Browne, D. A.; Buchea, B.; Buckley, B. B.; Buell, D. A.; Burger, T.; Burkett, B.; Bushnell, N.; Cabrera, A.; Campero, J.; Chang, H.-S.; Chen, Z.; Chiaro, B.; Claes, J.; Cleland, A. Y.; Cogan, J.; Collins, R.; Conner, P.; Courtney, W.; Crook, A. L.; Das, S.; Debroy, D. M.; De Lorenzo, L.; Barba, A. Del Toro; Demura, S.; Donohoe, P.; Dunsworth, A.; Earle, C.; Eickbusch, A.; Elbag, A. M.; Elzouka, M.; Erickson, C.; Faoro, L.; Fatemi, R.; Ferreira, V. S.; Burgos, L. Flores; Fowler, A. G.; Foxen, B.; Ganjam, S.; Gasca, R.; Giang, W.; Gidney, C.; Gilboa, D.; Giustina, M.; Gosula, R.; Dau, A. Grajales; Graumann, D.; Greene, A.; Habegger, S.; Hamilton, M. C.; Hansen, M.; Harrigan, M. P.; Harrington, S. D.; Heslin, S.; Heu, P.; Hill, G.; Hoffmann, M. R.; Huang, H.-Y.; Huang, T.; Huff, A.; Huggins, W. J.; Isakov, S. V.; Jeffrey, E.; Jiang, Z.; Jones, C.; Jordan, S.; Joshi, C.; Juhas, P.; Kafri, D.; Kang, H.; Kechedzhi, K.; Khaire, T.; Khattar, T.; Khezri, M.; Kieferová, M.; Kim, S.; Kitaev, A.; Klimov, P.; Korotkov, A. N.; Kostritsa, F.; Kreikebaum, J. M.; Landhuis, D.; Langley, B. W.; Laptev, P.; Lau, K.-M.; Guevel, L. Le; Ledford, J.; Lee, J.; Lee, K. W.; Lensky, Y. D.; Lester, B. J.; Li, W. Y.; Lill, A. T.; Liu, W.; Livingston, W. P.; Locharla, A.; Lundahl, D.; Lunt, A.; Madhuk, S.; Maloney, A.; Mandrà, S.; Martin, L. S.; Martin, O.; Martin, S.; Maxfield, C.; McClean, J. R.; McEwen, M.; Meeks, S.; Miao, K. C.; Mieszala, A.; Molina, S.; Montazeri, S.; Morvan, A.; Movassagh, R.; Neill, C.; Nersisyan, A.; Newman, M.; Nguyen, A.; Nguyen, M.; Ni, C.-H.; Niu, M. Y.; Oliver, W. D.; Ottosson, K.; Pizzuto, A.; Potter, R.; Pritchard, O.; Pryadko, L. P.; Quintana, C.; Reagor, M. J.; Rhodes, D. M.; Roberts, G.; Rocque, C.; Rosenberg, E.; Rubin, N. C.; Saei, N.; Sankaragomathi, K.; Satzinger, K. J.; Schurkus, H. F.; Schuster, C.; Shearn, M. J.; Shorter, A.; Shutty, N.; Shvarts, V.; Sivak, V.; Skruzny, J.; Small, S.; Smith, W. Clarke; Springer, S.; Sterling, G.; Suchard, J.; Szalay, M.; Sztein, A.; Thor, D.; Torres, A.; Torunbalci, M. M.; Vaishnav, A.; Vdovichev, S.; Villalonga, B.; Heidweiller, C. Vollgraff; Waltman, S.; Wang, S. X.; White, T.; Wong, K.; Woo, B. W. K.; Xing, C.; Yao, Z. Jamie; Yeh, P.; Ying, B.; Yoo, J.; Yosri, N.; Young, G.; Zalcman, A.; Zhu, N.; Zobrist, N.; Neven, H.; Babbush, R.; Boixo, S.; Hilton, J.; Lucero, E.; Megrant, A.; Kelly, J.; Chen, Y.; Smelyanskiy, V.; Vidal, G.; Roushan, P.; Läuchli, A. M.; Abanin, D. A.; Mi, X.

Thermalization and criticality on an analogue–digital quantum simulator

T. I. Andersen (), N. Astrakhantsev, A. H. Karamlou, J. Berndtsson, J. Motruk, A. Szasz, J. A. Gross, A. Schuckert, T. Westerhout, Y. Zhang, E. Forati, D. Rossi, B. Kobrin, A. Di Paolo, A. R. Klots, I. Drozdov, V. Kurilovich, A. Petukhov, L. B. Ioffe, A. Elben, A. Rath, V. Vitale, B. Vermersch, R. Acharya, L. A. Beni, K. Anderson, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, B. Ballard, J. C. Bardin, A. Bengtsson, A. Bilmes, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, D. A. Browne, B. Buchea, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, A. Cabrera, J. Campero, H.-S. Chang, Z. Chen, B. Chiaro, J. Claes, A. Y. Cleland, J. Cogan, R. Collins, P. Conner, W. Courtney, A. L. Crook, S. Das, D. M. Debroy, L. De Lorenzo, A. Del Toro Barba, S. Demura, P. Donohoe, A. Dunsworth, C. Earle, A. Eickbusch, A. M. Elbag, M. Elzouka, C. Erickson, L. Faoro, R. Fatemi, V. S. Ferreira, L. Flores Burgos, A. G. Fowler, B. Foxen, S. Ganjam, R. Gasca, W. Giang, C. Gidney, D. Gilboa, M. Giustina, R. Gosula, A. Grajales Dau, D. Graumann, A. Greene, S. Habegger, M. C. Hamilton, M. Hansen, M. P. Harrigan, S. D. Harrington, S. Heslin, P. Heu, G. Hill, M. R. Hoffmann, H.-Y. Huang, T. Huang, A. Huff, W. J. Huggins, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, S. Jordan, C. Joshi, P. Juhas, D. Kafri, H. Kang, K. Kechedzhi, T. Khaire, T. Khattar, M. Khezri, M. Kieferová, S. Kim, A. Kitaev, P. Klimov, A. N. Korotkov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, B. W. Langley, P. Laptev, K.-M. Lau, L. Le Guevel, J. Ledford, J. Lee, K. W. Lee, Y. D. Lensky, B. J. Lester, W. Y. Li, A. T. Lill, W. Liu, W. P. Livingston, A. Locharla, D. Lundahl, A. Lunt, S. Madhuk, A. Maloney, S. Mandrà, L. S. Martin, O. Martin, S. Martin, C. Maxfield, J. R. McClean, M. McEwen, S. Meeks, K. C. Miao, A. Mieszala, S. Molina, S. Montazeri, A. Morvan, R. Movassagh, C. Neill, A. Nersisyan, M. Newman, A. Nguyen, M. Nguyen, C.-H. Ni, M. Y. Niu, W. D. Oliver, K. Ottosson, A. Pizzuto, R. Potter, O. Pritchard, L. P. Pryadko, C. Quintana, M. J. Reagor, D. M. Rhodes, G. Roberts, C. Rocque, E. Rosenberg, N. C. Rubin, N. Saei, K. Sankaragomathi, K. J. Satzinger, H. F. Schurkus, C. Schuster, M. J. Shearn, A. Shorter, N. Shutty, V. Shvarts, V. Sivak, J. Skruzny, S. Small, W. Clarke Smith, S. Springer, G. Sterling, J. Suchard, M. Szalay, A. Sztein, D. Thor, A. Torres, M. M. Torunbalci, A. Vaishnav, S. Vdovichev, B. Villalonga, C. Vollgraff Heidweiller, S. Waltman, S. X. Wang, T. White, K. Wong, B. W. K. Woo, C. Xing, Z. Jamie Yao, P. Yeh, B. Ying, J. Yoo, N. Yosri, G. Young, A. Zalcman, N. Zhu, N. Zobrist, H. Neven, R. Babbush, S. Boixo, J. Hilton, E. Lucero, A. Megrant, J. Kelly, Y. Chen, V. Smelyanskiy, G. Vidal, P. Roushan, A. M. Läuchli, D. A. Abanin () and X. Mi ()
Additional contact information
T. I. Andersen: Google Research
N. Astrakhantsev: Google Research
A. H. Karamlou: Google Research
J. Berndtsson: Google Research
J. Motruk: University of Geneva
A. Szasz: Google Research
J. A. Gross: Google Research
A. Schuckert: NIST/University of Maryland
T. Westerhout: Radboud University
Y. Zhang: Google Research
E. Forati: Google Research
D. Rossi: University of Geneva
B. Kobrin: Google Research
A. Di Paolo: Google Research
A. R. Klots: Google Research
I. Drozdov: Google Research
V. Kurilovich: Google Research
A. Petukhov: Google Research
L. B. Ioffe: Google Research
A. Elben: Caltech
A. Rath: LPMMC
V. Vitale: LPMMC
B. Vermersch: LPMMC
R. Acharya: Google Research
L. A. Beni: Google Research
K. Anderson: Google Research
M. Ansmann: Google Research
F. Arute: Google Research
K. Arya: Google Research
A. Asfaw: Google Research
J. Atalaya: Google Research
B. Ballard: Google Research
J. C. Bardin: Google Research
A. Bengtsson: Google Research
A. Bilmes: Google Research
G. Bortoli: Google Research
A. Bourassa: Google Research
J. Bovaird: Google Research
L. Brill: Google Research
M. Broughton: Google Research
D. A. Browne: Google Research
B. Buchea: Google Research
B. B. Buckley: Google Research
D. A. Buell: Google Research
T. Burger: Google Research
B. Burkett: Google Research
N. Bushnell: Google Research
A. Cabrera: Google Research
J. Campero: Google Research
H.-S. Chang: Google Research
Z. Chen: Google Research
B. Chiaro: Google Research
J. Claes: Google Research
A. Y. Cleland: Google Research
J. Cogan: Google Research
R. Collins: Google Research
P. Conner: Google Research
W. Courtney: Google Research
A. L. Crook: Google Research
S. Das: Google Research
D. M. Debroy: Google Research
L. De Lorenzo: Google Research
A. Del Toro Barba: Google Research
S. Demura: Google Research
P. Donohoe: Google Research
A. Dunsworth: Google Research
C. Earle: Google Research
A. Eickbusch: Google Research
A. M. Elbag: Google Research
M. Elzouka: Google Research
C. Erickson: Google Research
L. Faoro: Google Research
R. Fatemi: Google Research
V. S. Ferreira: Google Research
L. Flores Burgos: Google Research
A. G. Fowler: Google Research
B. Foxen: Google Research
S. Ganjam: Google Research
R. Gasca: Google Research
W. Giang: Google Research
C. Gidney: Google Research
D. Gilboa: Google Research
M. Giustina: Google Research
R. Gosula: Google Research
A. Grajales Dau: Google Research
D. Graumann: Google Research
A. Greene: Google Research
S. Habegger: Google Research
M. C. Hamilton: Google Research
M. Hansen: Google Research
M. P. Harrigan: Google Research
S. D. Harrington: Google Research
S. Heslin: Google Research
P. Heu: Google Research
G. Hill: Google Research
M. R. Hoffmann: Google Research
H.-Y. Huang: Google Research
T. Huang: Google Research
A. Huff: Google Research
W. J. Huggins: Google Research
S. V. Isakov: Google Research
E. Jeffrey: Google Research
Z. Jiang: Google Research
C. Jones: Google Research
S. Jordan: Google Research
C. Joshi: Google Research
P. Juhas: Google Research
D. Kafri: Google Research
H. Kang: Google Research
K. Kechedzhi: Google Research
T. Khaire: Google Research
T. Khattar: Google Research
M. Khezri: Google Research
M. Kieferová: Google Research
S. Kim: Google Research
A. Kitaev: Google Research
P. Klimov: Google Research
A. N. Korotkov: Google Research
F. Kostritsa: Google Research
J. M. Kreikebaum: Google Research
D. Landhuis: Google Research
B. W. Langley: Google Research
P. Laptev: Google Research
K.-M. Lau: Google Research
L. Le Guevel: Google Research
J. Ledford: Google Research
J. Lee: Google Research
K. W. Lee: Google Research
Y. D. Lensky: Google Research
B. J. Lester: Google Research
W. Y. Li: Google Research
A. T. Lill: Google Research
W. Liu: Google Research
W. P. Livingston: Google Research
A. Locharla: Google Research
D. Lundahl: Google Research
A. Lunt: Google Research
S. Madhuk: Google Research
A. Maloney: Google Research
S. Mandrà: Google Research
L. S. Martin: Google Research
O. Martin: Google Research
S. Martin: Google Research
C. Maxfield: Google Research
J. R. McClean: Google Research
M. McEwen: Google Research
S. Meeks: Google Research
K. C. Miao: Google Research
A. Mieszala: Google Research
S. Molina: Google Research
S. Montazeri: Google Research
A. Morvan: Google Research
R. Movassagh: Google Research
C. Neill: Google Research
A. Nersisyan: Google Research
M. Newman: Google Research
A. Nguyen: Google Research
M. Nguyen: Google Research
C.-H. Ni: Google Research
M. Y. Niu: Google Research
W. D. Oliver: Google Research
K. Ottosson: Google Research
A. Pizzuto: Google Research
R. Potter: Google Research
O. Pritchard: Google Research
L. P. Pryadko: Google Research
C. Quintana: Google Research
M. J. Reagor: Google Research
D. M. Rhodes: Google Research
G. Roberts: Google Research
C. Rocque: Google Research
E. Rosenberg: Google Research
N. C. Rubin: Google Research
N. Saei: Google Research
K. Sankaragomathi: Google Research
K. J. Satzinger: Google Research
H. F. Schurkus: Google Research
C. Schuster: Google Research
M. J. Shearn: Google Research
A. Shorter: Google Research
N. Shutty: Google Research
V. Shvarts: Google Research
V. Sivak: Google Research
J. Skruzny: Google Research
S. Small: Google Research
W. Clarke Smith: Google Research
S. Springer: Google Research
G. Sterling: Google Research
J. Suchard: Google Research
M. Szalay: Google Research
A. Sztein: Google Research
D. Thor: Google Research
A. Torres: Google Research
M. M. Torunbalci: Google Research
A. Vaishnav: Google Research
S. Vdovichev: Google Research
B. Villalonga: Google Research
C. Vollgraff Heidweiller: Google Research
S. Waltman: Google Research
S. X. Wang: Google Research
T. White: Google Research
K. Wong: Google Research
B. W. K. Woo: Google Research
C. Xing: Google Research
Z. Jamie Yao: Google Research
P. Yeh: Google Research
B. Ying: Google Research
J. Yoo: Google Research
N. Yosri: Google Research
G. Young: Google Research
A. Zalcman: Google Research
N. Zhu: Google Research
N. Zobrist: Google Research
H. Neven: Google Research
R. Babbush: Google Research
S. Boixo: Google Research
J. Hilton: Google Research
E. Lucero: Google Research
A. Megrant: Google Research
J. Kelly: Google Research
Y. Chen: Google Research
V. Smelyanskiy: Google Research
G. Vidal: Google Research
P. Roushan: Google Research
A. M. Läuchli: Paul Scherrer Institute
D. A. Abanin: Google Research
X. Mi: Google Research

Nature, 2025, vol. 638, issue 8049, 79-85

Abstract: Abstract Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators1,2. Unlocking the full potential of such systems towards this goal requires flexible initial state preparation, precise time evolution and extensive probes for final state characterization. Here we present a quantum simulator comprising 69 superconducting qubits that supports both universal quantum gates and high-fidelity analogue evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. This hybrid platform features more versatile measurement capabilities compared with analogue-only simulators, which we leverage here to reveal a coarsening-induced breakdown of Kibble–Zurek scaling predictions3 in the XY model, as well as signatures of the classical Kosterlitz–Thouless phase transition4. Moreover, the digital gates enable precise energy control, allowing us to study the effects of the eigenstate thermalization hypothesis5–7 in targeted parts of the eigenspectrum. We also demonstrate digital preparation of pairwise-entangled dimer states, and image the transport of energy and vorticity during subsequent thermalization in analogue evolution. These results establish the efficacy of superconducting analogue–digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

Date: 2025
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DOI: 10.1038/s41586-024-08460-3

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