Quasi-Adiabatic Processing of Thermal States
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AbstractWe investigate the performance of an adiabatic evolution protocol when initialized from a Gibbs state at finite temperature. Specifically, we identify the diagonality of the final state in the energy eigenbasis, as well as the difference in energy and in energy variance with respect to the ideal adiabatic limit as key benchmarks for success and introduce metrics to quantify the off-diagonal contributions. Provided these benchmarks converge to their ideal adiabatic values, we argue that thermal expectation values of observables can be recovered, in accordance with the eigenstate thermalization hypothesis. For the transverse-field Ising model, we analytically establish that these benchmarks converge polynomially in both the quasi-adiabatic evolution time $T$ and system size. We perform numerical studies on non-integrable systems and find close quantitative agreement for the off-diagonality metrics, along with qualitatively similar behavior in the energy convergence.Featured image: Summary of the Quasi-Adiabatic Thermal Evolution protocolPopular summaryThe quantum adiabatic algorithm is a powerful strategy to probe ground state properties of many-body models: one initializes the system in the easily accessible ground state of a simple Hamiltonian and then slowly transforms the Hamiltonian into a more complex target model. If this transformation is sufficiently slow and the energy gap remains open, the system remains in its ground state. The same idea is also routinely used in ultra-cold atom experiments, where adiabatic ramps are widely used to explore equilibrium and near-equilibrium physics. However, in such realistic scenarios, systems are rarely prepared at exactly zero temperature. Instead, they typically start in thermal states that contain a mixture of energy eigenstates. Extending adiabatic ideas to this finite-temperature regime raises fundamental questions. Can a slowly driven thermal state remain thermal with respect to the final Hamiltonian, and under what conditions? How slow does the evolution need to be to preserve thermal properties? At first glance, the outlook seems pessimistic. In large quantum systems, the energy levels in the middle of the spectrum are extremely dense, with gaps that shrink exponentially with system size. Exact adiabatic following of individual excited states would therefore require exponentially long evolution times, which are impractical. However, exact tracking of each eigenstate may not be necessary. For many physical observables, it suffices that the final state is approximately diagonal in the energy basis and has energy fluctuations consistent with a thermal state. According to the eigenstate thermalization hypothesis, such conditions are already enough to reproduce thermal expectation values in generic systems. Motivated by this observation, we analyze a protocol we call quasi-adiabatic thermal evolution (QATE). In QATE, a thermal state is evolved unitarily under a slowly varying Hamiltonian, following the same interpolation as in the standard adiabatic algorithm. Rather than demanding exact adiabaticity, we ask a more physically relevant question: under realistic evolution times, can we still extract the thermal properties of the target Hamiltonian? To answer this, we develop quantitative benchmarks that measure how much mixing between energy eigenstates occurs and how far the final energy deviates from the ideal adiabatic limit. We prove polynomial convergence to the ideal adiabatic limit for the integrable transverse field Ising model and observe similar behavior numerically for non-integrable models.► BibTeX data@article{Irmejs2026quasiadiabatic, doi = {10.22331/q-2026-03-10-2018}, url = {https://doi.org/10.22331/q-2026-03-10-2018}, title = {Quasi-{A}diabatic {P}rocessing of {T}hermal {S}tates}, author = {Irmejs, Reinis and Ba{\~{n}}uls, Mari Carmen and Cirac, J. Ignacio}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2018}, month = mar, year = {2026} }► References [1] Google Quantum AI et al. ``Quantum error correction below the surface code threshold''. Nature 638, 920 (2024). https://doi.org/10.1038/s41586-024-08449-y [2] Minzhao Liu, Ruslan Shaydulin, Pradeep Niroula, Matthew DeCross, Shih-Han Hung, Wen Yu Kon, Enrique Cervero-Martín, Kaushik Chakraborty, Omar Amer, Scott Aaronson, Atithi Acharya, Yuri Alexeev, K. Jordan Berg, Shouvanik Chakrabarti, Florian J. Curchod, Joan M. Dreiling, Neal Erickson, Cameron Foltz, Michael Foss-Feig, David Hayes, Travis S. Humble, Niraj Kumar, Jeffrey Larson, Danylo Lykov, Michael Mills, Steven A. Moses, Brian Neyenhuis, Shaltiel Eloul, Peter Siegfried, James Walker, Charles Lim, and Marco Pistoia. ``Certified randomness using a trapped-ion quantum processor''. Nature 640, 343–348 (2025). https://doi.org/10.1038/s41586-025-08737-1 [3] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2024). https://doi.org/10.1038/s41586-023-06927-3 [4] Christian Gross and Immanuel Bloch. ``Quantum simulations with ultracold atoms in optical lattices''. Science 357, 995–1001 (2017). https://doi.org/10.1126/science.aal3837 [5] Julia Kempe, Alexei Kitaev, and Oded Regev. ``The complexity of the local hamiltonian problem''. Siam journal on computing 35, 1070–1097 (2006). https://doi.org/10.1137/S0097539704445226 [6] Max Born and Vladimir Fock. ``Beweis des adiabatensatzes''. Zeitschrift für Physik 51, 165–180 (1928). https://doi.org/10.1007/BF01343193 [7] Tosio Kato. ``On the adiabatic theorem of quantum mechanics''. Journal of the Physical Society of Japan 5, 435–439 (1950). https://doi.org/10.1143/JPSJ.5.435 [8] Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler. ``Bounds for the adiabatic approximation with applications to quantum computation''. Journal of Mathematical Physics 48, 102111 (2007). https://doi.org/10.1063/1.2798382 [9] Tameem Albash and Daniel A. Lidar. ``Adiabatic quantum computation''. Rev. Mod. Phys. 90, 015002 (2018). https://doi.org/10.1103/RevModPhys.90.015002 [10] Nikolai Il‘in, Anastasia Aristova, and Oleg Lychkovskiy. ``Adiabatic theorem for closed quantum systems initialized at finite temperature''. Physical Review A 104, L030202 (2021). https://doi.org/10.1103/PhysRevA.104.L030202 [11] Rafael L. Greenblatt, Markus Lange, Giovanna Marcelli, and Marcello Porta. ``Adiabatic evolution of low-temperature many-body systems''. Communications in Mathematical Physics 405 (2024). https://doi.org/10.1007/s00220-023-04903-6 [12] Yi Zuo, Qinghong Yang, Bang-Gui Liu, and Dong E. Liu. ``Work statistics for the adiabatic assumption in nonequilibrium many-body theory''. Phys. Rev. E 110, L022105 (2024). https://doi.org/10.1103/PhysRevE.110.L022105 [13] F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. Zambrini. ``Irreversible work and inner friction in quantum thermodynamic processes''. Phys. Rev. Lett. 113, 260601 (2014). https://doi.org/10.1103/PhysRevLett.113.260601 [14] G. Francica, J. Goold, and F. Plastina. ``Role of coherence in the nonequilibrium thermodynamics of quantum systems''. Phys. Rev. E 99, 042105 (2019). https://doi.org/10.1103/PhysRevE.99.042105 [15] Hadi Yarloo, Hua-Chen Zhang, and Anne E. B. Nielsen. ``Adiabatic time evolution of highly excited states''. PRX Quantum 5, 020365 (2024). https://doi.org/10.1103/PRXQuantum.5.020365 [16] Cécile Carcy, Gaétan Hercé, Antoine Tenart, Tommaso Roscilde, and David Clément. ``Certifying the adiabatic preparation of ultracold lattice bosons in the vicinity of the mott transition''.
Physical Review Letters 126 (2021). https://doi.org/10.1103/physrevlett.126.045301 [17] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. ``Many-body physics with ultracold gases''. Reviews of modern physics 80, 885–964 (2008). https://doi.org/10.1103/RevModPhys.80.885 [18] S Trotzky, L Pollet, F Gerbier, U Schnorrberger, Immanuel Bloch, NV Prokof’Ev, Boris Svistunov, and M Troyer. ``Suppression of the critical temperature for superfluidity near the mott transition''. Nature Physics 6, 998–1004 (2010). https://doi.org/10.1038/nphys1799 [19] W. Israel. ``Thermo-field dynamics of black holes''. Physics Letters A 57, 107–110 (1976). https://doi.org/10.1016/0375-9601(76)90178-X [20] Juan Maldacena. ``Eternal black holes in anti-de sitter''. Journal of High Energy Physics 2003, 021 (2003). https://doi.org/10.1088/1126-6708/2003/04/021 [21] Shira Chapman, Jens Eisert, Lucas Hackl, Michal P. Heller, Ro Jefferson, Hugo Marrochio, and Robert C. Myers. ``Complexity and entanglement for thermofield double states''. SciPost Phys. 6, 034 (2019). https://doi.org/10.21468/SciPostPhys.6.3.034 [22] William Cottrell, Ben Freivogel, Diego M. Hofman, and Sagar F. Lokhande. ``How to build the thermofield double state''. Journal of High Energy Physics 2019, 58 (2019). https://doi.org/10.1007/JHEP02(2019)058 [23] Daniel Faílde, Juan Santos-Suárez, David A. Herrera-Martí, and Javier Mas. ``Hamiltonian forging of a thermofield double''. Phys. Rev. A 111, 012432 (2025). https://doi.org/10.1103/PhysRevA.111.012432 [24] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W Hänsch, and Immanuel Bloch. ``Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms''. Nature 415, 39–44 (2002). https://doi.org/10.1038/415039a [25] Tilman Esslinger. ``Fermi-hubbard physics with atoms in an optical lattice''. Annual Review of Condensed Matter Physics 1, 129–152 (2010). https://doi.org/10.1146/annurev-conmatphys-070909-104059 [26] Albert Messiah. ``Quantum mechanics: two volumes bound as one''. Dover Publications, Inc. Garden City, New York (2020). [27] M. H. S. Amin. ``Consistency of the adiabatic theorem''. Phys. Rev. Lett. 102, 220401 (2009). https://doi.org/10.1103/PhysRevLett.102.220401 [28] Gheorghe Nenciu. ``Linear adiabatic theory. exponential estimates''. Communications in mathematical physics 152, 479–496 (1993). https://doi.org/10.1007/BF02096616 [29] George A Hagedorn and Alain Joye. ``Elementary exponential error estimates for the adiabatic approximation''. Journal of mathematical analysis and applications 267, 235–246 (2002). https://doi.org/10.1006/jmaa.2001.7765 [30] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. ``Thermalization and its mechanism for generic isolated quantum systems''. Nature 452, 854–858 (2008). https://doi.org/10.1038/nature06838 [31] Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics''. Advances in Physics 65, 239–362 (2016). https://doi.org/10.1080/00018732.2016.1198134 [32] Sirui Lu, Mari Carmen Bañuls, and J. Ignacio Cirac. ``Algorithms for quantum simulation at finite energies''. PRX Quantum 2 (2021). https://doi.org/10.1103/prxquantum.2.020321 [33] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Physical Review X 11 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [34] Koenraad M. R. Audenaert. ``Comparisons between quantum state distinguishability measures''. Quantum Info. Comput. 14, 31–38 (2014). https://doi.org/10.48550/arXiv.1207.1197 [35] A. A. Ovchinnikov, D. V. Dmitriev, V. Ya. Krivnov, and V. O. Cheranovskii. ``Antiferromagnetic ising chain in a mixed transverse and longitudinal magnetic field''. Phys. Rev. B 68, 214406 (2003). https://doi.org/10.1103/PhysRevB.68.214406 [36] Yimin Ge, András Molnár, and J. Ignacio Cirac. ``Rapid adiabatic preparation of injective projected entangled pair states and gibbs states''.
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This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-10 14:00:18: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractWe investigate the performance of an adiabatic evolution protocol when initialized from a Gibbs state at finite temperature. Specifically, we identify the diagonality of the final state in the energy eigenbasis, as well as the difference in energy and in energy variance with respect to the ideal adiabatic limit as key benchmarks for success and introduce metrics to quantify the off-diagonal contributions. Provided these benchmarks converge to their ideal adiabatic values, we argue that thermal expectation values of observables can be recovered, in accordance with the eigenstate thermalization hypothesis. For the transverse-field Ising model, we analytically establish that these benchmarks converge polynomially in both the quasi-adiabatic evolution time $T$ and system size. We perform numerical studies on non-integrable systems and find close quantitative agreement for the off-diagonality metrics, along with qualitatively similar behavior in the energy convergence.Featured image: Summary of the Quasi-Adiabatic Thermal Evolution protocolPopular summaryThe quantum adiabatic algorithm is a powerful strategy to probe ground state properties of many-body models: one initializes the system in the easily accessible ground state of a simple Hamiltonian and then slowly transforms the Hamiltonian into a more complex target model. If this transformation is sufficiently slow and the energy gap remains open, the system remains in its ground state. The same idea is also routinely used in ultra-cold atom experiments, where adiabatic ramps are widely used to explore equilibrium and near-equilibrium physics. However, in such realistic scenarios, systems are rarely prepared at exactly zero temperature. Instead, they typically start in thermal states that contain a mixture of energy eigenstates. Extending adiabatic ideas to this finite-temperature regime raises fundamental questions. Can a slowly driven thermal state remain thermal with respect to the final Hamiltonian, and under what conditions? How slow does the evolution need to be to preserve thermal properties? At first glance, the outlook seems pessimistic. In large quantum systems, the energy levels in the middle of the spectrum are extremely dense, with gaps that shrink exponentially with system size. Exact adiabatic following of individual excited states would therefore require exponentially long evolution times, which are impractical. However, exact tracking of each eigenstate may not be necessary. For many physical observables, it suffices that the final state is approximately diagonal in the energy basis and has energy fluctuations consistent with a thermal state. According to the eigenstate thermalization hypothesis, such conditions are already enough to reproduce thermal expectation values in generic systems. Motivated by this observation, we analyze a protocol we call quasi-adiabatic thermal evolution (QATE). In QATE, a thermal state is evolved unitarily under a slowly varying Hamiltonian, following the same interpolation as in the standard adiabatic algorithm. Rather than demanding exact adiabaticity, we ask a more physically relevant question: under realistic evolution times, can we still extract the thermal properties of the target Hamiltonian? To answer this, we develop quantitative benchmarks that measure how much mixing between energy eigenstates occurs and how far the final energy deviates from the ideal adiabatic limit. We prove polynomial convergence to the ideal adiabatic limit for the integrable transverse field Ising model and observe similar behavior numerically for non-integrable models.► BibTeX data@article{Irmejs2026quasiadiabatic, doi = {10.22331/q-2026-03-10-2018}, url = {https://doi.org/10.22331/q-2026-03-10-2018}, title = {Quasi-{A}diabatic {P}rocessing of {T}hermal {S}tates}, author = {Irmejs, Reinis and Ba{\~{n}}uls, Mari Carmen and Cirac, J. Ignacio}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2018}, month = mar, year = {2026} }► References [1] Google Quantum AI et al. ``Quantum error correction below the surface code threshold''. Nature 638, 920 (2024). https://doi.org/10.1038/s41586-024-08449-y [2] Minzhao Liu, Ruslan Shaydulin, Pradeep Niroula, Matthew DeCross, Shih-Han Hung, Wen Yu Kon, Enrique Cervero-Martín, Kaushik Chakraborty, Omar Amer, Scott Aaronson, Atithi Acharya, Yuri Alexeev, K. Jordan Berg, Shouvanik Chakrabarti, Florian J. Curchod, Joan M. Dreiling, Neal Erickson, Cameron Foltz, Michael Foss-Feig, David Hayes, Travis S. Humble, Niraj Kumar, Jeffrey Larson, Danylo Lykov, Michael Mills, Steven A. Moses, Brian Neyenhuis, Shaltiel Eloul, Peter Siegfried, James Walker, Charles Lim, and Marco Pistoia. ``Certified randomness using a trapped-ion quantum processor''. Nature 640, 343–348 (2025). https://doi.org/10.1038/s41586-025-08737-1 [3] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2024). https://doi.org/10.1038/s41586-023-06927-3 [4] Christian Gross and Immanuel Bloch. ``Quantum simulations with ultracold atoms in optical lattices''. Science 357, 995–1001 (2017). https://doi.org/10.1126/science.aal3837 [5] Julia Kempe, Alexei Kitaev, and Oded Regev. ``The complexity of the local hamiltonian problem''. Siam journal on computing 35, 1070–1097 (2006). https://doi.org/10.1137/S0097539704445226 [6] Max Born and Vladimir Fock. ``Beweis des adiabatensatzes''. Zeitschrift für Physik 51, 165–180 (1928). https://doi.org/10.1007/BF01343193 [7] Tosio Kato. ``On the adiabatic theorem of quantum mechanics''. Journal of the Physical Society of Japan 5, 435–439 (1950). https://doi.org/10.1143/JPSJ.5.435 [8] Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler. ``Bounds for the adiabatic approximation with applications to quantum computation''. Journal of Mathematical Physics 48, 102111 (2007). https://doi.org/10.1063/1.2798382 [9] Tameem Albash and Daniel A. Lidar. ``Adiabatic quantum computation''. Rev. Mod. Phys. 90, 015002 (2018). https://doi.org/10.1103/RevModPhys.90.015002 [10] Nikolai Il‘in, Anastasia Aristova, and Oleg Lychkovskiy. ``Adiabatic theorem for closed quantum systems initialized at finite temperature''. Physical Review A 104, L030202 (2021). https://doi.org/10.1103/PhysRevA.104.L030202 [11] Rafael L. Greenblatt, Markus Lange, Giovanna Marcelli, and Marcello Porta. ``Adiabatic evolution of low-temperature many-body systems''. Communications in Mathematical Physics 405 (2024). https://doi.org/10.1007/s00220-023-04903-6 [12] Yi Zuo, Qinghong Yang, Bang-Gui Liu, and Dong E. Liu. ``Work statistics for the adiabatic assumption in nonequilibrium many-body theory''. Phys. Rev. E 110, L022105 (2024). https://doi.org/10.1103/PhysRevE.110.L022105 [13] F. Plastina, A. Alecce, T. J. G. Apollaro, G. Falcone, G. Francica, F. Galve, N. Lo Gullo, and R. 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