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Adiabatic quantum state preparation in integrable models

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Researchers from the Max Planck Institute and University of Munich developed an adiabatic quantum algorithm to prepare high-energy eigenstates in integrable quantum models, significantly expanding the reach of quantum simulation beyond ground states. The team demonstrated polynomial circuit depth scaling for preparing eigenstates in the XXZ Heisenberg chain using the thermodynamic Bethe ansatz, outperforming previous exponential-time methods that relied on explicit integrability. A novel "parent Hamiltonian" construction leverages conserved quantities to isolate target eigenstates, preventing spectral crowding that typically disrupts adiabatic evolution for excited states. Rigorous proofs confirm efficiency for non-interacting XY spin chains, while numerical evidence shows polynomial scaling persists for interacting Richardson-Gaudin models, covering the full energy spectrum. This work bridges quantum integrability with algorithm design, offering a scalable pathway to simulate complex many-body systems on near-term quantum hardware.

Adiabatic quantum state preparation in integrable models

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AbstractWe propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits $N$, outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in $N$ for all eigenstates, despite the models being interacting.Featured image: An illustration how in integrable models, a suitably-gapped parent Hamiltonian $H_P$ for an adiabatic algorithm can be constructed from a set of integrals of motions $Q^{(k)}$ for an arbitrary eigenstate $v$, even if the eigenstate has exponentially close neighbours in the spectrum of each individual $Q^{(k)}$.Popular summaryQuantum computers hold significant promise for simulating quantum systems that are beyond the reach of classical computational methods. A central challenge in any such simulation is state preparation – getting the quantum computer into the right quantum state to begin with. For ground states of many physical systems, the adiabatic algorithm offers an established approach: by slowly deforming a simple Hamiltonian into the target one, the system remains in its lowest-energy state throughout the evolution. The success of this method relies on energy levels remaining well-separated throughout the process – if they crowd together, the system can jump to the wrong state. For highly excited states, exponentially many energy levels generically sit close together, which is why the adiabatic algorithm appears unsuitable in this setting. In this work, we demonstrate that for integrable models – a special class of exactly solvable quantum systems that possess an extensive number of conserved quantities – the adiabatic algorithm can be leveraged to efficiently prepare not only ground states, but arbitrary eigenstates under certain conditions. The central idea is to construct a tailored 'parent Hamiltonian' for each target eigenstate by penalizing deviations from the target values of all conserved quantities. Because integrable models have sufficiently many such quantities to uniquely distinguish every eigenstate, the parent Hamiltonian effectively pushes all other energy levels away from the target state, preventing the crowding that would otherwise obstruct the adiabatic algorithm. We first show that the standard adiabatic algorithm already prepares certain eigenstates of a prototypical integrable spin chain with polynomial circuit depth – an exponential improvement over previous integrability-based approaches. We then apply our parent Hamiltonian construction to both non-interacting and interacting integrable models, providing rigorous proofs and numerical evidence that the circuit depth remains polynomial across the entire spectrum. These results establish a new connection between quantum integrability and quantum algorithm design, opening a pathway toward efficiently accessing the full eigenstate structure of interacting integrable models on quantum hardware.► BibTeX data@article{Lutz2026adiabaticquantum, doi = {10.22331/q-2026-03-18-2032}, url = {https://doi.org/10.22331/q-2026-03-18-2032}, title = {Adiabatic quantum state preparation in integrable models}, author = {Lutz, Maximilian and Piroli, Lorenzo and Styliaris, Georgios 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 = {2032}, month = mar, year = {2026} }► References [1] See Supplemental Material for details. [2] Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabrese. Entanglement entropy of excited states. J. Stat. 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Reducing circuit depth in quantum state preparation for quantum simulation using measurements and feedforward. Phys. Rev. Appl., 23: 054066, May 2025. 10.1103/PhysRevApplied.23.054066. https://doi.org/10.1103/PhysRevApplied.23.054066Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-18 12:09:37: Could not fetch cited-by data for 10.22331/q-2026-03-18-2032 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-18 12:09:37: 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 propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits $N$, outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in $N$ for all eigenstates, despite the models being interacting.Featured image: An illustration how in integrable models, a suitably-gapped parent Hamiltonian $H_P$ for an adiabatic algorithm can be constructed from a set of integrals of motions $Q^{(k)}$ for an arbitrary eigenstate $v$, even if the eigenstate has exponentially close neighbours in the spectrum of each individual $Q^{(k)}$.Popular summaryQuantum computers hold significant promise for simulating quantum systems that are beyond the reach of classical computational methods. A central challenge in any such simulation is state preparation – getting the quantum computer into the right quantum state to begin with. For ground states of many physical systems, the adiabatic algorithm offers an established approach: by slowly deforming a simple Hamiltonian into the target one, the system remains in its lowest-energy state throughout the evolution. The success of this method relies on energy levels remaining well-separated throughout the process – if they crowd together, the system can jump to the wrong state. For highly excited states, exponentially many energy levels generically sit close together, which is why the adiabatic algorithm appears unsuitable in this setting. In this work, we demonstrate that for integrable models – a special class of exactly solvable quantum systems that possess an extensive number of conserved quantities – the adiabatic algorithm can be leveraged to efficiently prepare not only ground states, but arbitrary eigenstates under certain conditions. The central idea is to construct a tailored 'parent Hamiltonian' for each target eigenstate by penalizing deviations from the target values of all conserved quantities. Because integrable models have sufficiently many such quantities to uniquely distinguish every eigenstate, the parent Hamiltonian effectively pushes all other energy levels away from the target state, preventing the crowding that would otherwise obstruct the adiabatic algorithm. We first show that the standard adiabatic algorithm already prepares certain eigenstates of a prototypical integrable spin chain with polynomial circuit depth – an exponential improvement over previous integrability-based approaches. We then apply our parent Hamiltonian construction to both non-interacting and interacting integrable models, providing rigorous proofs and numerical evidence that the circuit depth remains polynomial across the entire spectrum. These results establish a new connection between quantum integrability and quantum algorithm design, opening a pathway toward efficiently accessing the full eigenstate structure of interacting integrable models on quantum hardware.► BibTeX data@article{Lutz2026adiabaticquantum, doi = {10.22331/q-2026-03-18-2032}, url = {https://doi.org/10.22331/q-2026-03-18-2032}, title = {Adiabatic quantum state preparation in integrable models}, author = {Lutz, Maximilian and Piroli, Lorenzo and Styliaris, Georgios 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 = {2032}, month = mar, year = {2026} }► References [1] See Supplemental Material for details. [2] Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabrese. Entanglement entropy of excited states. J. Stat. Mech., 2009 (10): P10020, 2009. 10.1088/1742-5468/2009/10/P10020. https://doi.org/10.1088/1742-5468/2009/10/P10020 [3] Tameem Albash and Daniel A Lidar. Adiabatic quantum computation. Rev. Mod. Phys., 90 (1): 015002, 2018. 10.1103/RevModPhys.90.015002. https://doi.org/10.1103/RevModPhys.90.015002 [4] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. 10.1126/science.1113479. https://doi.org/10.1126/science.1113479 [5] Alvise Bastianello, Lorenzo Piroli, and Pasquale Calabrese. Exact local correlations and full counting statistics for arbitrary states of the one-dimensional interacting bose gas. Phys. Rev. Lett., 120: 190601, May 2018. 10.1103/PhysRevLett.120.190601. https://doi.org/10.1103/PhysRevLett.120.190601 [6] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan. Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev., 120 (22): 12685–12717, 2020. 10.1021/acs.chemrev.9b00829. PMID: 33090772. https://doi.org/10.1021/acs.chemrev.9b00829 [7] Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and real computation. Springer Science & Business Media, 1998. 10.1007/978-1-4612-0701-6. https://doi.org/10.1007/978-1-4612-0701-6 [8] Jean-Sébastien Caux and Jorn Mossel. Remarks on the notion of quantum integrability. J. Stat. Mech., 2011 (02): P02023, feb 2011. 10.1088/1742-5468/2011/02/P02023. https://doi.org/10.1088/1742-5468/2011/02/P02023 [9] Pieter W. Claeys, Stijn De Baerdemacker, Mario Van Raemdonck, and Dimitri Van Neck. Eigenvalue-based method and form-factor determinant representations for integrable xxz richardson-gaudin models. Phys. Rev. B, 91: 155102, Apr 2015. 10.1103/PhysRevB.91.155102. https://doi.org/10.1103/PhysRevB.91.155102 [10] Pieter W Claeys, Dimitri Van Neck, and Stijn De Baerdemacker. Inner products in integrable richardson-gaudin models. SciPost Phys., 3 (4): 028, 2017. 10.21468/SciPostPhys. https://doi.org/10.21468/SciPostPhys [11] Claeys, Pieter. Richardson-Gaudin models and broken integrability. PhD thesis, Ghent University, 2018. [12] Fabian HL Essler, Holger Frahm, Frank Göhmann, Andreas Klümper, and Vladimir E Korepin. The one-dimensional Hubbard model.

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Adiabatic quantum state preparation in integrable models

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