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A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems

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A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems

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AbstractEstimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.Featured image: Gaps of the $\mathbb{Z}_3$ family of deformed Potts clock models. For a family of spin chain models parameterized by two parameters $(r,s)$, the lower bounds on the gap were computed at different values of the parameters using different methods: the Knabe and Gosset-Mozgunov finite size criteria, and our LTI SDP method.Popular summaryThe spectral gap in the thermodynamic limit plays a central role across quantum information, condensed matter physics, and classical sampling. Determining whether a system is gapped or gapless in this limit is, in general, undecidable. We introduce a hierarchy of simpler problems that yield certifiable lower bounds on the spectral gap directly in the thermodynamic limit. Our approach reformulates the problem by modifying the requirement of exact gap computation to the construction of a local, positive operator decomposition for a quadratic function of the Hamiltonian $H$. This formulation enables efficient optimization while preserving rigorous guarantees. A key feature of the method is the use of translation invariance, which allows one to solve a finite-size problem while still obtaining valid lower bounds in the thermodynamic limit. We benchmark the method on several spin-chain models, where it demonstrates substantial improvements over previous techniques. In particular, we observe both increased accuracy of the bounds and a significant expansion of the parameter regions that can be rigorously certified as gapped. Beyond one-dimensional systems, the framework naturally extends to higher dimensions, providing a systematic and scalable route toward gap certification. Overall, this work establishes a general methodology for rigorously bounding spectral gaps, with broad applicability across quantum many-body physics and related computational settings.► BibTeX data@article{Rai2026hierarchyofspectral, doi = {10.22331/q-2026-04-13-2065}, url = {https://doi.org/10.22331/q-2026-04-13-2065}, title = {A {H}ierarchy of {S}pectral {G}ap {C}ertificates for {F}rustration-{F}ree {S}pin {S}ystems}, author = {Rai, Kshiti Sneh and Kull, Ilya and Emonts, Patrick and Tura, Jordi and Schuch, Norbert and Baccari, Flavio}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2065}, month = apr, year = {2026} }► References [1] M B Hastings. An area law for one-dimensional quantum systems. 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Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group. Physical Review X, 14 (2): 021008, April 2024. ISSN 2160-3308. 10.1103/PhysRevX.14.021008. https://doi.org/10.1103/PhysRevX.14.021008 [55] Hamza Fawzi, Omar Fawzi, and Samuel O. Scalet. Entropy constraints for ground energy optimization. Journal of Mathematical Physics, 65 (3): 032201, March 2024. ISSN 0022-2488, 1089-7658. 10.1063/5.0159108. https://doi.org/10.1063/5.0159108 [56] Jie Wang, Jacopo Surace, Irénée Frérot, Benoı̂t Legat, Marc-Olivier Renou, Victor Magron, and Antonio Acín. Certifying ground-state properties of quantum many-body systems. Physical Review X, 14 (3): 031006, July 2024. ISSN 2160-3308. 10.1103/PhysRevX.14.031006. https://doi.org/10.1103/PhysRevX.14.031006 [57] Ramis Movassagh and Peter W. Shor. Supercritical entanglement in local systems: Counterexample to the area law for quantum matter. Proceedings of the National Academy of Sciences, 113 (47): 13278–13282, November 2016. ISSN 0027-8424, 1091-6490. 10.1073/pnas.1605716113. https://doi.org/10.1073/pnas.1605716113 [58] Zhao Zhang, Amr Ahmadain, and Israel Klich. Quantum phase transition from bounded to extensive entanglement entropy in a frustration-free spin chain, June 2016.Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 18:46:49: Could not fetch cited-by data for 10.22331/q-2026-04-13-2065 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 18:46:49: 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. AbstractEstimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate – a proof of a lower bound on the gap – is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.Featured image: Gaps of the $\mathbb{Z}_3$ family of deformed Potts clock models. For a family of spin chain models parameterized by two parameters $(r,s)$, the lower bounds on the gap were computed at different values of the parameters using different methods: the Knabe and Gosset-Mozgunov finite size criteria, and our LTI SDP method.Popular summaryThe spectral gap in the thermodynamic limit plays a central role across quantum information, condensed matter physics, and classical sampling. Determining whether a system is gapped or gapless in this limit is, in general, undecidable. We introduce a hierarchy of simpler problems that yield certifiable lower bounds on the spectral gap directly in the thermodynamic limit. Our approach reformulates the problem by modifying the requirement of exact gap computation to the construction of a local, positive operator decomposition for a quadratic function of the Hamiltonian $H$. This formulation enables efficient optimization while preserving rigorous guarantees. A key feature of the method is the use of translation invariance, which allows one to solve a finite-size problem while still obtaining valid lower bounds in the thermodynamic limit. We benchmark the method on several spin-chain models, where it demonstrates substantial improvements over previous techniques. In particular, we observe both increased accuracy of the bounds and a significant expansion of the parameter regions that can be rigorously certified as gapped. Beyond one-dimensional systems, the framework naturally extends to higher dimensions, providing a systematic and scalable route toward gap certification. Overall, this work establishes a general methodology for rigorously bounding spectral gaps, with broad applicability across quantum many-body physics and related computational settings.► BibTeX data@article{Rai2026hierarchyofspectral, doi = {10.22331/q-2026-04-13-2065}, url = {https://doi.org/10.22331/q-2026-04-13-2065}, title = {A {H}ierarchy of {S}pectral {G}ap {C}ertificates for {F}rustration-{F}ree {S}pin {S}ystems}, author = {Rai, Kshiti Sneh and Kull, Ilya and Emonts, Patrick and Tura, Jordi and Schuch, Norbert and Baccari, Flavio}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2065}, month = apr, year = {2026} }► References [1] M B Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007 (08): P08024–P08024, August 2007. ISSN 1742-5468. 10.1088/1742-5468/2007/08/P08024. https://doi.org/10.1088/1742-5468/2007/08/P08024 [2] Matthew B. Hastings and Tohru Koma. Spectral Gap and Exponential Decay of Correlations. Communications in Mathematical Physics, 265 (3): 781–804, August 2006. ISSN 0010-3616, 1432-0916. 10.1007/s00220-006-0030-4. https://doi.org/10.1007/s00220-006-0030-4 [3] Anurag Anshu, Itai Arad, and David Gosset. An area law for 2D frustration-free spin systems. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 12–18, June 2022. 10.1145/3519935.3519962. https://doi.org/10.1145/3519935.3519962 [4] Zeph Landau, Umesh Vazirani, and Thomas Vidick. A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nature Physics, 11 (7): 566–569, July 2015. ISSN 1745-2473, 1745-2481. 10.1038/nphys3345. https://doi.org/10.1038/nphys3345 [5] M. B. Hastings. Lieb-Schultz-Mattis in higher dimensions. Physical Review B, 69 (10): 104431, March 2004. ISSN 1098-0121, 1550-235X. 10.1103/PhysRevB.69.104431. https://doi.org/10.1103/PhysRevB.69.104431 [6] Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims. Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Communications in Mathematical Physics, 309 (3): 835–871, February 2012. ISSN 0010-3616, 1432-0916. 10.1007/s00220-011-1380-0. https://doi.org/10.1007/s00220-011-1380-0 [7] Yimin Ge, András Molnár, and J. Ignacio Cirac.

Rapid Adiabatic Preparation of Injective Projected Entangled Pair States and Gibbs States.

Spectral Gap Optimization for Enhanced Adiabatic State Preparation, September 2024. [9] Dorit Aharonov and Amnon Ta-Shma.

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A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems

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