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Exploring Variational Entanglement Hamiltonians

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Exploring Variational Entanglement Hamiltonians

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AbstractRecent advances in analog and digital quantum-simulation platforms have enabled exploration of the spectrum of entanglement Hamiltonians via variational algorithms. In this work we analyze the convergence properties of the variationally obtained solutions and compare them to numerically exact calculations in quantum critical systems. We demonstrate that interpreting the cost functional as an integral permits the deployment of iterative quadrature schemes, thereby reducing the required number of measurements by more than an order of magnitude even in the presence of noise. We further show that a modified ansatz captures deviations from the Bisognano-Wichmann form in lattice models, improves convergence, improves trainability and provides a cost-function-level diagnostic for quantum phase transitions. Finally, we establish that a low cost value does not by itself guarantee convergence in trace distance. Nevertheless, it faithfully reproduces degeneracies and spectral gaps, which are essential for applications to topological phases.Popular summaryThe entanglement Hamiltonian is an effective Hamiltonian whose spectrum contains detailed information about how a subsystem is entangled with the rest of a quantum many-body system. Its low-energy levels can reveal robust structures such as degeneracies, entanglement gaps, and edge-mode-like features. These quantities are especially useful in topological systems, where they can diagnose properties that are not visible in local order parameters. Our work studies a variational method for learning entanglement Hamiltonians without reconstructing the full quantum state. The method proposes a parametrized entanglement Hamiltonian and optimizes it using measurements of time evolution, making it relevant for quantum simulators. Using quantum critical spin chains as a benchmark, we show that ansätze going beyond the standard Bisognano–Wichmann form can more accurately reproduce the low-lying entanglement spectrum of lattice systems. We also introduce an iterative quadrature scheme that substantially reduces the number of measurements needed to evaluate the variational cost function, while remaining robust to noise. Overall, the results provide a more practical route to entanglement spectroscopy in quantum simulators, with potential applications to identifying topological phases through their entanglement spectra.► BibTeX data@article{Kind2026exploring, doi = {10.22331/q-2026-06-12-2133}, url = {https://doi.org/10.22331/q-2026-06-12-2133}, title = {Exploring {V}ariational {E}ntanglement {H}amiltonians}, author = {Kind, Yanick S. and Fauseweh, Benedikt}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2133}, month = jun, year = {2026} }► References [1] Richard P. Feynman. ``Simulating physics with computers''. Int. J. Theor. Phys. 21, 467–488 (1982). https://doi.org/10.1007/BF02650179 [2] Seth Lloyd. ``Universal Quantum Simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [3] Andrew J. 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This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-06-12 11:25:46).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. AbstractRecent advances in analog and digital quantum-simulation platforms have enabled exploration of the spectrum of entanglement Hamiltonians via variational algorithms. In this work we analyze the convergence properties of the variationally obtained solutions and compare them to numerically exact calculations in quantum critical systems. We demonstrate that interpreting the cost functional as an integral permits the deployment of iterative quadrature schemes, thereby reducing the required number of measurements by more than an order of magnitude even in the presence of noise. We further show that a modified ansatz captures deviations from the Bisognano-Wichmann form in lattice models, improves convergence, improves trainability and provides a cost-function-level diagnostic for quantum phase transitions. Finally, we establish that a low cost value does not by itself guarantee convergence in trace distance. Nevertheless, it faithfully reproduces degeneracies and spectral gaps, which are essential for applications to topological phases.Popular summaryThe entanglement Hamiltonian is an effective Hamiltonian whose spectrum contains detailed information about how a subsystem is entangled with the rest of a quantum many-body system. Its low-energy levels can reveal robust structures such as degeneracies, entanglement gaps, and edge-mode-like features. These quantities are especially useful in topological systems, where they can diagnose properties that are not visible in local order parameters. Our work studies a variational method for learning entanglement Hamiltonians without reconstructing the full quantum state. The method proposes a parametrized entanglement Hamiltonian and optimizes it using measurements of time evolution, making it relevant for quantum simulators. Using quantum critical spin chains as a benchmark, we show that ansätze going beyond the standard Bisognano–Wichmann form can more accurately reproduce the low-lying entanglement spectrum of lattice systems. We also introduce an iterative quadrature scheme that substantially reduces the number of measurements needed to evaluate the variational cost function, while remaining robust to noise. Overall, the results provide a more practical route to entanglement spectroscopy in quantum simulators, with potential applications to identifying topological phases through their entanglement spectra.► BibTeX data@article{Kind2026exploring, doi = {10.22331/q-2026-06-12-2133}, url = {https://doi.org/10.22331/q-2026-06-12-2133}, title = {Exploring {V}ariational {E}ntanglement {H}amiltonians}, author = {Kind, Yanick S. and Fauseweh, Benedikt}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2133}, month = jun, year = {2026} }► References [1] Richard P. Feynman. ``Simulating physics with computers''. Int. J. Theor. Phys. 21, 467–488 (1982). https://doi.org/10.1007/BF02650179 [2] Seth Lloyd. ``Universal Quantum Simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [3] Andrew J. 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In Topological Aspects of Condensed Matter Physics: Lecture Notes of the Les Houches Summer School: Volume 103, August 2014.

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Exploring Variational Entanglement Hamiltonians

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