We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of a k-sparse fermionic Hamiltonian with strictly q-local terms, as well as a 1/(4k+1)-approximation when the Hamiltonian has both 2-local and 4-local terms. More generally we obtain a 1/O(qk²)-approximation for k-sparse fermionic Hamiltonians with terms of locality at most q. Our techniques also yield analogous approximations for k-sparse, q-local qubit Hamiltonians with small hidden constants and improved dependence on q.
@InProceedings{hothem_et_al:LIPIcs.TQC.2023.6, author = {Hothem, Daniel and Parekh, Ojas and Thompson, Kevin}, title = {{Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians}}, booktitle = {18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)}, pages = {6:1--6:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-283-9}, ISSN = {1868-8969}, year = {2023}, volume = {266}, editor = {Fawzi, Omar and Walter, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.6}, URN = {urn:nbn:de:0030-drops-183163}, doi = {10.4230/LIPIcs.TQC.2023.6}, annote = {Keywords: Approximation algorithms, Extremal eigenvalues, Sparse Hamiltonians, Fermionic Hamiltonians, Qubit Hamiltonians} }
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