Optimizing sparse fermionic Hamiltonians

Yaroslav Herasymenko1,2, Maarten Stroeks2,3, Jonas Helsen1, and Barbara Terhal2,3

1QuSoft & CWI, Science Park 123 1098 XG Amsterdam, The Netherlands
2QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
3EEMCS Faculty, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE, Delft, The Netherlands

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We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense case [1, 2], we prove that strictly $q$-local $\rm {\textit {sparse}}$ fermionic Hamiltonians have a constant Gaussian approximation ratio; the result holds for any connectivity and interaction strengths. Sparsity means that each fermion participates in a bounded number of interactions, and strictly $q$-local means that each term involves exactly $q$ fermionic (Majorana) operators. We extend our proof to give a constant Gaussian approximation ratio for sparse fermionic Hamiltonians with both quartic and quadratic terms. With additional work, we also prove a constant Gaussian approximation ratio for the so-called sparse SYK model with strictly $4$-local interactions (sparse SYK-$4$ model). In each setting we show that the Gaussian state can be efficiently determined. Finally, we prove that the $O(n^{-1/2})$ Gaussian approximation ratio for the normal (dense) SYK-$4$ model extends to SYK-$q$ for even $q\gt4$, with an approximation ratio of $O(n^{1/2 – q/4})$. Our results identify non-sparseness as the prime reason that the SYK-$4$ model can fail to have a constant approximation ratio [1, 2].

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[2] Yaroslav Herasymenko, Anurag Anshu, Barbara M. Terhal, and Jonas Helsen, "Fermionic Hamiltonians without trivial low-energy states", Physical Review A 109 5, 052431 (2024).

[3] Babak Tarighi, Reyhaneh Khasseh, and M. A. Rajabpour, "Efficient representation of Gaussian fermionic pure states in noncomputational bases", Physical Review A 109 6, 062214 (2024).

[4] Kostas Vilkelis, Antonio L. R. Manesco, Juan Daniel Torres Luna, Sebastian Miles, Michael Wimmer, and Anton R. Akhmerov, "Fermionic quantum computation with Cooper pair splitters", SciPost Physics 16 5, 135 (2024).

[5] Babak Tarighi, Reyhaneh Khasseh, and M. A. Rajabpour, "Efficient Representation of Gaussian Fermionic Pure States in Non-Computational Bases", arXiv:2403.03289, (2024).

[6] Chi-Fang (Anthony) Chen, Andrew Lucas, and Chao Yin, "Speed limits and locality in many-body quantum dynamics", Reports on Progress in Physics 86 11, 116001 (2023).

[7] Yaroslav Herasymenko, Anurag Anshu, Barbara Terhal, and Jonas Helsen, "Fermionic Hamiltonians without trivial low-energy states", arXiv:2307.13730, (2023).

[8] J. Eisert, "A note on lower bounds to variational problems with guarantees", arXiv:2301.06142, (2023).

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