Optimizing sparse fermionic Hamiltonians
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 ho...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2023-08-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2023-08-10-1081/pdf/ |
| _version_ | 1797745674208411648 |
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| author | Yaroslav Herasymenko Maarten Stroeks Jonas Helsen Barbara Terhal |
| author_facet | Yaroslav Herasymenko Maarten Stroeks Jonas Helsen Barbara Terhal |
| author_sort | Yaroslav Herasymenko |
| collection | DOAJ |
| description | 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]. |
| first_indexed | 2024-03-12T15:27:29Z |
| format | Article |
| id | doaj.art-551e6f2b34a848c48fc7da973fbeeab4 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-12T15:27:29Z |
| publishDate | 2023-08-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-551e6f2b34a848c48fc7da973fbeeab42023-08-10T12:47:59ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2023-08-017108110.22331/q-2023-08-10-108110.22331/q-2023-08-10-1081Optimizing sparse fermionic HamiltoniansYaroslav HerasymenkoMaarten StroeksJonas HelsenBarbara TerhalWe 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].https://quantum-journal.org/papers/q-2023-08-10-1081/pdf/ |
| spellingShingle | Yaroslav Herasymenko Maarten Stroeks Jonas Helsen Barbara Terhal Optimizing sparse fermionic Hamiltonians Quantum |
| title | Optimizing sparse fermionic Hamiltonians |
| title_full | Optimizing sparse fermionic Hamiltonians |
| title_fullStr | Optimizing sparse fermionic Hamiltonians |
| title_full_unstemmed | Optimizing sparse fermionic Hamiltonians |
| title_short | Optimizing sparse fermionic Hamiltonians |
| title_sort | optimizing sparse fermionic hamiltonians |
| url | https://quantum-journal.org/papers/q-2023-08-10-1081/pdf/ |
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