Hybrid quantum-classical algorithms for approximate graph coloring
We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX- k -CUT, the problem of finding an approximate k -vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (n...
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| Vydáno v: | Quantum (Vienna, Austria) Ročník 6; s. 678 |
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Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
30.03.2022
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| ISSN: | 2521-327X, 2521-327X |
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| Abstract | We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-
k
-CUT, the problem of finding an approximate
k
-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level
p
: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-
1
QAOA and level-
1
RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-
1
QAOA and RQAOA with up to
300
qutrits applied to ensembles of randomly generated
3
-colorable constant-degree graphs. We find that level-
1
RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices. |
|---|---|
| AbstractList | We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level $p$: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-$1$ QAOA and level-$1$ RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-$1$ QAOA and RQAOA with up to $300$ qutrits applied to ensembles of randomly generated $3$-colorable constant-degree graphs. We find that level-$1$ RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices. We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX- k -CUT, the problem of finding an approximate k -vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level p : the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level- 1 QAOA and level- 1 RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level- 1 QAOA and RQAOA with up to 300 qutrits applied to ensembles of randomly generated 3 -colorable constant-degree graphs. We find that level- 1 RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices. |
| ArticleNumber | 678 |
| Author | Kliesch, Alexander Tang, Eugene Bravyi, Sergey Koenig, Robert |
| Author_xml | – sequence: 1 givenname: Sergey surname: Bravyi fullname: Bravyi, Sergey organization: IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA – sequence: 2 givenname: Alexander surname: Kliesch fullname: Kliesch, Alexander organization: Zentrum Mathematik, Technical University of Munich, 85748 Garching, Germany – sequence: 3 givenname: Robert surname: Koenig fullname: Koenig, Robert organization: Institute for Advanced Study & Zentrum Mathematik, Technical University of Munich, 85748 Garching, Germany – sequence: 4 givenname: Eugene surname: Tang fullname: Tang, Eugene organization: Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125 |
| BackLink | https://www.osti.gov/biblio/1860281$$D View this record in Osti.gov |
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| CitedBy_id | crossref_primary_10_1007_s10586_024_04686_y crossref_primary_10_1016_j_engappai_2023_106668 crossref_primary_10_1103_PhysRevA_110_052435 crossref_primary_10_1088_1367_2630_acb5bc crossref_primary_10_1109_TQE_2022_3151137 crossref_primary_10_1007_s10766_024_00781_0 crossref_primary_10_1007_s11433_022_2057_y crossref_primary_10_3390_math11153423 crossref_primary_10_1007_s10288_023_00549_1 crossref_primary_10_2166_ws_2025_075 crossref_primary_10_1063_5_0237599 crossref_primary_10_1038_s41534_025_00982_6 crossref_primary_10_1109_TQE_2024_3386753 crossref_primary_10_1103_PhysRevResearch_7_023141 crossref_primary_10_1007_s11128_024_04438_2 crossref_primary_10_1103_PRXQuantum_5_020327 crossref_primary_10_1038_s41534_023_00733_5 crossref_primary_10_1038_s41598_022_20853_w crossref_primary_10_1140_epjqt_s40507_023_00214_w crossref_primary_10_1134_S0021364023604256 crossref_primary_10_1016_j_future_2024_107632 crossref_primary_10_1007_s42484_023_00125_0 crossref_primary_10_1007_s11128_024_04286_0 crossref_primary_10_1103_PhysRevResearch_5_033193 crossref_primary_10_1016_j_physrep_2024_03_002 crossref_primary_10_1103_3l96_41xf crossref_primary_10_1109_TQE_2024_3481280 crossref_primary_10_1145_3711935 crossref_primary_10_1088_2058_9565_ad200a crossref_primary_10_1109_TQE_2024_3443660 crossref_primary_10_1007_s00723_023_01544_9 crossref_primary_10_1088_2058_9565_ac7f4f crossref_primary_10_1007_s10479_024_06253_5 crossref_primary_10_1109_TWC_2024_3523135 |
| Cites_doi | 10.1109/12.736440 10.48550/arXiv.1812.04170 10.1145/380752.380838 10.1103/PhysRevLett.125.260505 10.48550/arXiv.1411.4028 10.48550/arXiv.2010.14021 10.1103/PRXQuantum.2.030312 10.1007/BF02523688 10.48550/arXiv.2002.01068 10.1137/S0097539794270248 10.1002/rsa.20096 10.48550/arXiv.1712.05771 10.48550/arXiv.2005.08747 10.4230/OASIcs.SOSA.2018.13 10.1016/S0095-8956(81)80022-6 10.1137/S0097539705447372 10.1016/0012-365X(81)90023-6 10.1103/PhysRevA.97.022304 10.1145/227683.227684 10.4086/cjtcs.1997.002 10.1038/s41467-018-07090-4 10.1023/B:JOCO.0000038911.67280.3f 10.22331/q-2021-06-17-479 10.1109/HPEC.2019.8916288 |
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| Snippet | We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-
k
-CUT, the problem of finding an approximate
k
-vertex coloring... We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring... |
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| Title | Hybrid quantum-classical algorithms for approximate graph coloring |
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