A Short Path Quantum Algorithm for Exact Optimization

We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D ; this problem is called weighted MAX-E D -LIN2....

Full description

Saved in:
Bibliographic Details
Published inQuantum (Vienna, Austria) Vol. 2; p. 78
Main Author Hastings, M. B.
Format Journal Article
LanguageEnglish
Published Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften 26.07.2018
Online AccessGet full text
ISSN2521-327X
2521-327X
DOI10.22331/q-2018-07-26-78

Cover

Abstract We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D ; this problem is called weighted MAX-E D -LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D ; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.
AbstractList We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D ; this problem is called weighted MAX-E D -LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D ; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.
We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree $D$; this problem is called weighted MAX-E$D$-LIN2. We require that the optimal solution be unique for odd $D$ and doubly degenerate for even $D$; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.
ArticleNumber 78
Author Hastings, M. B.
Author_xml – sequence: 1
  givenname: M. B.
  surname: Hastings
  fullname: Hastings, M. B.
  organization: Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA, Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA
BookMark eNqFkEtLAzEUhYMo-Ore5fyB0eTO5DHLUnwUBBUV3IU7maRNmZnUTIqPX29tRcSFrs7hwvddOIdktw-9JeSE0VOAomBnzzlQpnIqcxC5VDvkADiwvAD5tPuj75PRMCwopaCkEKo8IHyc3c9DTNktpnl2t8I-rbps3M5C9GneZS7E7PwVTcpulsl3_h2TD_0x2XPYDnb0lUfk8eL8YXKVX99cTifj69yUTKbcIha8cLZ2dSkZFxUF6ixgU6sKaG0555JZAxVzBoWtmXHKOVkDa9bhquKITLfeJuBCL6PvML7pgF5vDiHONMbkTWu1q6RpGgFNo2xZcESDzqlSUAvOAZNrF9u6Vv0S316wbb-FjOrNjPpZf86oqdQgtFRrRmwZE8MwROu08WmzQIro279A-gv899cHGwOJIQ
CitedBy_id crossref_primary_10_1103_PhysRevA_110_032434
crossref_primary_10_1038_s42254_023_00603_1
crossref_primary_10_2151_sola_2025_006
crossref_primary_10_1126_sciadv_adm6761
crossref_primary_10_1016_j_biosystems_2023_105037
crossref_primary_10_1103_PRXQuantum_1_020312
crossref_primary_10_22331_q_2021_12_06_597
crossref_primary_10_22331_q_2019_11_11_201
crossref_primary_10_22331_q_2019_05_20_145
crossref_primary_10_22331_q_2023_09_14_1111
Cites_doi 10.1103/physrevlett.35.1792
10.1103/physrevlett.43.1754
10.1103/physrevlett.101.130504
10.4086/cjtcs.2018.002
10.1103/PhysRevLett.114.090502
10.1137/s0097539796300933
10.1016/j.disopt.2007.08.001
10.1126/science.1057726
10.1090/conm/305
10.1090/gsm/047
10.1103/physrevlett.50.1946
10.2307/2373688
10.1103/PhysRevLett.118.010501
10.1016/j.tcs.2014.01.010
10.1103/physreva.94.022309
10.1103/physrevlett.103.150502
10.1145/2591796.2591854
10.1103/physrevlett.96.197204
10.1073/pnas.1002116107
10.1016/j.tcs.2005.09.023
10.1137/15m1050902
10.1145/1007352.1007428
10.1109/FOCS.2015.54
10.1007/978-3-540-69507-3_22
10.1017/cbo9781139814782.002
10.1145/237814.237866
10.1103/physreve.65.046137
10.1016/0003-4916(78)90222-1
ContentType Journal Article
DBID AAYXX
CITATION
ADTOC
UNPAY
DOA
DOI 10.22331/q-2018-07-26-78
DatabaseName CrossRef
Unpaywall for CDI: Periodical Content
Unpaywall
DOAJ Directory of Open Access Journals
DatabaseTitle CrossRef
DatabaseTitleList CrossRef
Database_xml – sequence: 1
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 2
  dbid: UNPAY
  name: Unpaywall
  url: https://proxy.k.utb.cz/login?url=https://unpaywall.org/
  sourceTypes: Open Access Repository
DeliveryMethod fulltext_linktorsrc
EISSN 2521-327X
ExternalDocumentID oai_doaj_org_article_f97cdd62dd8e435aacaff8460e2ff217
10.22331/q-2018-07-26-78
10_22331_q_2018_07_26_78
GroupedDBID AAFWJ
AAYXX
AFPKN
ALMA_UNASSIGNED_HOLDINGS
CITATION
GROUPED_DOAJ
M~E
ADTOC
UNPAY
ID FETCH-LOGICAL-c417t-eaa353febfb471569020fe2adb8920be55571ec291fca6eb1cf8ff7b21dff7f93
IEDL.DBID UNPAY
ISSN 2521-327X
IngestDate Fri Oct 03 12:37:06 EDT 2025
Tue Aug 19 21:12:31 EDT 2025
Tue Jul 01 01:59:48 EDT 2025
Thu Apr 24 23:10:23 EDT 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Language English
License https://creativecommons.org/licenses/by/4.0
cc-by
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c417t-eaa353febfb471569020fe2adb8920be55571ec291fca6eb1cf8ff7b21dff7f93
OpenAccessLink https://proxy.k.utb.cz/login?url=https://quantum-journal.org/papers/q-2018-07-26-78/pdf/
ParticipantIDs doaj_primary_oai_doaj_org_article_f97cdd62dd8e435aacaff8460e2ff217
unpaywall_primary_10_22331_q_2018_07_26_78
crossref_citationtrail_10_22331_q_2018_07_26_78
crossref_primary_10_22331_q_2018_07_26_78
ProviderPackageCode CITATION
AAYXX
PublicationCentury 2000
PublicationDate 2018-07-26
PublicationDateYYYYMMDD 2018-07-26
PublicationDate_xml – month: 07
  year: 2018
  text: 2018-07-26
  day: 26
PublicationDecade 2010
PublicationTitle Quantum (Vienna, Austria)
PublicationYear 2018
Publisher Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
Publisher_xml – name: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
References 22
23
24
25
26
27
28
29
30
31
10
32
11
33
12
34
13
14
15
16
17
18
19
1
2
3
4
5
6
7
8
9
20
21
References_xml – ident: 18
– ident: 17
  doi: 10.1103/physrevlett.35.1792
– ident: 26
  doi: 10.1103/physrevlett.43.1754
– ident: 32
  doi: 10.1103/physrevlett.101.130504
– ident: 16
  doi: 10.4086/cjtcs.2018.002
– ident: 7
  doi: 10.1103/PhysRevLett.114.090502
– ident: 4
  doi: 10.1137/s0097539796300933
– ident: 31
  doi: 10.1016/j.disopt.2007.08.001
– ident: 28
– ident: 10
  doi: 10.1126/science.1057726
– ident: 8
  doi: 10.1090/conm/305
– ident: 19
  doi: 10.1090/gsm/047
– ident: 27
  doi: 10.1103/physrevlett.50.1946
– ident: 13
  doi: 10.2307/2373688
– ident: 23
  doi: 10.1103/PhysRevLett.118.010501
– ident: 12
  doi: 10.1016/j.tcs.2014.01.010
– ident: 20
– ident: 22
– ident: 33
  doi: 10.1103/physreva.94.022309
– ident: 3
– ident: 15
  doi: 10.1103/physrevlett.103.150502
– ident: 6
  doi: 10.1145/2591796.2591854
– ident: 25
  doi: 10.1103/physrevlett.96.197204
– ident: 2
  doi: 10.1073/pnas.1002116107
– ident: 34
  doi: 10.1016/j.tcs.2005.09.023
– ident: 30
– ident: 1
  doi: 10.1137/15m1050902
– ident: 29
  doi: 10.1145/1007352.1007428
– ident: 5
  doi: 10.1109/FOCS.2015.54
– ident: 11
  doi: 10.1007/978-3-540-69507-3_22
– ident: 24
  doi: 10.1017/cbo9781139814782.002
– ident: 14
  doi: 10.1145/237814.237866
– ident: 9
  doi: 10.1103/physreve.65.046137
– ident: 21
  doi: 10.1016/0003-4916(78)90222-1
SSID ssj0002876684
Score 2.219522
Snippet We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the...
SourceID doaj
unpaywall
crossref
SourceType Open Website
Open Access Repository
Enrichment Source
Index Database
StartPage 78
SummonAdditionalLinks – databaseName: DOAJ Directory of Open Access Journals
  dbid: DOA
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1NS8MwGA6yi3oQRcX5RQ5eFMqapPnoccrGEPxCB7uFNE0csk_pUP-9b9o6d3IXT4WSkvZ90zzPkzbPi9BFnlPGEpdEACUqSohnkcoEj1JHjPMiAd0cdiPf3YteP7kd8MFKqa_wT1hlD1wFruVTafNc0DxXDqDdGGu8B9CMHfUe-HSYfWOVroipt3LJSAqhkuq7JCAgI605DAiiwrIcFVGoqraCQ6Vd_zbaXExm5uvDjEYrGNPdRTs1OcTt6qb20Iab7CPexs9DIMn4EcgaflpAKBZj3B69TkHXD8cYWCfufBpb4Ad4_cf1vsoD1O92Xm56UV3sILIJkUXkjGGceZf5DPCCg2ilsXfU5JlKaZw5zrkkztKUeGsEzLDWK-9lRkkOB5-yQ9SYTCfuCGELEsrzjMngbW5sbBjhkipDmaSJIqKJWj-Prm3tBB4KUow0KIIyWHquQ7B0LDUVWqomulxeMatcMP5oex2iuWwX_KvLE5BVXWdVr8tqE10tc7G2x-P_6PEEbf2Oj1PUKN4X7gz4RpGdl0PrG7CD0aY
  priority: 102
  providerName: Directory of Open Access Journals
Title A Short Path Quantum Algorithm for Exact Optimization
URI https://quantum-journal.org/papers/q-2018-07-26-78/pdf/
https://doaj.org/article/f97cdd62dd8e435aacaff8460e2ff217
UnpaywallVersion publishedVersion
Volume 2
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAON
  databaseName: DOAJ Directory of Open Access Journals
  customDbUrl:
  eissn: 2521-327X
  dateEnd: 99991231
  omitProxy: true
  ssIdentifier: ssj0002876684
  issn: 2521-327X
  databaseCode: DOA
  dateStart: 20170101
  isFulltext: true
  titleUrlDefault: https://www.doaj.org/
  providerName: Directory of Open Access Journals
– providerCode: PRVHPJ
  databaseName: ROAD: Directory of Open Access Scholarly Resources
  customDbUrl:
  eissn: 2521-327X
  dateEnd: 99991231
  omitProxy: true
  ssIdentifier: ssj0002876684
  issn: 2521-327X
  databaseCode: M~E
  dateStart: 20170101
  isFulltext: true
  titleUrlDefault: https://road.issn.org
  providerName: ISSN International Centre
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwELZgewAOPASI5VH5wAUkd2M7fuS4oFYVEqUIVlpOll9DEbvbtErE49czeXSBHkBwihSNHWvG8Xyf7Zkh5GlKQsoylwxdiWUlB8ls0IpVmfsMukTe3EUjvz7Sh4vy1VItx62LPoq_xQ7bNRs12Z_l175GKDQ7Q4ty2-2rCc2MndUJZlfJjlYIwidkZ3F0PP_QlZJDh8SkMMvhUBLdn-SX2_7mhPpc_TfItXZT-29f_Gr1i4M5uEWWF0Mb7pV83mubsBe_X8ra-B9jv01ujqCTzodZcodcyZu7RM3puxME3_QYQSB9O_RK56uPp-efmpM1RTRL97_62NA3uKysx3jNe2RxsP_-5SEbiyiwWHLTsOy9VBJygIB-SCEZFgVk4VOwlShCVkoZnqOoOESvceWOYAFMEDzhAyp5n0w2p5v8gNCI1AxUkKbLme5j4SVXRlgvpBGl5XpKZhdadXHMMN4Vulg5ZBq9HdyZ6_TgCuOEdsZOybNti3rIrvEH2RedobZyXV7s_gWq2Y2_mYPKxJS0SMlmBILeRw-AEKvIAgDZ15Q835r5r198-C_Cj8j1nyZ-TCbNeZufIF5pwm7P83fHGfoDHFbnRg
linkProvider Unpaywall
linkToUnpaywall http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwEB7B9gAceAgQy0s-cAHJ3diOHzkuqFWFRCmClZaT5WeL2N2mVSIev57Jowv0AIJTpGjsWDOO5xvb8w3Asxi5EGUqKboSQ0uWBTVeSVol5lJWJcbNXTbym0N1sChfL-Vy3Lros_hb7LBd01GT_Vl-7WqEQrMztCgz3b4aV1SbWR3z7CrsKIkgfAI7i8Oj-ceulBw6JCq4Xg6Hkuj-BLvc9jcn1HP134Br7aZ237641eoXB7N_C5YXQxvulXzebRu_G75fYm38j7Hfhpsj6CTzYZbcgStpcxfknLw_QfBNjhAEkndDr2S-Oj49_9ScrAmiWbL31YWGvMVlZT3ma96Dxf7eh1cHdCyiQEPJdEOTc0KKnHz26IckBsO8yIm76E3FC5-klJqlwCuWg1O4codsctaes4iPXIn7MNmcbtIDIAFDsyy90B1nuguFE0xqbhwXmpeGqSnMLrRqw8gw3hW6WFmMNHo72DPb6cEW2nJltZnC822LemDX-IPsy85QW7mOF7t_gWq2429mc6VDjIrHaBICQeeCyxkhVpF4zhh9TeHF1sx__eLDfxF-BNd_mvgxTJrzNj1BvNL4p-Pc_AG2I-ZR
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Short+Path+Quantum+Algorithm+for+Exact+Optimization&rft.jtitle=Quantum+%28Vienna%2C+Austria%29&rft.au=Hastings%2C+M.+B.&rft.date=2018-07-26&rft.issn=2521-327X&rft.eissn=2521-327X&rft.volume=2&rft.spage=78&rft_id=info:doi/10.22331%2Fq-2018-07-26-78&rft.externalDBID=n%2Fa&rft.externalDocID=10_22331_q_2018_07_26_78
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2521-327X&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2521-327X&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2521-327X&client=summon