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Computing the Number of the Equivalence Classes for Reversible Logic Functions

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Abstract

Reversible logic function classification plays an important role in reversible logic synthesis. This paper studies the calculation of the number of the equivalence classes for reversible logic functions. In order to do this work, an n-qubit reversible function is expressed as a permutation in the symmetric group \(S_{2^{n}}\), so that a universal formula for calculating the number of equivalence classes of reversible logic functions is derived based on group theory. Based on the calculation method of the number of conjugacy classes of permutation groups, an improved method for calculating the number of equivalence classes of reversible logic functions is obtained. In the experiments with GAP software on a laptop, we can compute the NPNP-equivalence classes for up to 13 qubits, LL-equivalence classes for up to 10 qubits and AA-equivalence classes for up to 10 qubits. Experiment results indicate that our scheme for calculating these equivalence classes of more than 6 qubits over previous published methods is a significant advancement.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 61772006, the Natural Science Foundation of Guangxi under Grant No. 2019GXNSFAA185033, and the Special Fund for Bagui Scholars of Guangxi.

Author information

Authors and Affiliations

  1. Department of Computer Science and Technology, School of Information Engineering, Hubei Minzu University, Enshi, 44500, China

    Qing-bin Luo

  2. College of Mathematics and Physics, Guangxi University for Nationalities, Nanning, 530006, China

    Qing-bin Luo & Jin-zhao Wu

  3. Big Data Research Center & School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Guo-wu Yang & Chen Lin

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  1. Qing-bin Luo
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  2. Guo-wu Yang
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  3. Jin-zhao Wu
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  4. Chen Lin
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Correspondence to Guo-wu Yang.

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Appendices

Appendix A: The Number of NPNP-Equivalence Classes Nn for \(7\leqslant n\leqslant 9\)

n

N n

7

92657180627206416894321404628764148109892835203836255610293338454709

 

39846119691972208974475504942689614094494621525282857760050715322261

 

59798384649365038150756391193832099360403268370959565678719148097536

8

80514493230861009237180944923666507887764199637062952723945947443966

 

39336469781392178991125348445814144898619375477049676743164349688969

 

09750674475093684508311194857009782770003011248081667377303152594375

 

96361339675344041376451086904224356069541201097649777439162059207315

 

03998560158517859605432623177410740126091376088030486252408645889861

 

89385348144718856291494232389720029875144657172444459702631192412195

 

70813826590147122163621259142438166886775118356122550588441967965284

 

79430356919910400

9

10073372596960961994140361512434725047571881090945885196993555296023

 

52889219582381224803943626430295712735582948249269869109001215786285

 

92926272993911897343726541554451059415404946457953089369976892842084

 

04403545444557500430238326130667257762646197664312166772708711358804

 

65434907773460154333707020918035044837887096225703088231170444065562

 

41366786979256253054579797566664614119973340822314369481739546745500

 

60887290560201984944743874722637595269048887669378353935770175948064

 

77822910399643957447086698934758004324009902162397453962618800879121

 

19195882938262316617897911837367132594223268381533261594772631947239

 

12614700847485835492603852026371683137416860402778790713390798589961

 

50372239057214403117025249080243456225342345053302866865940150709724

 

59365271374860259670769030309253827407439604472633293928914882162418

 

44768862636916583076259639814801951471948564537230020808534302441648

 

05314969537956278605488965240755299099272178924087823257363679992945

 

82200080237285273587999726879369422309027990373228308928653659072197

 

16211059446534264873133381637269635978705904929294140287485550498365

 

602477789344570879452123046246915561048364618954057990537216000

Appendix B: The Number of LL-Equivalence Classes Nn for \(7\leqslant n\leqslant 9\)

n

N n

7

14363730219634710882862287300142339538811536049418741112973109479380

 

50124843861485805578077759084336972978578341447325405221566011476826

 

6334924465455884656892188635369821546456709033078816

8

29991755513339381424498183455040895913390943610417801351090424352005

 

93540857067728194477910125280244575393676453366763352401056535623227

 

84650797285198597633256714777503750173888398351210460811422903925993

 

06446814541510477713192909336671052049969769453791200180218494467009

 

56811663239032601387629688168390156041532935732958508026886772801748

 

58834741094771360186408640391531263322815507449625490081381371566791

 

67946732141320623894379740121606057347547038059792534008719856

9

71043770189524106678142056211356566119538077124753178707024721735463

 

31377908223853793028242379142209886530531242560761421483691625143568

 

96535038902120049684933593659424958357822840996725381419210860081968

 

31347179995076452944017252000984232759768492631959787053935391456111

 

21958385095185309562363842460153702425434384090185203538792102388085

 

43854687323050504170144249214715551724721421249002954108331534092210

 

52751275286003155690753120381732101816412165165809512047781462891221

 

26726055113140350325054874129917617958079772309748975940870751691722

 

37888977847849897397215039948188111845263558503160533650611887013993

 

18277321569373700946174535178947738815342755828523187256336357471379

 

17583020320075358897550346647994213048982475054068499564464452233670

 

44013221381845864302947174595154584725093616964234386972540100955943

 

50364405016937581286301529814895977999690442534960580680114133314009

 

35114424614843650251489259996321121503529117546355158794037340764383

 

12413500636799575781215617169240191143987393771377027233743115636273

 

19983721628365057793392794158040557423542081945789567717151115232620

 

9318959378024626476590092648244

Appendix C: The Number of AA-Equivalence Classes Nn for \(7\leqslant n\leqslant 9\)

n

N n

7

87669251828825139665907515259657834099191504207878058550861263912234

 

50469643811864931355527712138322589212547692969638955078555531883869

 

08204901473878725895316404384316228470586133104

8

45763787099211702613064855125489648305345067764919740831131628955087

 

18171473797192679562240791748420067434198689829655994264307457921185

 

06852412849729305470667594570165634422121146541561779731391648457099

 

47260103898192877520201619190942444753324369227601873447213962999125

 

31217827335549231728564445454029537174639987903380054136159817385092

 

16600422072967645280863434558442606389215899998769021202491246305077

 

896267823028683004485116177896801466590934522276904039712

9

27101047588166849776512930378477693984809141969586631281671417898354

 

84076655664006726466462089211353258716785904907517021745182657296588

 

50301757393692035554860532249231322615746628187837746207889884979998

 

89887687681227284600836659241098111251742741635116495915960461218304

 

14565424001764415574021851524411660165952447544168549933926430659517

 

45550036362858010929162692724119396867645805835343534133928686777311

 

86780625443114812405746448146864079024084270156142338423438262793302

 

28867945064536450190575132367620691716105889935793500201182980894600

 

45569033453841242296619948614221729468917554098907293148197436703072

 

66070103122745161251620505107771249505860349434668960370694169809491

 

96358086495902313551009378970776884790913450568855613683375749227737

 

46310253321096589609465356263003874163734214551176299006906872012827

 

70403179181178270297392708082184542696924136177354499912062787082402

 

41695023353060045262989228660539935994722587058100641354611783036187

 

84837419680776436514681082930197247591658063299513081393718362657611

 

97804168381173843020485581429782258064328979166734993368978867958637

 

90079697382469010782524820

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Luo, Qb., Yang, Gw., Wu, Jz. et al. Computing the Number of the Equivalence Classes for Reversible Logic Functions. Int J Theor Phys 59, 2384–2396 (2020). https://doi.org/10.1007/s10773-020-04508-y

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