Quaternionic Computing

Inotherwords,itsomehowkeepstrackofthephase(angle)oftherepresentationof|Φ inrebitspacewithrespecttothesesubspaces.TheCNOTgate(oranyotherrealgate)doesnotchangethisphasefactor.However,asarbitrarygateswithcomplextransitionamplitudesa ectthisphasefactor,theire ectissimulatedby“recording”thischangeinthetoprebit.Howweinitialisethetoprebitgivesanarbitraryinitialphasetotherepresentationof|Φ ,butaswesaw,thisinitialphasedoesnota ectstatisticsofthebottomwires,andthuscanbesettoanyvalue.However,howthisphasehasbeenchangedbypreviouscomplexgateswilla ectthebottomrebitsinsubsequentcomplexgates,inasimilarfashionasthephasekickbackphenomenoninmanyquantumalgorithms7.Thatiswhythattoprebitisneeded.

4QuaternionicComputing

ThissectioncloselymimicsSection3.Firstwede newhatwemeanbyquaternioniccomputing,makingsurethatitisasensiblemodel.Wethenproveanequivalencetheoremwithquantumcomputing,byusingthesametechniquesasthoseofTheorem2.

4.1

4.1.1De nitionsQuaternions

QuaternionswereinventedbytheIrishmathematicianWilliamRowanHamiltonin1843,asageneralisationofcomplexnumbers.Theyformanon-commutative,associativedivisionalgebra.Aquaternionisde nedas

α =a0+a1i+a2j+a3k(34)

wherethecoe cientsaarerealnumbersandi,j,andkobeytheequations

ii=jj=kk=ijk= 1(35)

Multiplicationofquaternionsisde nedbyformallymultiplyingtwoexpressionsfromEqua-tion34,andrecombiningthecrosstermsbyusingEquation35.Itisveryimportanttonotethatwhileallnon-zeroquaternionshavemultiplicativeinversestheyarenotcommutative8.Thus,theyformwhatiscalledadivisionalgebra,sometimesalsocalledaskew eld.Thequaternionconjugationoperationisde nedasfollows:

α =a0 a1i a2j a3k(36)

whereforclarity,werepresentwiththe(non-standard)symbol( )inordertodistinguishitfromcomplexconjugationrepresentedwith( ).Withthisconjugationrule,wecande nethemodulusofaquaternionas

|α |=√222a20+a1+a2+a3(37)