Quaternionic Computing

Proof.AsintheproofofLemma1,allwerequirearethematrixmultiplicationrulesofEqua-tion12

OC|Ψ0 =(T UC)(T0 |Ψ0 ) Re(UC)= Im(UC)Im(|Ψ0 ) Re(UC)Re(|Ψ0 )+Im(UC)Im(|Ψ0 )=

=T0 (UC|Ψ0 )=T0 |Φ =|Φ0 Im(UC|Ψ0 )(26)

Withthesamemethod,wecanobtainasimilarexpressionforΦ1,i.e.

OC|Ψ1 =(T UC)(T1 |Ψ1 ) Re(UC)= Im(UC)

=T1 |Φ =|Φ1 Re(|Ψ1 )=...

(27)

Letusassumeforamoment—andinfact,thisiswithoutlossofgenerality—thattheoriginalcircuitwastobeinitialisedwithsomebasevector|x ,witha nalstate|Φ =U|x .Again,therearetwopossiblechoicesforinitialisingthecorrespondingrealcircuit,namely|x0 =|0 |x and|x1 =|1 |x .Whatwouldthenbetheoutputofthesimulatedcircuitineithercase?Intheveryspecialcasethat|Φ isalsoabasevector,thenwewouldhave|Φ0 =|0 |Φ and|Φ1 =|1 |Φ ,andthus,ineithercase,thebottomn-wireswouldcontaintherightanswerandwecanignorethetopwire.Butwhen|Φ issomearbitrarypurestate,neitherpurelyrealnorpurelyimaginary,wecannotgivesuchanicesemantictothetopwire.Inparticular,itmightbeentangledwiththerestofthewires,andhencewecannotfactorthe nalstate.

Nonetheless,whatissurprisingisthatifwetraceoutthetopwire,inallcaseswewillgetthesamestatisticsandfurthermorethatwewillobtaintherightstatistics,i.e.thesameasifwehadusedtheoriginalquantumcircuitC.Moreformally,wehave

Lemma5.Let|Φ beanarbitraryn-qubitpurestate,andletρ0=Tr1|Φ0 Φ0|andρ1=Tr1|Φ1 Φ1|representthepartialtracesobtainedbytracingout(i.e.forgettingabout)thetopwire.Thenwehavethat

ρ0=ρ1,

Diag(ρ0)=Diag(ρ1)=Diag(|Φ Φ|).(28)(29)

Proof.Thepartialtraceofthe rstwireofanarbitrarydensityoperatorgiveninblockmatrixform Aρ=C