Quaternionic Computing

isgivenby,

Tr1(ρ)=[In|0]ρ[In|0] +[0|In]ρ[0|In]

=A+D

Inparticular,wehavethat

|Φ0 Φ0|=(T0 |Φ )(T0 |Φ )t

whichbyapplyingtranspositionrulesforblockmatricesandEquation12gives

=

= Re(|Φ )Re(|Φ )Re( Φ|)

Im(|Φ )Re( Φ|)Im( Φ|) (30)

Im(|Φ )Re( Φ|)

Re(|Φ )Re( Φ|) (32)

Bysymmetry,wethushavethesameexpressionforbothpartialtraces

ρ0=ρ1=Re(|Φ )Re( Φ|) Im(|Φ )Im( Φ|)

=Re(|Φ Φ|)(33)

Since|Φ Φ|ishermitian,itsdiagonalentriesareallreal,andthereforeithasthesamediagonalentriesasρ0andρ1.

Inotherwords,combiningthiswithLemma4,wearrivetotheconclusionthatitdoesnotmatterwhatwesetastheinitialvalueofthetopwire,|0 or|1 .Furthermore,itiseasytoverifythatany1-rebitstatewilldo,whetherpureoreventotallymixed,aslongasitisunentangledanduncorrelatedwiththebottomwires.

3.4

3.4.1FurtherConsiderationsandConsequencesComplexityofsimulation

Ingeneral,ifweinitiallyhavead-qubitgate,thenewgatewillbea(d+1)-rebitgate.However,ifUgcontainsonlyrealentries,thenOg=I Ug,whichmeansthatinthisparticularcasethetoprebitneednotbeinvolved,andthereforethenewgateisthesameastheoriginal.Ifthewholequantumcircuitwearegivenisconstructedwithsuchrealgates,thenweareinluckandwedonotrequiretheextrarebitatall.Inthegeneralcomplexcase,however,thecircuitwidthisatmostonemorethanthatoftheoriginalcircuit.