A Lecture on Quantum Logic Gates

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1、ALectureonQuantumLogicGatesKazuyukiFUJII∗DepartmentofMathematicalSciencesYokohamaCityUniversityYokohama,236-0027JapanAbstractInthisnotewemakeashortreviewofconstructionsofn-repeatedcontrolledunitarygatesinquantumlogicgates.arXiv:quant-ph/0101054v113Jan2001∗E-mailaddress:fujii@math.yokoh

2、ama-cu.ac.jp01IntroductionThisisoneofmylectuersentitled“IntroductiontoQuantumComputation”givenatGraduateSchoolofYokohamaCityUniversity.Thecontentsoflecturearebasedonthebook[1]andreviewpapers[2],[3].ThecontrolledNOTgate(moregenerally,controlledunitarygates)playsveryimportantroleinquantu

3、mlogicgatestoproveauniversality.Theconstructionsofcontrolledunitarygatesorcontrolled-controlledunitarygatesareclearandeasytounderstand.Buttheconstructionofgeneralcontrolledunitarygates(n-repeatedcontrolledunitarygates)seem,inmyteachingexperience,noteasytounderstandforyounggraduatestude

4、nts.Ithoughtoutsomemethodtomaketheproofmoreaccessibletothem.Iwillintroduceitinthisnote.Maybeitis,moreorless,well-knowninsomefieldinPureMathematics,butwearetoobusytostudysuchafieldleisurely.Ibelievethatthisnotewillmakenon-expertsmoreaccessibletoquantumlogicgates.2SomeIdentityonZ2Letusstar

5、twiththemod2operationinZ2:forx,y∈Z2x⊕y=x+y(mod2).(1)Fromtherelations0⊕0=0,0⊕1=1,1⊕0=1,1⊕1=0,itiseasytoseex⊕y=x+y−2xy,orx+y−x⊕y=2xy.(2)Wenotethatx⊕0=x,x⊕1=1−x,x⊕x=2x−2x2=2x(1−x)=0.Fromx+y−x⊕y=2xywehavex+y+z−(x⊕y+x⊕z+y⊕z)+x⊕y⊕z=4xyz(3)1forx,y,z∈Z2.Theproofiseasy,soweleaveittothereaders.M

6、oreoverwehavex+y+z+w−(x⊕y+x⊕z+x⊕w+y⊕z+y⊕w+z⊕w)+(x⊕y⊕z+x⊕y⊕w+x⊕z⊕w+y⊕z⊕w)−x⊕y⊕z⊕w=8xyzw(4)forx,y,z,w∈Z2.Butisthisproofsoeasy?NowwedefineafunctionXnXnXnFn(x1,···,xn)=xi−xi⊕xj+xi⊕xj⊕xk−···i=1i

7、(x1,x2)=2x1x2andF3(x1,x2,x3)=4x1x2x3andF4(x1,x2,x3,x4)=8x1x2x3x4.FromtheserelationsitiseasytoconjecturePropositionAn−1Fn(x1,x2,···,xn)=2x1x2···xn.(6)Thisiswell-known[4],butIdon’tknowtheusualproof(in[4]thereisnoproof).Thisproofmaybenoteasyfornon-expertsagainsttheclaiminthebook[1],seep