Teleportation-Based Quantum Computation, Extended Temperley-Lieb Diagrammatical Approach and Yang--Baxter Equation

0.56 MB·English

by  Yong Zhang

Teleportation-Based Quantum Computation, Extended Temperley–Lieb
DiagrammaticalApproach and Yang–BaxterEquation
YongZhanga,1,KunZhanga,2,andJinglongPangb,3
a SchoolofPhysicsandTechnology,WuhanUniversity,P.R.China430072
bDepartmentofPhysics,PekingUniversity,P.R.China100871
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Abstract
0
Thispaperfocusesonthestudyoftopologicalfeaturesinteleportation-basedquantumcom-
2
putationaswellasaimsatpresentingadetailedreviewonteleportaiton-basedquantumcom-
n
putation(GottesmanandChuang,Nature402,390,1999). IntheextendedTemperley–Lieb
a
J diagrammatical approach, we clearly show that such topological features bring about the
fault-tolerantconstructionofbothuniversalquantumgatesandfour-partiteentangledstates
8
2 moreintuitiveandsimpler. Furthermore,wedescribetheYang–Baxtergatebyitsextended
Temperley–Lieb configuration, and then study teleportation-based quantum circuit models

] using the Yang–Baxter gate. Moreover, we discuss the relationship between the extended
h
p Temperley–Liebdiagrammaticalapproachand the Yang–Baxtergate approach. With these
- research results, we propose a worthwhile subject, the extended Temperley–Liebdiagram-
t
n maticalapproach,forphysicistsinquantuminformationandquantumcomputation.
a
u
q
[ KeyWords: Teleportation,QuantumComputation,Temperley–Liebalgebra,Yang–BaxterEquation

1 PACSnumbers: 03.67.Lx,03.65.Ud,02.10.Kn
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[email protected]
[email protected]
[email protected]
1 Introduction
Quantum information and computation [1, 2] is an interdisciplinary research field of applying
fundamentalprinciplesof quantummechanicsto informationscience and computerscience. It
representsa furtherdevelopmentof quantummechanics,andindeedhelpsusto achievedeeper
understandingsonquantumphysics. Quantumentanglement[3,4]asafundamentalconceptof
distinguishingclassicalphysicsfromquantumphysicshasbecomeawidelyexploitedresourcein
quantuminformationandcomputation. Ithasbeenexperimentallyverifiedthat,undertheassis-
tanceofquantumentanglement,anunknownqubitcanbetransmittedfromAlicetoBobwithout
anynon-localphysicalinteractionbetweenthem,usingthequantuminformationprotocolcalled
quantumteleportation[5,6,7,8]. Hencebothquantumentanglementandquantumteleportation
obligequantumphysiciststothinkabouttherelationshipbetweenEinstein’slocalityandquantum
non-locality[3,4].
Fault-tolerantquantumcomputation[1, 2] is requiredin practice to overcomedecoherence,
andteleportation-basedquantumcomputation[9,10,11,12,13]isapowerfulapproachtofault-
tolerantquantumcomputationinwhichuniversalquantumgatesetisprotectedfromnoiseusing
the teleportation protocol. Besides this, teleportation-based quantum computation is an exam-
ple of measurement-based quantum computation which exploits quantum measurement as the
maincomputingresourcetodeterminewhichquantumgateistobeperformed. Inquantumme-
chanics, quantum measurementbreaks quantumcoherenceand is usually performedat the end
ofanexperiment,someasurement-basedquantumcomputationessentiallychangesourconven-
tionalviewpointonquantummeasurementandalsoonthestandardquantumcircuitmodel[10]
inwhichacoherentunitarydynamicsismainlyinvolved.
Asthetitleofthispaperclaims,weapplytheextendedTemperley–Liebdiagrammaticalap-
proachandtheYang–Baxtergateapproachtothereformulationofteleportation-basedquantum
computation. The Temperley–Liebalgebra[14] is a well knownconceptin bothstatistical me-
chanicsandlowdimensionaltopology[15],andtheYang–Baxterequation[16]arisesinexactly
solvingbothsome1+1-dimensionalquantummany-bodysystemsandvertexmodelsinstatistics
physics. The fact [15] that a type of solutionsof the Yang–Baxterequation can be constructed
usingtheTemperley–Liebalgebramotivatestheauthorstoexploreteleportation-basedquantum
computationusingthetworelatedapproaches.
TheextendedTemperley–Liebdiagrammaticalapproach[17,18]isanextensionofthestan-
darddiagrammaticalrepresentationoftheTemperley–Liebalgebra,andwithit,aquantuminfor-
mation protocol involving bipartite maximally entangled states, such as quantum teleportation,
hasaverynicetopologicaldiagrammaticalinterpretation. TheYang–Baxtergatesarenontrivial
unitarysolutionsofthe Yang–Baxterequation,andtheYang–Baxtergateapproach[19,20, 21]
toquantuminformationandcomputationisanalgebraicmethodoriginallymotivatedbytheob-
servationthattopologicalentanglements(likebraidingconfigurations[15])andquantumentan-
glements may have a kind of connection. Therefore, our research is expected to be interesting
for physicists in quantum information and computation, which shows that teleportation-based
quantumcomputationadmitsbothtopologicalandalgebraicdescriptionsbesidesitsstandardde-
scriptioninquantumcircuitmodel[9,10,11].
Aswehaveintroduced,wearegoingtogodirectlyintoatleastthreedifferentresearchsub-
jectsinthispaper,includingquantuminformationandcomputation,theTemperley–Liebalgebra
andtheYang–Baxterequation. Theguidingprincipleofthiskindofinterdisciplinaryresearchis
that we want to explorethe nature of quantumentanglement(quantumnon-locality[3, 4]). As
amatteroffact,nowadays,nobodyisabletostatethatthenatureofquantumentanglementhas
beenfullyunderstood[1,2]. Inaccordancewith[22],westudythenatureofquantumentangle-
mentbysettingupalinkbetweenquantumentanglement(quantumnon-locality)andtopological
entanglement(topologicalnon-locality[15]).
Topologicalentanglementin the paper is representedby the extended Temperley–Liebdia-
grammatical configuration or the Yang–Baxter gate configuration (the braiding configuration),
2
while quantum entanglement is represented by bipartite maximally entangled two-qubit pure
states, i.e., the Bell states [1, 2]. In the extended Temperley–Lieb diagrammatical approach
[17, 18], both the Bell states and Bell measurementshave the Temperley–Liebdiagrammatical
configurationsonwhichquantumgatesareallowedtomovefromthisqubittothatqubit. Inthe
Yang–Baxtergateapproach[19,20,21],thealgebraicformulationofteleportation-basedquantum
computationadmits the braiding configurationon which quantumgates are permitted to move.
Especially,therelationshipbetweensuchthetwoapproacheswillbeclarifiedinthispaper.
Thecompleteschemeofteleportation-basedquantumcomputationwasproposedbyGottes-
man and Chuang [9] in a very brief style, so it is necessary firstly to make a detailed review
on teleportation-basedquantumcomputation, topics including quantumteleportation, the fault-
tolerantconstructionofuniversalquantumgatesetandthefault-tolerantpreparationoffour-qubit
entangledstates. Thenwepresentatopologicaldiagrammaticaldescriptionofsuchthereviewed
topics in the extended Temperley–Lieb diagrammatical approach. Afterwards, we concentrate
ontheYang–BaxtergateswhicharetheBelltransform,aunitarybasistransformationfromthe
product basis to the Bell states, and with the help of our previous research on teleportation-
based quantumcomputationusing the Belltransform[13], we gointo the algebraicdescription
ofteleportation-basedquantumcomputationin termsofthe Yang–Baxtergate. Finally, inview
oftheextendedTemperley–LiebdiagrammaticalrepresentationoftheYang–Baxtergate,weset
up a transparent link between the extended Temperley–Lieb diagrammatical approach and the
Yang–Baxtergateapproach.
Our study in this paper is meaningful and useful in the following sense. First, as we have
emphasized,wetrytodigoutthenatureofquantumnon-localityfromtheviewpointoftopolog-
ical non-locality. Second, the topologicalfeatures in teleportation-basedquantum computation
makethefault-tolerantconstructionofbothuniversalquantumgatesetandfour-qubitentangled
states more intuitive and simpler. Third, we develop the conceptof teleportation operator [13]
in the Yang–Baxtergate approachto catch the intrinsic characteristic of quantumteleportation,
whichis capableof includingthe algebraicformulationsof allpossible teleportaitonprocesses.
Fourth,themethodologiesunderlyingourresearchareexpectedtobeappliedtoothersubjectsin
quantuminformationandcomputation,seeSection7forconcludingremarksonfurtherresearch.
Fifth, our research is expected to shed a light on further research in mathematical physics, see
AppendixA.
It is obvious that what we have done in this paper is an interdisciplinary research among
quantuminformationandcomputation,thelow-dimensionaltopologyandmathematicalphysics
suchastheYang–Baxterequation. ButweaimatintroducingboththeTemperley–Liebalgebra
and the Yang–Baxter equation to physicists in quantum information and computation, in other
words, readerscan go throughthe entire paper withoutthe preliminaryknowledgeon concepts
in mathematicalphysics. All the resultsaboutboththe Temperley–Liebalgebraandthe Yang–
BaxterequationarecollectedinAppendixA.Furthermore,forexpertsinmathematicalphysics,
AppendixA.6indeedpresentsalistofopenproblemsforfurtherresearchwhicharebasedonour
reformulationofteleportation-basedquantumcomputation.
Thispaperisorganizedasfollows.Section2isareviewonteleportation-basedquantumcom-
putation [9]. Section 3 focuses on the topological diagrammaticaldescription of teleportation-
basedquantumcomputationintheextendedTemperley–Liebdiagrammaticalapproach. Section
4 presents the extendedTemperley–Liebconfigurationsof two types of Yang–Baxter gates de-
rivedinAppendixA.Section5describestheYang–Baxtergateapproachtoteleportation-based
quantumcomputation.Section6clarifiestherelationshipbetweentheextendedTemperley–Lieb
diagrammaticalapproachandtheYang–Baxtergateapproach.Tomakethepaperself-consistent,
wepresentfiveappendicesforreaders’conveniences. AppendixAreviewsboththeTemperley–
Lieb algebra and the Yang–Baxter equation as well as presents a detailed construction on the
Yang–Baxter gate via the Temperley–Lieb algebra. Appendix B collects interesting properties
ofthepermutation-likeYang–Baxtergates. BothAppendixCandAppendixDpresentthetopo-
logical diagrammatical construction of four-qubit entangled states respectively associated with
3
the CNOT gate and CZ gate [1]. Appendix E is about a method of calculating the extended
Temperley–Liebdiagrammaticalrepresentationoftheteleportationoperator.
2 Review on teleportation-based quantum computation
In this section, we make a review on teleportation-based quantum computation [9]. The key
topics include the standard description of quantumteleportation, the fault-tolerantconstruction
ofuniversalquantumgateset, andthe fault-tolerantconstructionoffour-qubitentangledstates.
Meanwhile,wesetupournotationandconventionforthestudyinthepaper.
2.1 Notation
Asingle-qubitHilbertspace hasanorthonormalbasisdenotedby i ,i = 0,1,andasingle-
2
H | i
qubitstate α isgivenby α =a0 +b1 withcomplexnumbersaandb. Theunitmatrix11
2
| i | i | i | i
andthePauligatesX andZ taketheform
1 0 0 1 1 0
11 = , X = , Z = (1)
2 (cid:18) 0 1 (cid:19) (cid:18) 1 0 (cid:19) (cid:18) 0 1 (cid:19)
−
withZ 0 = 0 andZ 1 = 1 ,andthePauligateY isdefinedasY = ZX. Asingle-qubit
| i | i | i −| i
gateisan elementoftheunitarygroupU(2), andatypicalsingle-qubitgateW inthe present
ij
paperhastheform
W =XiZj, i,j =0,1, (2)
ij
satisfyingWiTj =Wi†j,inwhichtheupperindexT denotesthematrixtransposeconjugationand
theupperindex denotesthematrixHermitianconjugation.
†
Atwo-qubitHilbertspace hasanorthonormalproductbasisdenotedby ij ,i,j =
2 2
H ⊗H | i
0,1.TheEPRstate Ψ [3,4]takestheform
| i
1
Ψ = (00 + 11 ) (3)
| i √2 | i | i
andithasaverynicealgebraicproperty
(11 U)Ψ =(UT 11 )Ψ (4)
2 2
⊗ | i ⊗ | i
withU denotinganysingle-qubitgate.ThesetoftheorthonormalBellstates ψ(ij) givenby
| i
ψ(ij) =(11 W )Ψ , i,j =0,1 (5)
2 ij
| i ⊗ | i
iscalledtheBellbasis[1,2]ofthetwo-qubitHilbertspace . Obviously, Ψ = ψ(00) .
2 2
H ⊗H | i | i
2.2 Quantum teleportation
LetAliceandBobsharetheEPRstate Ψ ,andAlicewantstotransferanunknownquantumstate
| i
α toBob.AliceandBobpreparethequantumstate α Ψ whichcanbeformulatedas
| i | i⊗| i
1
1
α Ψ = ψ(ij) W α (6)
ij
| i⊗| i 2 | i⊗ | i
iX,j=0
whichwascalledtheteleportationequationin[18]. Then,AliceperformstheBellmeasurements
ψ(ij) ψ(ij) 11 onthepreparedstate α Ψ ,
2
| ih |⊗ | i⊗| i
1
(ψ(ij) ψ(ij) 11 )(α Ψ )= ψ(ij) W α , (7)
2 ij
| ih |⊗ | i⊗| i 2| i⊗ | i
4
|αi i •
BELL
j
•
|Ψi•
W† |αi
ij
Figure 1: Quantum circuit for quantumteleportation as a diagrammaticalrepresentation of the
teleportationequation(6). An unknownqubit α is sentfromAlice to Bob. Thetop two lines
| i
represent Alice’s system and the bottom line represents Bob’s system. The single-lines denote
qubitsandthe double-linesdenoteclassical bits. The triangleboxrepresentsthe Bell measure-
mentperformedbyAliceandthesquareboxrepresentsthelocalunitarycorrectionoperatorWi†j
performedbyBobtoobtainthetransmittedstate α .
| i
andsheinformsBobhermeasurementresultslabeledas(i,j). Finally,Bobappliestheunitary
correctionoperatorWi†j onhisstate
(112⊗112⊗Wi†j)(|ψ(ij)i⊗Wij|αi)=|ψ(ij)i⊗|αi (8)
toobtainthetransmittedquantumstate α . SeeFigure1forthedescriptionoftheteleportation
| i
protocolinthelanguageofquantumcircuit.
NotethattheteleportationprotocolofBobtransmittinganunknownqubit α toAliceadmits
| i
anothertypeoftheteleportationequation
1
1
Ψ α = WT α ψ(ij) (9)
| i⊗| i 2 ij| i⊗| i
iX,j=0
whichwascalledthetransposeteleportationequationin[13]andistobeusedinthefault-tolerant
constructionoftwo-qubitgatesinSubsubsection2.3.4.
2.3 Fault-tolerantconstruction ofuniversal quantum gateset
Quantumgates[23]aredefinedasunitarytransformationmatricesactingonquantumstates,and
thesetofalln-qubitgatesformsarepresentationoftheunitarygroupU(2n).
2.3.1 Universalquantumgateset
Anentanglingtwo-qubitgate[24]withsingle-qubitgatesiscalledauniversalquantumgateset
whichcanperformuniversalquantumcomputationinthecircuitmodel[1]ofquantumcomputa-
tion. Allsingle-qubitgatescanbegeneratedbyboththeHadamardgateH givenby
1
H = (X +Z), (10)
√2
andtheπ/8gate[25]givenby
1 0
T = . (11)
(cid:18) 0 eiπ4 (cid:19)
Anentanglingtwo-qubitgate[24]isdefinedasatwo-qubitgatecapableoftransformingatensor
productoftwosingle-qubitstatesintoanentanglingtwo-qubitstate. Forexamples,theCNOT
gate
CNOT = 0 0 11 + 1 1 X, (12)
2
| ih |⊗ | ih |⊗
5
andtheCZ gate
CZ = 0 0 11 + 1 1 Z. (13)
2
| ih |⊗ | ih |⊗
Theyaremaximallyentanglinggates[26],namely,withsingle-qubitgates,theycangeneratethe
maximallyentanglingstatessuchastheBellstates(5)fromtheproductstates. TheCNOT gate
isthewellknowntwo-qubitgateinquantuminformationandcomputationwhichisthequantum
analogueoftheExclusiveORgateinclassicalcomputation[1].TheCZgateiswidelyusedinthe
one-wayquantumcomputation[27],therepresentativeexampleofmeasurement-basedquantum
computation,anditisrelatedtotheCNOT gateintheway
CZ =(11 H)CNOT(11 H) H CNOT H (14)
2 2 2 2
⊗ ⊗ ≡
inwhichthesubscriptoftheHadamardgateH meansthatitisactingonthesecondqubit.Hence
thesetoftheCNOT gate(12)(ortheCZ gate(13))withsingle-qubitgatesH (10)andT (11)
iscalledauniversalquantumgatesettoperformuniversalquantumcomputation[1].
2.3.2 Cliffordgatesandfault-tolerantquantumcomputation
The Pauli group gatesC are generatedby tensor productsof the Pauli matrices X, Z and the
1
identitymatrix11 withglobalphasefactors 1, i.QuantumgatesU [1]areclassifiedby
2
± ±
C U UC U C (15)
k k 2 † k 1
≡{ | − ⊆ − }
whereC denotesthe Cliffordgatespreservingthe Pauligroupgatesunderconjugation. Obvi-
2
ously,theHadamardgateH isaCliffordgate,namelyH C ,dueto
2
∈
HXH =Z, HZH =X, (16)
andtheπ/8gateT isnotaCliffordgate,T C ,since
3
∈
X √ 1Y
TXT†= − − , TZT†=Z (17)
√2
inwhich(X √ 1Y)/√2isaCliffordgate[1,28]. Hereisanotherequivalentdefinitionofthe
− −
Cliffordgate[1,28]. IfaquantumgatecanberepresentedasatensorproductoftheHadamard
gate(10),thephasegate
1 0
S = , (18)
(cid:18) 0 i (cid:19)
andtheCNOT gate(12),thenitisaCliffordgate.NotethatthephasegateSisthesquareofthe
π/8gate(11),S =T2,andthesquareofthephasegateS isthePauliZ gate,S2 =Z.
Infault-tolerantquantumcomputation[28],thePauligroupgatesC andCliffordgatesC can
1 2
beeasilyperformedinprinciple,buttheC gatesmaybedifficultlyrealized. Theteleportation-
3
basedquantumcomputation[9]isfault-tolerantquantumcomputationinthefollowingsense: to
performaC gate,itpreparesaquantumstatewiththeactionofC gateandthenappliesC or
3 3 1
C gatestosuchthequantumstateusingtheteleportationprotocol.
2
2.3.3 Fault-tolerantconstructionofsingle-qubitgates
Toperformasingle-qubitgateU C (15)ontheunknownstate α ,Alicepreparesthequantum
k
∈ | i
state Ψ givenby
U
| i
Ψ =(11 U)Ψ (19)
U 2
| i ⊗ | i
6
|αi i •
BELL
j
•
|Ψ i•
U
U R† U|αi
ij
Figure2:Quantumcircuitforthefault-tolerantconstructionofthesingle-qubitgateU associated
withthe teleportationequation(20). Withthe suchdiagrammaticalrepresentation,itisobvious
thatfault-tolerantlyimplementingthesingle-qubitU becomeshowtofault-tolerantlyperformthe
Ri†j gateandpreparethepreliminaryquantumstate|ΨUi(19).
andreformulate α Ψ as
U
| i⊗| i
1
1
α Ψ = ψ(ij) R U α , (20)
U ij
| i⊗| i 2 | i⊗ | i
iX,j=0
wherethesingle-qubitgateR hastheformR =UW U withW definedin(2). ThenAlice
ij ij ij † ij
makesBellmeasurements ψ(ij) ψ(ij) 11 andinformsBobhermeasurementresultslabeled
2
| ih |⊗
Sbyee(iF,ijg)u.reFi2nafollry,thBeoqbupaenrtfuomrmcsirtchueituonfitfaaruyltc-otorlreercatniotlnyoppeerrfaotromriRngi†jth∈eCsikn−g1le(-1q5u)bittogaattteaiUnUus|αinig.
theteleportationprotocol.
Asthesingle-qubitgateU istheHadamardgateH (10),thesingle-qubitgateR isgivenby
ij
R (H) = WT, whichisa tensorproductofPauligates. Andasthesingle-qubitgateU isthe
ij ji
π/8gate(11),thesingle-qubitgateR isgivenby
ij
X √ 1Y
R (T)=( − − )iZj, (21)
ij
√2
whichcanbeeasilyfault-tolerantlyperformed,becauseitisa Cliffordgate. Notethatthediffi-
cultyofperformingasingle-qubitgateU C becomeshowtofault-tolerantlypreparethestate
k
∈
|ΨUiandperformthesingle-qubitgateRi†j ∈Ck−1.
2.3.4 Fault-tolerantconstructionoftwo-qubitCliffordgates
Toperformatwo-qubitCliffordgateCU suchastheCNOT gate(12)andtheCZ gate(13)on
twounknownsingle-qubitstates α and β ,weprepareafour-qubitentangledstate Ψ ,
CU
| i | i | i
Ψ =(11 CU 11 )(Ψ Ψ ), (22)
CU 2 2
| i ⊗ ⊗ | i⊗| i
withtheactionoftheCU gate,andreformulatethepreparedstate α Ψ β as
CU
| i⊗| i⊗| i
α Ψ β
CU
| i⊗| i⊗| i
1 1 1 (23)
= (11 Q P 11 )(ψ(i j ) CU αβ ψ(i j ) )
4 4 1 1 2 2
4 ⊗ ⊗ ⊗ | i⊗ | i⊗| i
i1X,j1=0i2X,j2=0
with11 = 11 11 ,whichiscalledtheteleportationequationassociatedwiththefault-tolerant
4 2 2
⊗
constructionofthetwo-qubitCliffordgateCU.
Thesingle-qubitgatesQandP intheteleportationequation(23)arecalculatedby
Q P =CU(W WT )CU . (24)
⊗ i1j1 ⊗ i2j2 †
7
|αi i1 •
BELL
j1
•
•

 Q†

|ΨCUi CU CU|αβi
P†
 
•

 i2 •
BELL
|βi j2 •
Figure3: Quantumcircuitforthefault-tolerantconstructionofatwo-qubitCliffordgateCU,as
adiagrammaticalrepresentationoftheteleportationequation(23). Specifically,thesingle-qubit
gatesQandP arecalculatedwiththeformula(24).
AstheCU gateistheCNOT gate,thegatesQandP areexpressedas
Q=Zj2Xi1Zj1, P =Zj2Xi2Xi1. (25)
AstheCU gateistheCZ gate,thegatesQandP havetheform
Q=Zi2Xi1Zj1, P =Zj2Xi2Zi1. (26)
NotethattheQandP gates(24)maybenotsingle-qubitgatesiftheCU gateisnotaClifford
gateinaccordancewiththedefinitionofaCliffordgate[1,28].
Next,weperformtheBellmeasurementsgivenby
ψ(i j ) ψ(i j ) 11 11 ψ(i j ) ψ(i j ), (27)
1 1 1 1 2 2 2 2 2 2
| ih |⊗ ⊗ ⊗| ih |
andwiththemeasurementresultslabeledby(i ,j )and(i ,j ),weperformtheunitarycorrec-
1 1 2 2
tionoperator,Q P ,toobtaintheexactactionoftheCliffordgateCU onthetwo-qubitstate
† †
⊗
α β ,namely,CU αβ .
| i⊗| i | i
The quantum circuit for the fault-tolerant construction of the two-qubit Clifford gate CU
usingtheteleportationprotocolisdepictedinFigure3. Notethatthetwo-qubitCliffordgateCU
westudyheremaybenotthecontrolled-operationtwo-qubitgatessuchastheCNOT gate(12)
andtheCZ gate(13).
2.4 Construction offour-qubit entangled states
FromSubsection2.3,wealreadyknowthat,theconceptualpointinthefault-tolerantconstruction
ofasingle-qubitquantumgateU (oratwo-qubitCliffordgateCU)usingquantumteleportation
isthefault-tolerantpreparationofthetwo-qubitentangledstate(19)(orthefour-qubitentangled
state(22)). Inthissubsection,wefocusonthefault-tolerantpreparationofthefour-qubitentan-
gledstate Ψ (22)usingtheteleportationprotocol,whentheCU gateistheCNOT gate(12)
CU
| i
andtheCZ gate(13)respectively.
In Gottesman and Chuang’s original proposal[9] of teleportation-basedquantum computa-
tion, a four-qubitentangledstate Ψ is preparedin the followingsteps. First, we preparea
CU
| i
priorentangledsix-qubitstateasatensorproductofathree-qubitGHZstate Υ [29],
| i
1
Υ = (000 + 111 ) (28)
| i √2 | i | i
8
X
|Υi• Z X
|Ψi•
i
•
•
= BELL
H j •
|Ψi•
|Υi• H
H
Figure4: Quantumcircuitfortheconstructionoffour-qubitentangledstate Ψ (29),asa
CNOT
| i
diagrammaticalrepresentationoftheteleportationequation(30). The Υ stateisthethree-quibt
| i
GHZstate(28)andthesingle-qubitgatedenotedasH istheHadamardgate(10). TheCNOT
gate(12)takestheconventionalconfigurationinquantumcircuitmodel[1].
and anotherthree-qubitGHZ state Υ with the localaction of the Hadamardgate H. Second,
| i
we performthe Bell measurementson such the six-qubitentangled state. Third, after classical
communicationoftheBellmeasurementoutcomes,weperformtheunitarycorrectionoperators
toobtainthefour-qubitentangledstate Ψ .
CU
| i
2.4.1 Constructionofthefour-qubitentangledstate Ψ
CNOT
| i
Thefour-qubitentangledstate Ψ isthetargetstatewewanttoconstruct,
CNOT 1256
| i
Ψ =(11 CNOT 11 )(Ψ Ψ ) (29)
CNOT 1256 2 25 2 12 56
| i ⊗ ⊗ | i ⊗| i
in which the CNOT gate requires the i-th qubit as the control qubit and the j-th qubit as
ij
the targetqubit. We preparethe six-qubitentangledstate H H H Υ Υ , which canbe
4 5 6 123 456
| i | i
reformulatedas
1
H H H Υ Υ = ψ(ij) ZjXiXi Ψ , (30)
4 5 6| i123| i456 | i34 2 1 2| CNOTi1256
iX,j=0
thenmakethejointBellmeasurementonboththethirdandfourthqubitsgivenby
11 11 ψ(ij) ψ(ij) 11 11 . (31)
2 2 34 2 2
⊗ ⊗| i h |⊗ ⊗
Withthemeasurementoutcome(i,j),weperformtheunitarycorrectionoperator
Xi XiZj 11 11 11 11 , (32)
2 2 2 2
⊗ ⊗ ⊗ ⊗ ⊗
ontheresultantquantumstatetoobtainthefour-qubitentangledstate Ψ (29). Thequan-
CNOT
| i
tumcircuitfortheconstructionofthe Ψ stateisshowninFigure4.
CNOT
| i
Notethattheteleportationequation(30)admitsanequivalentform
1
H H H Υ Υ = ψ(ij) XiZjZj Ψ , (33)
4 5 6| i123| i456 | i34 5 5 6| CNOTi1256
iX,j=0
so thatwehavethe secondmethodofconstructingthe Ψ state (29) usingtheteleporta-
CNOT
| i
tionprotocol. Thesetwomethodsarefoundtobeequivalentinthelowdimensionaltopological
diagrammaticalapproach,seeSubsubsection3.4.2.
9
2.4.2 Constructionoffour-qubitentangledstates|Ψ↑CNOTiand|ΨCZi
In Gottesman and Chuang’s original study [9], the fault-tolerant construction of the four-qubit
entangled state |Ψ↑CNOTi (180) was presented instead of the state |ΨCNOTi (29) that we are
working on. For readers’ convenience, the construction of the |Ψ↑CNOTi state is discussed in
Appendix C. Besides |ΨCNOTi and |Ψ↑CNOTi, since the CZ gate (13) has a wide application
in measurement-based quantum computation [27], we work out the construction of four-qubit
entangledstate Ψ (188)inAppendixD.
CZ
| i
3 The extended Temperley–Lieb diagrammatical approach to
teleportation-based quantum computation
Inthissection4,weaimatexhibitingtopologicalfeaturesofteleportation-basedquantumcompu-
tationinthetransparentstyle.IntheextendedTemperley–Liebdiagrammaticalapproach[17,18],
westudythetopologicaldiagrammaticalconstructionofbothuniversalquantumgatesetandfour-
qubitentangledstates in teleportation-basedquantumcomputation[9, 10, 11], whose algebraic
counterpartshavebeenpresentedinSection2.
TheextendedTemperley–Liebdiagrammaticalconfiguration[17,18] isakindoftheexten-
sionofthestandardTemperley–Liebconfigurationonwhichbothsingle-qubitgatesandtwo-qubit
gatesareallowedtomovealongthelinesundercertainrules. Thedefinitionofthe Temperley–
LiebalgebraanditsdiagrammaticalrepresentationisshownupinAppendixA.1andA.2.
3.1 The extended Temperley–Lieb diagrammaticalapproach
A single-qubit state vector ϕ is denoted by a vertical line followed by the symbol on the
| i ∇
bottom,
ϕ = (34)
| i
∇
andthelinewiththesymbol onthetoprepresentsthecovector φ,soaverticallinewithboth
△ h |
symbols and ontheboundarydenotestheinnerproduct φϕ . Averticallinewithasolid
∇ △ h | i
pointrepresentsasingle-qubitgateU actingonthestate ϕ ,
| i
r
U ϕ = U (35)
| i
∇
inwhichthealgebraicexpressionisreadfromtherighttotheleftandthediagrammaticalrepre-
sentationisreadfromthebottomtothetop.
The diagrammatical configuration of the Bell state ψ(ij) (5) is the core of the extended
| i
Temperley–Liebdiagrammaticalapproach,andit isrepresentedbya cupwith a solid pointde-
notingthesingle-qubitgateW (2),
ij
ψ(ij) = r (36)
| i Wij
so thata cup withouta solid pointdenotesthe EPRstate Ψ (3). The adjointofthe Bell state,
| i
ψ(ij),isrepresentedbyacap
h |
hψ(ij)|= rWi†j (37)
4This section is an extended version of the authors’ unpublished paper [12] in which topological features of
teleportation-basedquantumcomputationareclaimedtobeassociatedwithtopologicalfeaturesofspace-time.
10