Efficient Computation of Decoherent Quantum Walks through Eigenvalue Perturbation

0.09 MB·

by  Andrea Torsello

Efficient Computation of Decoherent Quantum Walks
through Eigenvalue Perturbation
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AndreaTorsello LucaRossi
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Universita`Ca’FoscariVenezia AstonUniversity
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[email protected] [email protected]
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n
a
J Abstract

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A numberof recentstudieshave investigatedthe introductionofdecoherencein

quantum walks and the resulting transition to classical random walks. Interest-
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h ingly,ithasbeenshownthatalgorithmicpropertiesofquantumwalkswithdeco-
p herencesuchasthespreadingratearesometimesbetterthantheirpurelyquantum
- counterparts.Notonlyquantumwalkswithdecoherenceprovideageneralization
t
n ofquantumwalksthatnaturallyencompassesboththequantumandclassicalcase,
a buttheyalsogiverisetonewanddifferentprobabilitydistribution. Theapplica-
u
tionofquantumwalkswithdecoherencetolargegraphsislimitedbythenecessity
q
of evolvinga state vectorwhose size is quadraticin the numberof nodesof the
[
graph, as opposedto the linear state vector of the purely quantum(or classical)

1 case. Inthis technicalreport, weshow howtouse perturbationtheoryto reduce
v the computationalcomplexityof evolvinga continuous-timequantumwalk sub-
1 jecttodecoherence.Morespecifically,givenagraphovernnodes,weshowhow
9 to approximatethe eigendecompositionof the n2 ×n2 Lindblad super-operator
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fromtheeigendecompositionofthen×ngraphHamiltonian.
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.
1 1 Introduction
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1 Quantum walks on graphs represent the quantum mechanical analogue of classical random
: walks [1, 2, 3]. Despite being similar in the definition, the dynamics of the two types of walks
v
canberemarkablydifferent,withquantumwalkspossessinganumberofinterestingpropertiesnot
i
X exhibitedbytheirclassicalcounterparts.Intheclassicalcase,theevolutionofthewalkisdescribed
by a real-valuedprobabilityvector. In the quantumcase, the state is characterizedby a complex-
r
a valued amplitude vector. An interesting consequence of this is that different paths are naturally
allowedtodestructively(constructively)interferewitheachother.
Most of the work in the literature has consideredpure quantumdynamics[1, 2, 3, 4, 5, 6, 7], i.e,
fullycoherentquantumwalks. However,ithasbeenshownthattheintroductionofdecoherencecan
result in some algorithmic propertiesof the walk, such as the spreading rate, being better than in
thepurelyquantumcasecase[8,9]. Mostimportantly,quantumwalkswithdecoherencerepresent
a generalizationofquantumwalksthatencompassesbothclassicalandquantumwalks, aswellas
newtypesofwalksthatresultindifferentprobabilitydistributions[9].
Recallthatdecoherenceistheprocessbywhichaquantumsystemisalteredbyitsinteractionwith
theenvironment. Theresultofthisprocessisatransitionofthesystemfromquantumtoclassical.
For example, a quantumwalk subjectto decoherencetransitions to a classical randomwalk, with
a speedthatdependsonthedecoherencerate. Unfortunately,whileinthe fullyclassicalandfully
quantumcasesthesizeofthestatevectorisn,wherendenotesthenumberofnodesofthegraph,
in the decoherentcase the size of the state vector is n2. The Hamiltonian operator acting on the
state vector of a unitary quantum walk is represented by a n × n matrix. With the addition of
decoherence,on the other hand, the evolution is defined by the Lindbladsuper-operator,which is
1
representedbyan2×n2matrix.Thisclearlylimitsthepossibilityofanalysinglargegraphstructures
usingdecoherentquantumwalks.
Inthistechnicalreportweproposetouseperturbationtheory[10,11]toreducethecomputational
complexityof evolvinga continuous-timequantumwalk subjectto decoherence. In Section 2 we
introduce the necessary quantum mechanicalbackground. In Section 3 we review the eigenvalue
perturbationproblemandinSection4weshowhowthiscanbeappliedtotheproblemathand.
2 Continuous-time Quantum WalkswithDecoherence
2.1 Continuous-TimeQuantumWalks
The continuous-time quantum walk is the quantum analogous of the continuous-time random
walk [1]. Let G = (V,E) denote an undirected graph with n nodes. If p(t) ∈ Rn denotes the
stateofwalkattimet,inacontinuous-timerandomwalkthestatevectorevolvesaccordingtothe
equationp(t)=e−Ltp(0),wherethegraphLaplacianListheinfinitesimalgeneratormatrixofthe
underlyingcontinuous-timeMarkovprocess.
Similarly to its classical counterpart, the state space of the continuous-timequantumwalks is the
vertexsetofthegraph.Theclassicalstatevectorisreplacedbyavectorofcomplexamplitudesover
V whosesquarednormsumstounity,andassuchthestateofthesystemisnotconstrainedtoliein
aprobabilityspace,thusallowinginterferencetotakeplace. Thegeneralstateofthewalkattimet
isacomplexlinearcombinationofthebasisstates|ui,i.e.,
|ψ(t)i= α (t)|ui, (1)
u
u∈V
X
where the amplitude α (t) ∈ C and |ψ(t)i ∈ C|V| are both complex. Moreover, we have
u
that α (t)α∗(t) gives the probability that at time t the walker is at the vertex u, and thus
u u
α (t)α∗(t)=1andα (t)α∗(t)∈[0,1],forallu∈V,t∈R+.
u∈V u u u u
TPheevolutionofthewalkisgovernedbytheSchro¨dingerequation
∂
|ψ(t)i=−iH|ψ(t)i, (2)
∂t
wherewe denotethe time-independentHamiltonianasH. Generallyspeaking, a continuous-time
quantumwalkisinducedwheneverthestructureofthegraphsisreflectedbythe(0,1)patternofthe
Hamiltonian. Forexample,we couldtaketheadjacencymatrixortheLaplacian. Inthefollowing
weassumeH =L.
Given an initial state |ψ(0)i, solving the Schro¨dinger equation gives the expression of the state
vectorattimet,
|ψ(t)i=e−iLt|ψ(0)i. (3)
ThiscanbeconvenientlyexpressedintermsofthespectraldecompositionoftheHamiltonianH =
ΦΛΦ⊤,i.e.,|ψ(t)i=Φ⊤e−iΛtΦ|ψ(0)i,whereΦdenotesthen×nmatrixΦ=(φ1|φ2|...|φj|...|φn)
withtheorderedeigenvectorsφjsofH ascolumnsandΛ=diag(λ1,λ2,...,λj,...,λn)isthen×n
diagonalmatrixwiththeorderedeigenvaluesλ ofH aselements,andwehavemadeuseofthefact
j
thatexp[−iLt]=Φ⊤exp[−iΛt]Φ.
2.2 QuantumWalkswithDecoherence
The density matrix is introduced in quantum mechanics to describe a system whose state is an
ensemble of pure quantumstates |ψ(i)i, each with probabilityp(i) [12]. The density operator of
suchasystemisdefinedas
ρ= p(i)|ψ(i)ihψ(i)| . (4)
i
X
Fora quantumwalk with state vector|ψ(t)i, thecorrespondingdensitymatrixattime t is ρ(t) =
hψ(t)|ψ(t)i. SimilarlytotheSchro¨dingerequation,theLiouville-vonNeumannequationdescribes
howadensityoperatorevolvesintime
∂
ρ(t)=−i[L,ρ(t)], (5)
∂t
2
whereListhegraphLaplacianand[A,B]=AB−BAdenotesthecommutator.
Wecanaddnon-unitarydecoherencetothesystembywriting[8]
∂
ρ(t)=−i[L,ρ(t)]−pρ(t)+pPρ(t), (6)
∂t
where p is the rate per unit time with which we add decoherence to the walk, and Pρ(t) =
P ρ(t)P† representsthe effectof noiseonρ(t), where{P }is aset ofprojectors. Intuitively,
j j j j
theeffectoftheextratermsistoreducetheoff-diagonalelementsofρ(t),i.e.,thecoherenceterms,
P
ataratepperunittime[8],whileleavingthediagonalelementsunaffected. Morespecifically,let
vec ρ(t) bethevectorizationofthedensitymatrixρ(t). Thenwecanwrite
(cid:0) ∂(cid:1)
vec ρ(t) = −i(L⊗I +I⊗−L)+p E ⊗E −I⊗I vec ρ(t) , (7)
vv vv
∂t
" !#
v∈V
(cid:0) (cid:1) X (cid:0) (cid:1)
whereE isthematrixwhichis1in(v,v)and0elsewhere,i.e.,theprojectoronthenodev.
vv
3 Eigenvalue Perturbation
Let A(t) be a n×n complex matrix parametrized by t ∈ T ⊆ R. Further, assume that A(t) is
diagonalizableforallvaluesoft,i.e.,thereexisttwon×nparametricmatricesX(t),Y(t),anda
diagonaln×nmatrixΛ(t)suchthatforallt∈T
A(t)X(t) = X(t)Λ(t) (8)
Y(t)A(t) = Λ(t)Y(t). (9)
Withoutlackofgenerality,assume
Y(t)X(t) = I (10)
diag X(t)†X(t) = 1. (11)
(cid:0) (cid:1)
WewanttoreconstructX(t),Y(t),andΛ(t)tothefirstorder:
X(t) ≈ X+tX′ (12)
Y(t) ≈ Y +tY′ (13)
Λ(t) ≈ Λ+tΛ′, (14)
whereX,X′,Y,Y′,Λ,andΛ′arecomputedattimet=0.
Tothisend,sinceA(t)isdiagonalizable,X andY havefullrank,sowecanwrite
X′ =XB (15)
Y′ =CY, (16)
forsomematricesBandC.
3.1 DistinctEigenvalues
In the case where allthe eigenvaluesare distinct onecan computethe eigenvalueandeigenvector
derivativesdirectly.Differentiating(8),wehave
A′X +AX′ =X′Λ+XΛ′. (17)
Left-multiplyingbothsidesbyY andrecallingthatYAX =Λ,wehave
YA′X +ΛB =BΛ+Λ′, (18)
fromwhich
diag(Λ′)=diag(YA′X)+diag(ΛB−BΛ)=diag(YA′X), (19)
fromwhichwehavetheeigenvaluederivatives
λ′ =y†A′x , (20)
i i i
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where x and y are respectivelythe rightand lefteigenvectorscorrespondingto eigenvalueλ of
i i i
A(0).
FortheEigenvectors,from(18)wehave,fori6=j
(YA′X) +(ΛB) = (BΛ) +(Λ′) (21)
ij ij ij ij
y†A′x +λ b = b λ +0, (22)
i j i ij ij j
fromwhich
y†A′x
b = i j. (23)
ij
λ −λ
j i
Differentiating(11),weobtain
0 = diag X(t)†X(t) ′ =diag (X′)†X +X†X′ (24)
= diag((cid:0)B†X†X +X(cid:1) †XB)=(cid:0)2Re diag(X†X(cid:1)B)
so,foralli,wehave (cid:0) (cid:1)
Re (X†X) b =0, (25)
ik ki
k
X (cid:0) (cid:1)
or,extractingthetermfork=i,andrecallingthat(X†X) =1
ii
Re(b )=− Re (X†X) b (26)
ii ik ki
k6=i
X (cid:0) (cid:1)
Asfortheimaginarypartofb ,recallthatevenafterthenormalizationconstraint(11)thereisstill
ii
adegreeoffreedominthechoiceoftheglobalphaseoftheeigenvectorsx whichisreflectedinan
i
arbitrarinessinthechoiceofIm(b ). HerewesetIm(b )=0.
ii ii
Note,also,thatdifferentiating(10)weobtain
0= Y(t)X(t) ′ =Y′X+YX′ =CYX +YXB =C+B, (27)
fromwhichweobtain(cid:0) (cid:1)
C =−B. (28)
3.2 RepeatedEigenvalues
Inthecaseofrepeatedeigenvalueswehaveanadditionaldegreeoffreedomfromthechoiceofthe
eigenbasis. Any linear combination of eigenvectorscorrespondingto the same eigenvalue is still
aneigenvectorofthe matrix,thustheobservedeigenvectorsx canindeedbe linearcombinations
i
of the limiting eigenvectorsof A(t) as t → 0, resulting in a discontinuity. This can be solved by
assumingthatthereisanunknowneigenvectorbasisX thatiscontinuousintandexpressingitin
termsoftheobservedeigenvectormatrixXˆ:
X =XˆΓ. (29)
Substitutinginto(8)andleft-multiplyingbyYˆ =Xˆ−1wehave
YˆAXˆΓ = XˆΓΛ (30)
ΛΓ = ΓΛ, (31)
Hence,Γisco-diagonalizablewithΛ. RecallthatΛisdiagonal,thusΓmustbeblockdiagonalwith
theblockscorrespondingtotherepeatedvaluesofΛ.
Let λ¯ be one such repeated eigenvalue, repeated with multiplicity r. We can partition the eigen-
value/eigenvectormatricesasfollows:
Λ= Λ01 λ¯0I Γ= Γ01 Γ02 B = BB1211 BB1222 (32)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Y
Xˆ =(X1 X2) Yˆ = Y1 . (33)
2
(cid:18) (cid:19)
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From(18)weobtain
Y1A′X1Γ1 Y1A′X2Γ2 = Γ1(B11Λ1−Λ1B11−Λ′1) Γ1(λ¯I−Λ1)B12 . (34)
Y2A′X1Γ1 Y2A′X2Γ2 −Γ2B21(λ¯I −Λ1) Γ2Λ′2
(cid:18) (cid:19) (cid:18) (cid:19)
Hence,theblock-diagonalelementΓ2canbeobtainedbysolvingtheeigenvalueproblem
Y2A′X2Γ2 =Γ2Λ′2, (35)
wherethederivativesΛ′2 ofthe repeatedeigenvaluesλ¯ are theeigenvaluesofY2A′X2Γ2. Ifthese
eigenvaluesaredistinctthematrixΓ2isuniqueuptoamultiplicativefactor.
Ifweassumethatboththeobservedandthecontinuouseigenvectorsarenormalized,i.e.,
diag X(t)†X(t) =1 diag Γ†X(t)†X(t)Γ =1, (36)
it is clear that on non-repeated eigenvalues, the corresponding diagonal element of Γ must have
(cid:0) (cid:1) (cid:0) (cid:1)
norm1. Asusualthephaseremainsarbitrary,butwecanpickγ =1withoutlossofgenerality.
ii
Theequation
Y1A′X1Γ1 =Γ1(B11Λ1−Λ1B11−Λ′1) (37)
is equivalent to (18) on the reduced eigenvalue set and can be recursively partitioned if Λ1 still
containsrepeatedeigenvaluesandsolvedasin thecaseofnon-repeatedeigenvalues. Ontheother
hand,thevaluesofB andB canbecomputedfromthefollowingequations
12 21
Y1A′X2Γ2 = Γ1(λ¯I−Λ1)B12 (38)
Y2A′X1Γ1 = −Γ2B21(λ¯I−Λ1). (39)
To computeB22, we differentiate(17) one more time, setting X′′ = XD, we left-multiplyby Y
andweconcentrateonthesub-matrixcorrespondingtotherepeatedeigenvalues:
Y2A′′X2Γ2+2Y2A′Γ2B22+λ¯Γ2D =Γ2Dλ¯+2Γ2B22Λ′2+γ2Λ′2′. (40)
RecallingthatΓ2isasolutiontotheeigenvalueproblem(35),wehave
Y2A′′X2Γ2+2Γ2Λ′2B22 =2Γ2B22Λ′2+γ2Λ′2′ (41)
or
2(B22Λ′2−Λ′2B22)=Γ−21Y2A′′X2Γ2−Λ′2′, (42)
fromwhichwecanextracttheoff-diagonalelementsofB :
22
(B22)ij = Γ−221(Yλ¯2′A−′′Xλ¯′2)Γ2. (43)
j i
NotethatinthespecialcasewherethematrixA(t)isalinearfunctionoft, i.e.,A(t) = A+tA′,
thenA′′ =0andthus(B22)ij =0.
Asforthenon-repeatedeigenvaluecase,thediagonalofB iscomputedfromtheconstraint
22
diag Γ†X(t)†X(t)Γ =1, (44)
resultingin
(cid:0) (cid:1)
Re(b )=− Re (Γ†X†XΓ) b (45)
ii ik ki
k6=i
X (cid:0) (cid:1)
and,withoutlossofgeneralityIm(b )=0.
ii
3.3 HermitianMatrices
IfAisHermitian,thenX isunitaryandY =X†. Withthisinmind,inthedistincteigenvaluecase,
wehave
λ′ = x†A′x (46)
i i i
x†A′x
b = i j (47)
ij
λ −λ
j i
b = 0. (48)
ii
Thus,ifalsoA′isHermitian,B isskew-symmetric.
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4 Applicationto Quantum WalkswithDecoherence
Recall that the evolution of a quantum walk with decoherence expressed in terms of the density
matrixρisgivenbyEq.7. Inordertocomputetheevolutionofρ(t),weanalyzethebehaviorofthe
eigenvaluesandeigenvectorsof
A(p)=−i(L⊗I+I ⊗−L)+p E ⊗E −I ⊗I =iA+pA′ (49)
vv vv
!
v∈V
X
asafunctionofthedecoherenceratep.
Forp=0theeigenvaluesofA(p)arealloftheform
π =i(λ −λ ), (50)
jk k j
whereλ andλ areeigenvaluesofL. Thecorrespondingeigenvectorsareoftheform
k j
ξ =φ ⊗φ , (51)
jk j k
whereφ isaneigenvectorofLcorrespondingtoλ andφ isaneigenvectorcorrespondingtoλ .
j j k k
NotethatthereisatleastonerepeatedeigenvalueinA(0),namely0withmultiplicityatleastn. In
fact, foralli = 1,...,n we haveπ = i(λ −λ ) = 0whichisan eigenvaluewitheigenvector
jj j j
ξ = φ ⊗φ . Inthe followingwe makethesimplifyingassumptionthatthisis theonlycase of
jj j j
repeatedeigenvalue,namelythattheeigenvaluegapsλ −λ inLarealluniqueforj 6=k.
j k
Using(20)wecancomputetheeigenvaluederivativesπ ′forj 6=k:
jk
π ′ = ξT A′ξ =(φ ⊗φ )T E ⊗E −I⊗I (φ ⊗φ )
jk jk jk j k vv vv j k
!
v∈V
X
= (φTE φ )⊗(φTE φ )−1=− 1− φ2 φ2 , (52)
j vv j k vv k jv kv
!
v∈V v∈V
X X
wherethequantity
o = φ2 φ2 (53)
jk jv kv
v∈V
X
is the probability of co-observation of the standing waves φ and φ . This means that the (real)
j k
decayofthemixedeigenvectorξ introducedbythedecoherenceisproportionaltotheprobability
jk
thatthetwocomponentsφ andφ arenotobservedonthesamenode.
j k
For the eigenvector derivative, we compute the mixing proportion blm for j 6= k, l 6= m, and
jk
(j,k)6=(l,m). Intuitively,thistellsushowmuchofξ goesintoξ′
jk lm
blm = ξjTkA′ξlm = (φj ⊗φk)T v∈V Evv⊗Evv −I⊗I (φl⊗φm)
jk π −π i(λ +λ −λ −λ )
lm jk (cid:0)P m j l k (cid:1)
(φTE φ )⊗(φTE φ ) φ φ φ φ
= v∈V j vv l k vv m = v∈V jv lv kv mv. (54)
i(λ +λ −λ −λ ) i(λ +λ −λ −λ )
P m j l k Pm j l k
Hence,themixingisproportionaltotheprobabilityofco-observationofthestandingwavesφ ,φ ,
j k
φ ,andφ .
l m
Forj =kwehaverepeatedeigenvalues,soweneedtosolvethefollowingeigensystem:
ΞΓ=ΓΛ(0)′, (55)
withΞ=(ξ )
jk
ξ =ξ A′ξ = (φ ⊗φ )T E ⊗E −I ⊗I (φ ⊗φ )
jk jj kk j j vv vv k k
!
v∈V
X
= φ2 φ2 −δ , (56)
jv kv jk
v∈V
X
6
thuswehavethatΞ = O−I whereOisthematrixofco-observationsofthestandingwaves. Itis
easytoshowthatOisdoublystochastic,infact
o = φ2 φ2 = φ2 φ2 = φ2 =1 (57)
jk jv kv  jv kv kv
j j v∈V v∈V j v∈V
X XX X X X
 
o = φ2 φ2 = φ2 φ2 = φ2 =1, (58)
jk jv kv jv kv jv
!
k k v∈V v∈V k v∈V
X XX X X X
thus, Ξ has all real negative eigenvalue with the exception of at least one zero eigenvalue corre-
spondingtothesteadystateofO.
4.1 ComputationalComplexity
We concludethistechnicalreportwithsomeremarksonthecomputationalcomplexityofthepro-
posed approach. To this end, note that we first need to compute the eigendecomposition of the
Laplacian matrix L, which has complexity O(n3), where n is the number of nodes of the graph.
Similarly,solvingtheeigensystemofEq.55hascomplexityO(n3),whereΞisareal-valuedsym-
metricmatrixandΓisorthogonal.
ThecomputationofB12inEq.38requiresinvertingΓ1,whichinourcaseistheidentitymatrix,and
adiagonalmatrix,i.e.,(λ¯I −Λ1). Similarly,solvingEq.39forB21 requiresinverting(λ¯I −Λ1)
andΓ2. SinceΓisorthogonalandblock-diagonal,weconcludethatΓ2 isanorthogonalmatrix. In
general,notethatB isann2×n2 matrixandthereforethecomplexityofconstructingitisatleast
O(n4). Inparticular,fromEq.54itfollowsthatthecomplexityofcomputingthen4elementsofB
isO(n5). Weshouldstress,however,thatthecomputationoftheblmcanbeeasilyparallelized.
jk
Asaresult,weconcludethatthecomplexityoftheproposedapproachisdominatedbytheO(n5)
computation of the matrix B. This should be contrasted with the cost of directly computing the
eigendecompositionof the n2 ×n2 super-operatorA(p), which is O(n6). Finally, note thatfor a
generic p > 0, A(p) is not Hermitian and thereforetechniqueslike singular value decomposition
cannotbeemployed.
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