Quantum Stochastic Walk models for quantum state discrimination

Physics Letters A 384 (2020) 126195

Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Quantum

Stochastic

Walk

models

for

quantum

state

discrimination

Nicola Dalla Pozza

∗

,

Filippo Caruso

DipartimentodiFisicaeAstronomia,UniversitàdegliStudidiFirenze,I-50019SestoFiorentino,Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline11December2019 CommunicatedbyM.G.A.Paris Keywords:

QuantumStochasticWalk Quantumstatediscrimination Quantumnetwork

Decoherence

QuantumStochasticWalks(QSW)allowforageneralizationofbothquantumandclassicalrandomwalks bydescribingthedynamicevolutionofanopenquantumsystemonanetwork,withnodescorresponding toquantumstatesofafixedbasis.Weconsidertheproblemofquantumstatediscriminationonsucha system, and we solve it byoptimizing the network topologyweights. Finally,we test iton different quantumnetworktopologiesandcompareitwithoptimaltheoreticalbounds.

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Quantum Stochastic Walks (QSW) have been proposed as a frameworktoincorporatedecoherenceeffectsintoquantumwalks (QW), which, on the contrary, allow only for a purely unitary evolution of the state [1–3]. Unitary dynamics has been suffi-cienttoprovethecomputationaluniversality[4,5] andthe advan-tages provided by QW for quantum computation and algorithm design [4–7], but it is worthwhile the possibility to include ef-fectsofdecoherence.Indeed,thebeneficialimpactofdecoherence has already been proved in a variety of systems [8–10], in par-ticular in light-harvesting complexes [11–13]. As a consequence, shortlyaftertheirformalizationQSWhavebeeninvestigatedin re-lationtopropagationspeed[14,15],learningspeed-up[16],steady stateconvergence[17,18] andenhancementofexcitationtransport [19–23].

Effectively,thisframeworkallowstointerpolatebetween quan-tumwalksandclassicalrandomwalks[24].TheevolutionofQSW is defined by a Gorini–Kossakowski–Sudarshan–Lindblad master equation[25–27],writtenas d

ρ

dt

= −(

1

−

p

)

i [H

,

ρ

]

+

p



k



Lk

ρ

L†_k

−

1 2



L_k†Lk

,

ρ



.

(1)

In(1),theHamiltonianH accountsforthecoherentevolution,the setofLindbladoperators Lkaccountsfortheirreversibleevolution, whilethesmoothingparameter p definesalinearcombinationof thetwo.Forp

=

0 weobtainaquantumwalk,forp

=

1 weobtain aclassicalrandomwalk(CRW).

*

Correspondingauthor.

E-mailaddress:nicola.dallapozza@unifi.it(N. Dalla Pozza).

We consider the problem of discriminating a set of known quantumstates

{

ρ

(n)

_}

n preparedwithapriori probabilities

{

pn

}

n. Byoptimizing theset ofPositive Operator-ValuedMeasure

{

n

}

n we aimatestimatingthepreparedquantumstate withthe high-est probability ofcorrect detection Pc

=



npnTr





n

ρ

(n)

.In its generalformulationtheproblemrequiresnumericalmethods[28], butcloseanalyticalsolutionsareavailableinthecaseofsymmetric states [29–31]. For instance,in thebinary discrimination ofpure statestheoptimal P(copt) isknownasHelstromboundandit eval-uates Pc(opt)

= (

1

+

1

−

4p1p2Tr



ρ

(1)

_ρ

(2)

_)/

_{2.Forareviewofthe}

results of quantum state discrimination in quantum information theory we refer to [32–38], and to [39] for a recent connection withmachinelearning.Inhereweconsidertheproblemina quan-tumsystemrepresentedbyanetwork,andwetestdifferent mod-elsoptimizingthediscriminationonmultipletopologies.

2. QuantumStochasticWalksonnetworks

Weconsideraquantumsystemwhichisrepresentedbya net-work

G = (N ,

E)

, where the nodes

N

i

∈

N

corresponds to the quantumstatesofafixedbasisandthelinks

N

i

→

N

j

∈

E

depend onthehoppingratesbetweenthenodes.Asinclassicalgraph the-ory, the adjacency matrix A indicates whether a link is present ( Aj,i

=

1), or not ( Aj,i

=

0). In weighted graphs, Ai,j isa weight associated to the link. A symmetric adjacency matrix Ai,j

=

Aj,i referstoanundirected graph,otherwisethegraphissaidtobe di-rected.

In thispaperwe willfocus on thecontinuous version of ran-domwalks.CRWsaredefinedwithatransition-probabilitymatrix

T which represents the possible transitions of a walker from a node onto the connected neighbours.From the adjacency matrix it ispossible todefine T

=

A D−1, where D isthe (diagonal) de-https://doi.org/10.1016/j.physleta.2019.126195

2 N. Dalla Pozza, F. Caruso / Physics Letters A 384 (2020) 126195

Fig. 1. ExamplesofM−N−O networkmodels.Ineachsubfigure,theleftlayer (bluecircles)collectstheinputnodes,intherightlayer(purplecircles)the out-putnodesandinbetween(orangecircles)theintermediatenodes.Directedlinks specifyirreversibleprocessestowardsthesinks,plainlinesindicatenon-zero en-triesinAi,j.Bydefault,allthenodesinalayerareconnectedwithalltheother

nodesinthesamelayerandwiththenextone,exceptforthesinknodesthatare connectedonlythroughtheirsinkernode.Whenconsideringareduced connectivity inthegraph,weindicatear nearthenumberofnodes,asin(II).(For interpreta-tionofthecoloursinthefigures,thereaderisreferredtothewebversionofthis article.)

greematrix,with Di

=



jAj,irepresentingthenumberofnodes connectedtoi.Theprobabilitydistributionofthenodeoccupation, writtenasacolumnvector



q

(

t

)

,isevaluatedforacontinuoustime randomwalkas d_dtq

= (

T

−

I

)

q.



QuantumwalksaresimplydefinedbyposingH

=

A,and defin-ingtheevolution as d_dtρ

= −

i [H

,

ρ].

Thepopulationon thenodes isobtainedapplyingaprojectiononthebasisoftheHilbertspace associatedwiththenodes.

InthecaseofaQSW,thedynamicsisdefinedfromEq.(1) with

H andLk

=

Li,j

=



Ti,j

|

i



j

|

,with0

≤

Ti,j

≤

1,



iTi,j

=

1.While thisissufficientto definea properLindbladequation,anditalso givesthecorrectlimitingcaseforp

→

0 and p

→

1,thereare dif-ferentwaystodefine H andT from A.Forinstance,inboth[20,

24] wehaveH

=

A,T

=

A D−1,withthedifferencethatin[20] the weightscanonlybeunitaryornull.Herewewanttocomparethe performanceofthesemodelswiththecasewhereH andT are re-latedwithA byAi,j

=

0

=⇒

Hi,j

=

0,Ti,j

=

0,withH beingareal symmetricmatrixwithnulldiagonaland T verifying0

≤

Ti,j

≤

1,



iTi,j

=

1.Alternatively, wecan thinkthat Ai,j iseither1 or0, anditactsasamarkeronthelinksthatwewanttooptimize(1) orswitchoff(0).Thisschemerelaxestheconstraintson Hi,j,Ti,j, allowing for additionaldegrees offreedom to exploit inthe dis-crimination.

3. Networkmodel

To define the quantum system, we consider a network orga-nized in layers, which mimics the structure of neural networks [40–42],withM inputnodes, N intermediateancillarynodesand

O output nodesinthemodel M

−

N

−

O (seeFig.1). The quan-tum system is initialized only on the input nodes with

ρ

(

0

)

∈

{

ρ

(m)

_}

M

m=1.Thus,inputnodesdefineasub-spaceofsize M usedto accessthenetwork,withM ingeneralnotrelatedto

M

.The quan-tumsystemthenevolvesaccordingto(1) throughtheintermediate nodesandintotheoutputnodes.Suchnodesaresink nodeswhere thepopulationgets trapped,that is,thenetworkrealizesan irre-versible one-waytransfer ofpopulation froma sinker node sn to then-thsinkviatheoperatorLn

= |

n



sn

|

.Weaddthesumtotal

2



s M



n=1

|

n



sn

|

ρ

|

sn



n

| −

1 2

{|

sn



sn

|,

ρ

}

(2)

ontheright–handsideofEq.(1),setting



s

=

1 inoursimulations. We considera sinkernode foreach sink, butmore sinkers con-nectedtothesamesinkcouldbeintroduced.However,thenumber ofsinks O mustbeequalto orgreaterthan

M

tohaveasinkfor each hypothesis onthepreparedstates.Infact, attheendofthe time evolutiona measurementis performedby projecting onthe nodesbasis, andiftheoutcome n corresponding tothen-thsink isobtainedweestimatethatthequantumstate

ρ

(n)_hasbeen

pre-pared. Thenetworkwillbeoptimizedsuchthattheseestimations work as best aspossible, and if an outcome corresponding to a nodethatisnotasinkoccurs,weconsideritinconclusive. 4. Results

WeconsiderthetwonetworkmodelsrepresentedinFig.1and setuptheoptimizationoftheparametersin H ,T usingfourQSW schemes: (a) Hi,j

≥

0,T

=

H D−1 asin[24],(b) Hi,j

∈ {

0

,

1

}

,T

=

H D−1 as in [20], (c) Hi,j

= −

max

{

Ti,j

,

Tj,i

}

, Hi,i

= −



j=iHj,i andT un–normalizedasin[43] and(d)H ,T independently opti-mized,withtheoptimizationvariablesindicatedby A.

Forthemodel2

−

2

−

2,wediscriminatebetweenthepurestate

ρ

(1)_{andthemixedstate}

_ρ

(2)_{,writtenintheinputsub-spacebasis}

|

1



=



1₀

,

|

2



=



0₁

as

ρ

(1)

=



2+√2 4 1+i4 1−i 4 2−√2 4



,

ρ

(2)

=



0

.

68

−

0

.

13

−

0

.

13i

−

0

.

13

+

0

.

13i 0

.

32



,

p1

=

p2

.

Inthecaseofthemodel4r

−

4

−

4,wediscriminatebetweenfour equallyprobablequantumstates

ρ

(m)

_{= (}

₁

₋

_α

₎

I

4

+

α

|

ϕ

m



ϕ

m

|

, de-fined as a linearcombination ofthe completely mixedstate and a pure state

|

ϕ

m



=



k4=1e

−i 2πmk 4

2

|

k



, withm

=

1

,

. . .

4 being the

m-th stateinthe mutuallyunbiasedbasis oftheinput nodes

|

k



. For

α

=

1 wewouldhavethediscriminationofthepurestates

|

k



, which wouldhaveatheoretical bound Pc(opt)

=

1 sincethestates areorthogonal.Wetake

α

=

0

.

7 tosimulateanoisypreparationof

|

ϕ

m



.

The optimal probability of correct decision can be evaluated using semi-definite programming [28], andevaluates to Pc(opt)

=

0

.

7795 in the binary case, andto Pc(opt)

=

0

.

7750 inthe

M

-ary discrimination.Weruntheoptimizationforp

∈ [

0

,

1

]

andfor mul-tiplevaluesofthetotalevolutiontime

τ

.Asexpected,for increas-ing valuesof

τ

the performancesalsoincrease sincemore popu-lationcan betransferred fromtheinput nodestothesinknodes. However, theperformances saturate, and we plotthe asymptotic behaviour asafunctionof p inFig.2.Forlower

τ

thetrendin p

issimilar.

5. Discussionandconclusions

We have considered the problem of discriminating a set of quantum states prepared in a quantum system whose dynamics is described by QSW on a network. We have investigated four different schemes to define the GKSL master equation from the network,andoptimizedthecoefficientsoftheHamiltonian H and theLindbladoperatorcoefficientsT asafunctionofp andthe to-talevolutiontime

τ

.Wehavereportedtheasymptoticprobability of correctdetection, where aclear gapcan be seen amongst the four schemes. The setup (d) with H , T independently optimized gives the best performance on the whole range of p due to the increasednumberofdegreesoffreedombutattheexpenses ofa highercomputational costfortheoptimization. Further investiga-tions shouldconsider howtheperformance scales inthenumber

N. Dalla Pozza, F. Caruso / Physics Letters A 384 (2020) 126195 3

Fig. 2. Asymptoticprobabilityofcorrectdetection(τ=102_{s)forscheme(a)inyellow(squaredmarkers),(b)inblue(triangularmarkers),(c)inred(circularmarkers)and}

(d)ingreen(fulldisks).InreddashedlinethevalueofP(opt)c .

ofnodesandlayers toidentify thebestnetwork topology,which mayalso depend on the quantum statesto discriminate. Finally, ourresults couldbe tested throughalready experimentally avail-able benchmark platforms such as photonics-based architectures [22] andcoldatomsinopticallattices[44].

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

This work was financially supported from Fondazione CR Firenzethroughthe projectQ-BIOSCANandQuantum-AI,PATHOS EU H2020 FET-OPEN grant no. 828946, and UNIFI grant Q-CODYCES.

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