Quantum Search on Bounded-Error Inputs

PeterHyer 1;?

,Mi heleMos a 2;??

, andRonalddeWolf 3;???

1

Dept.ofComputerS ien e,Univ.ofCalgary,Alberta,Canada.

hoyer ps .u algary. a

2

Dept.ofCombinatori s&Optimization,Univ.ofWaterloo,andPerimeter

InstituteforTheoreti alPhysi s,Ontario,Canada.mmos auwaterloo. a

3

CWI.Kruislaan413,1098SJ,Amsterdam,theNetherlands.rdewolf wi.nl

Abstra t. Suppose we have n algorithms, quantumor lassi al, ea h

omputing some bit-value with bounded error probability. We des ribe

a quantum algorithm that uses O(

p

n ) repetitions of the base algo-

rithmsandwithhighprobabilityndstheindexofa1-bitamongthese

n bits (if there is su han index). This shows that it is not ne essary

torstsigni antlyredu e theerrorprobabilityinthebase algorithms

toO(1=poly(n))(whi hwouldrequireO(

p

nlogn)repetitions intotal).

Ourte hniqueisare ursiveinterleavingofamplitudeampli ationand

error-redu tion,andmaybeofmoregeneralinterest.Essentially,itshows

thatquantumamplitudeampli ation anbemadetoworkalsowitha

bounded-error verier.Asa orollaryweobtainoptimalquantumupper

bounds of O(

p

N) queries for all onstant-depth AND-ORtrees on N

variables,improvinguponearlierupperboundsofO(

p

Npolylog(N)).

1 Introdu tion

Oneofthemainsu essesofquantum omputingisGrover'salgorithm[10,7℄.It

ansear hann-elementspa einO(

p

n)steps,whi hisquadrati allyfasterthan

any lassi alalgorithm.Thealgorithmassumesora lea ess totheelementsin

the spa e, meaning that in unit time it an de ide whether the ith element

is a solution to its sear h problem or not. In some more realisti settings we

aneÆ ientlymakesu hanora leourselves.Forinstan e,ifwewanttode ide

satisability of an m-variable Boolean formula, the sear h spa e is the set of

all n = 2 m

truth assignments, and we an eÆ iently de ide whether a given

assignmentsatises the formula. However, in these asesthe de ision is made

withoutanyerrorprobability.Inthispaperwestudythe omplexityofquantum

sear hifweonlyhavebounded-error a esstotheelementsinthespa e.

Morepre isely, supposethatamong nBooleanvaluesf

1

;:::;f

n

wewantto

ndasolution(ifoneexists),i.e.,anindexjsu hthatf

j

=1.Forea hiwehave

?

SupportedinpartbytheAlbertaIngenuityFundandthe Pa i Institutefor the

Mathemati alS ien es.

??

Supported by St. Jerome's University, the Canada Resear h Chair programme,

NSERC(CROandDis overyGrant),CFI,OIT,PREA,ORDCFandMITACS.

???

Thisresear hwas(partially)fundedbyproje tsQAIP(IST{1999{11234)andRESQ

i i i

is1thenthealgorithmoutputs1withprobability,say,atleast9=10,andiff

i

=0

thenitoutputs0withprobabilityatleast9=10.Grover'salgorithmisnolonger

appli ableinthisbounded-errorsetting,atleastnotdire tly,be ausetheerrors

inea hstepwillqui klyadduptosomethingun ontrollablylarge.A ordingly,

weneedtodosomethingdierenttogetaquantumsear halgorithmthatworks

here.Wewillmeasurethe omplexityofourquantum sear halgorithmsbythe

numberoftimesthey alltheunderlyingalgorithmsF

i

.Clearly,the( p

n)lower

bound for the standard error-lesssear h problem, due to Bennett, Bernstein,

Brassard,andVazirani[4℄,also appliestoourmoregeneralsetting. Ouraimis

togiveamat hingupperbound.

Anobviousbut sub-optimal quantum sear halgorithm isthe following.By

repeating F

i

k = O(logn) times and outputting the majority value of the k

out omes,we an omputef

i

witherrorprobabilityatmost1=100n.Ifwethen

opytheanswertoasafepla eandreversethe omputationto leanup(mostof)

theworkspa e,thenwegetsomethingthatissuÆ iently\ lose"toperfe tora le

a esstothef

i

bitstojusttreatitassu h.Nowwe anapplyGrover'salgorithm

ontopofthis, andbe ausequantum omputationalerrorsaddlinearly[5℄,the

overalldieren ewithperfe tora lea esswillbenegligiblysmall.Thissolves

thebounded-errorquantumsear hproblemusingO(

p

nlogn)repetitionsofthe

F

i

's,whi hisanO(logn)-fa torworsethanthelowerbound.Belowwewillrefer

tothisalgorithm as\thesimplesear halgorithm".

Arelativelystraightforwardimprovementoverthesimplesear halgorithmis

thefollowing.Partitionthesear hspa einton=log 2

nblo ksofsizelog 2

nea h.

Pi konesu hblo katrandom.We anndapotentialsolution(anindexjinthe

hosenblo ksu hthatf

j

=1,ifthereissu haj)in omplexityO(lognloglogn)

using the simplesear h algorithm, and then verifythat itis indeed 1 wither-

rorprobabilityatmost1=nusinganotherO(logn)invo ationsofF

j

.Applying

Groversear h onthespa eof alln=log 2

nblo ks,weobtainanalgorithmwith

omplexityO(

q

n=log 2

n)
O(lognloglogn+logn)=O(

p

nloglogn).

A further improvement omes from doing the splitting re ursively:we an

use the improved upper bound to do the omputation of the \inner" blo ks,

insteadofthesimplesear halgorithm.UsingT(n)todenotethe omplexityon

sear hspa eof sizen,thisgivesusthere ursion

T(n)d



T(log 2

n) r

n

log 2

n

+logn



for some onstantd >0. This re ursion resolvesto omplexity O(

p

n
log



n

)

for some onstant > 0.It is similar to (and inspired by) the ommuni ation

omplexityproto olforthedisjointnessproblemofHyeranddeWolf[11℄.

Apartfrombeingrathermessy,thisimprovedalgorithmisstillnotoptimal.

Themainresultofthispaperistogivearelatively leanalgorithmthatusesthe

optimalnumberO(

p

n)ofrepetitionstosolvethebounded-errorsear hproblem.

Ouralgorithmuses akindof\ arrot-and-sti k"approa hthat maybeofmore

i

bran hesin ludesolutions,buttheyalsoin lude\falsepositives":bran hes or-

responding to the 1=10 errorprobability of F

i

's where f

i

= 0.We then \push

these ba k"bytesting whethera1-bran hisarealpositiveorafalseone(i.e.,

whether f

i

=1ornot) andremoving mostof thefalseones.Interleavingthese

amplifyandpush-ba kstepsproperly,we anamplifytheweightofthesolutions

toa onstantusingO(

p

n)repetitions.Atthispointwejustdoameasurement,

seeapotentialsolutionj, andverifyit lassi allybyrunningF

j

afewtimes.

As anappli ation of ourbounded-error quantum sear h algorithm,in Se -

tion4wegiveoptimalquantumalgorithmsfor onstant-depthAND-ORtreesin

thequery omplexitysetting.Forany onstantd,weneedonlyO(

p

N)queries

forthed-levelAND-ORtree,improvingupontheearlierO(

p

N(logN) d 1

)algo-

rithmsofBuhrman,Cleve,andWidgerson[9℄.Mat hinglowerboundsof( p

N)

werealreadyshownforsu hAND-ORtrees,usingAmbainis'quantumadversary

method[1,2℄.Finally,inSe tion5weindi atehowtheideaspresentedhere an

be astmoregenerallyin termsofamplitudeampli ation.

2 Preliminaries

Herewebrie ysket hthebasi sandnotationofquantum omputation,referring

to thebook byNielsen andChuang[12℄ for moredetail.An m-qubit stateisa

linear ombinationofall lassi alm-bitstates

ji= X

i2f0;1g m

i jii;

wherejiidenotesthebasisstatei(a lassi alm-bitstring),theamplitude 

i is

a omplexnumber,and P

i j

i j

2

=1.Weviewjiasa2 m

-dimensional olumn

ve tor.Ameasurementof statejiwill givejii withprobabilityj

i j

2

, andthe

statewillthen ollapsetotheobservedjii.Anon-measuringquantumoperation

orrespondstoapplyingaunitary(=linearandnorm-preserving)transformation

U to theve torof amplitudes.Ifjiand j i arequantumstateson mand m 0

qubits, respe tively,then the two-registerstatejij i=jij i orresponds

to the2 m+m

0

-dimensionalve torthatisthetensorprodu tofjiandj i.

Thesettingofquery omplexityisasfollows.Forinputx2f0;1g n

,aquery

orresponds tothe unitarytransformationO that maps ji;b;zi!ji;bx

i

;zi.

Here i 2 [n℄ and b 2 f0;1g; the z-part orresponds to the workspa e, whi h

is notae ted by thequery. A T-queryquantum algorithmhas theform A=

U

T OU

T 1

OU

1 OU

0

, wherethe U

k

areunitarytransformations,independent

of x. This A depends on x only via the T appli ations of O. The algorithm

startsin initialall-zerostatej0ianditsoutput(whi hisarandomvariable)is

In this se tion we des ribe our quantum algorithm for bounded-error sear h.

Thefollowingtwofa tsgeneralize,respe tively,theGroversear handtheerror-

redu tionusedinthealgorithmswesket hedin theintrodu tion.

Fa t1(Amplitude ampli ation[8℄) LetS

0

betheunitarythat putsa`-' in

front of the all-zero state j0i, and S

1

be the unitary that puts a`-' in front of

all basis states whose last qubit isj1i. Let Aj0i =sin()j

1

ij1i+ os()j

0 ij0i

whereangle is su hthat0=2 andsin 2

()equals theprobability thata

measurementofthelastregisterofstateAj0iyieldsa'1'.SetG= AS

0 A

1

S

1 .

Then GAj0 i=sin(3)j

1

ij1i+ os(3)j

0 ij0i.

Amplitude ampli ation is a pro ess that is used in many quantum algo-

rithms to in rease the su ess probability. Amplitude ampli ation ee tively

implementsarotationbyanangle2inatwo-dimensionalspa e(aspa edier-

entfromtheHilbertspa ea tedupon)spannedbyj

1

ij1iandj

0

ij0i.Notethat

we analwaysapplyamplitudeampli ationregardlessofwhether theangle 

isknownto usornot.

Fa t2(Error-redu tion) Suppose Aj0i = p

pj

b ijbi+

p

1 pj

1 b

ij1 bi,

where b 2 f0;1g and p  9=10. Then using O(log(1=")) appli ations of A

and majority-voting, we an build a unitary E su h that Ej0i = p

qj

b ijbi+

p

1 qj

1 b

ij1 bi with q  1 ", and j

b=1 b

i possibly of larger dimension

thanj

b=1 b

i(be auseofextraworkspa e).

Wewillre ursivelyinterleavethese twofa ts to getaquantum sear halgo-

rithmthatsear hesthespa ef

1

;:::;f

n

2f0;1g.Weassumeea hf

i

is omputed

byunitaryF

i

withsu essprobabilityatleast9=10.Let =fj:f

j

=1gbethe

set ofsolutions,and t=j j itssize (whi h isunknown toour algorithm).The

goalistondanelementin ift1,andtooutput `nosolutions'ift=0.

Wewill buildanalgorithmthat hasasuperpositionof allj 2[n℄in itsrst

register,agrowingse ondregisterthat ontainsworkspa eandotherjunk,and

a 1-qubit third registerindi ating whether somethingis deemed a solution or

not.Thealgorithmwillsu essivelyin reasetheweightofthebasisstatesthat

simultaneouslyhaveasolutionintherstregisteranda1in thethird.

Consideranalgorithm Athat runs all F

i

on e in superposition, produ ing

thestateAj0i,whi hwerewriteas

1

p

n n

X

i=1 jii



p

p

i j

i;1 ij1i+

p

1 p

i j

i;0 ij0i



=sin()j

1

ij1i+ os()j

0 ij0i;

where p

i

is the probability that F

i

outputs 1, the states j

i;b

i des ribe the

workspa eoftheF

i

,andsin() 2

= P

n

i=1 p

i

9t=10n.

Theideaistoapplyaround ofamplitudeampli ationtoAtoamplifythe

j1i-partfromsin()tosin(3).Thiswillamplifyboththegoodstatesjjij1ifor

toredu etheamplitudeofthefalsepositives,setting\most"ofitsthirdregister

to 0.These twostepstogetherform anew algorithmthat puts almost 3times

asmu hamplitudeonthesolutionsasAdoes,andthat putslessamplitudeon

thefalsepositivesthanA.Wethenrepeattheamplify-redu estepsonthisnew

algorithmto getanevenbetteralgorithm,andsoon.

Letusbemorepre ise.Ouralgorithmwill onsistofanumberofrounds.In

roundk wewillhaveaunitaryA

k

thatprodu es

A

k j0i=

k j

k

ij1i+

k j

k ij1i+

q

1 

2

k 

2

k jH

k ij0i;

where

k

;

k

arenon-negativereals,j

k

iisaunitve torwhoserstregisteronly

ontains j 2 , j

k

i is aunit ve torwhose rst registeronly ontainsj 62 ,

andjH

k

iisaunit ve tor.Ifwemeasure therstregisteroftheabovestate,we

willseeasolution(i.e.somej2 )withprobabilityatleast 2

k .A

1

istheabove

algorithm A, whi h runs the F

i

in superposition. Initially,  2

1

 9t=10nsin e

ea hsolution ontributesatleast9=10n.Wewanttomakethegoodamplitude

k

growbyafa torofalmost3inea hround.

Amplitude ampli ation step.Forea h roundk, dene

k

2[0;=2℄by

sin(

k )

2

=  2

k +

2

k

. Applying amplitude ampli ation (G

k

= A

k S

0 A

1

k S

1 )

givesusthestateG

k A

k

j0i,whi hwemaywriteas

sin(3

k )

sin(

k )

k j

k ij1i+

sin(3

k )

sin(

k )

k j

k ij1i+

s

1



sin(3

k )

sin(

k )



2

( 2

k +

2

k )jH

k ij0i:

WeappliedA

k

twi eandA 1

k

on e,sothe omplexitygoesupbyafa torof3.

Error-redu tionstep.Conditionalonthequbitin thethirdregisterbeing

1,theerror-redu tionstepE

k

nowdoesmajorityvotingonO(k)runs oftheF

j

(forallj insuperposition)tode idewitherroratmost1=2 k +5

whether f

j

=1.

Itaddsone0-qubitasthenewthirdregisterandmaps (ignoringitsworkspa e,

whi hisaddedtothese ondregister)

E

k

jjij1ij0i=a

jk

jjij1ij1i+ q

1 a 2

jk

jjij1ij0i

E

k

jjij0ij0i=jjij0ij0i

where a 2

jk

1 1=2 k +5

if f

j

=1 and a 2

jk

 1=2 k +5

if f

j

= 0. This way, E

k

removesmostofthefalsepositives.

Putting A

k +1

=E

k G

k A

k

anddening 

k +1 , 

k +1 ,j

k +1 i,j

k +1

i, andjH

k +1 i

appropriately,wenowhave

A

k +1 j0i=

k +1 j

k +1 ij1i+

k +1 j

k +1 ij1i+

q

1 

2

 2

jH

k +1 ij0i:

stepE

k

,aswellasbythequbitthatpreviouslywasthethirdregister.Thegood

amplitudehasgrowninthepro ess:

k +1



k sin(3

k )

sin(

k )

q

1 1=2 k +5

:

Sin ex x 3

=6sin(x)x,wehave

sin(3

k )

sin(

k )

3 9

2

k

=2:

A ordingly,aslongas

k

issmall,

k

willgrowbyafa torofalmost3inea h

round.Ontheotherhand,theweightofthefalsepositivesgoesdownrapidly:

k +1



k sin(3

k )

sin(

k )

1

p

2 k +5

:

We nowanalyze thenumberm of roundsthat weneed to makethe good am-

plitudelarge.Ingeneral,wehavesin(

k )

2

= 2

k +

2

k

,hen e 2

k

2(

2

k +

2

k )for

thedomainweareinterestedin.Here 2

k

9 k 1

 2

1 and

2

k

 1

10 (9=2

6

) k 1

.Note

m 1

X

k =1

 2

k

2 m 1

X

k =1 

2

k +

2

k

2 m 1

X

k =1 9

k 1

 2

1 +2

m 1

X

k =1 1

10 (9=2

6

) k 1

2
9 m 1

 2

1 +1=4:

Therefore,mroundsoftheabovepro essamplies thegoodamplitude

k to

m



1 m 1

Y

k =1 sin(3

k )

sin(

k )

q

1 1=2 k +5



1 m 1

Y

k =1

3 9

2

k

=2

1 1=2 k +5

=

1 3

m 1 m 1

Y

k =1

1 3

2

k

=2

1 1=2 k +5



1 3

m 1

1 3

2 m 1

X

k =1

 2

k m 1

X

k =1 1

2 k +5

!



1 3

m 1



1 3

2 (2
9

m 1

 2

1

+1=4) 1=16





1 3

m 1

1=2 3
9 m 1

 2

:

[n=9 m+1

;n=9 m

℄,equivalently9 m

2[n=9t;n=t℄,thenwehave

1

3 m

p

10

 r

9t

10n



1

 r

t

n

 1

3 m

:

Thisimplies

m

0:04;

so the probability of seeing a solution after m rounds is at least 0:0016. By

repeating this lassi ally a onstantnumber of times, say 1000 times, we an

bring the su ess probability lose to 1 (note to avoid onfusion: these 1000

repetitions arenotpartofthedenition ofA

m itself).

The omplexityC

k

oftheoperationA

k

,intermsofnumberofrepetitionsof

theF

i

algorithms,isgivenbythere ursion

C

1

=1andC

k +1

=3C

k

+O(k);

where the 3C

k

is the ost of amplitude ampli ation and O(k) is the ost of

error-redu tion.This impliesC

m

=O(

P

m 1

k =1 k
3

m k 1

)=O(3 m

):

Wenowgivethefullalgorithmwhenthenumberofsolutionsisunknown:

Algorithm:Quantumsear h on bounded-errorinputs

1. form=0todlog

9

(n)e 1do:

(a) runA

m

1000times

(b) verifythe1000measurementresults,ea hbyO(logn)runsofthe orre-

sponding F

j

( ) ifasolutionhasbeenfound,thenoutputasolutionandstop

2. Output`nosolutions'

Thisndsasolutionwithhighprobabilityifoneexists.The omplexityis

dlog

9 (n)e 1

X

m=0

1000
O(3 m

)+1000
O(logn)=O(3 log

9 (n)

)=O(

p

n):

Ifweknowthatthereisatleastonesolutionbutwedon'tknowhowmanythere

are,then,usingamodi ationofouralgorithmasin[7℄,we anndasolution

usinganexpe tednumberofrepetitionsinO(

p

N=t),wheretisthe(unknown)

numberofsolutions.Thisisquadrati allyfasterthan lassi ally,andoptimalfor

anyquantumalgorithm.

4 Optimal Upper Bounds for AND-OR Trees

A d-levelAND-OR tree on N Boolean variables is a Boolean fun tion that is

inputvariable,andiff

1

;:::;f

n

allared-levelAND-ORtreesonmvariables,ea h

withanAND(resp.OR) asroot,thenOR(f

1

;:::;f

n

)(resp.AND) isa(d+1)-

levelAND-ORtreeonN =nmvariables.AND-ORtrees anbe onvertedeasily

intoOR-ANDtreesandvi eversausingDeMorgan'slaws,ifweallownegations

to beaddedtothetree.

Considerthetwo-leveltree onN =n 2

variableswith anORasroot,ANDs

asits hildren,andfanoutninbothlevels.Ea hAND-subtree anbequantum

omputedbyGrover'salgorithm withone-sidederrorusingO(

p

n)queries(we

letGroversear hfora`0',andoutput1ifwedon'tnd any),andthevalueof

theOR-ANDtreeisjusttheORofthosenvalues.A ordingly,the onstru tion

ofthepreviousse tiongivesanO(

p

n
p

n)=O(

p

N)algorithmwithtwo-sided

error.Thisisoptimalupto a onstantfa tor[1℄.

Moregenerally,ford-levelAND-ORtreeswe anapplytheabovealgorithm

re ursivelyto obtainanalgorithm withO(

d 1 p

N)queries.Here isthe on-

stant hidden in the O(
) of the result of the previous se tion. For ea h xed

d, this omplexity is O(

p

N), whi h is optimal up to a onstant fa tor [2℄. It

improvesupontheO(

p

N(logN) d 1

)algorithmgivenin [9℄.

Our query omplexity upper bound also implies that the minimal degree

among N-variatepolynomialsapproximatingAND-OR isO(

p

N)[3℄.Whether

thisupperbound onthedegreeisoptimalremainsopen.Thebest knownlower

boundforthe2-level aseis(N 1=4

p

logN)[13℄.

5 Amplitude Ampli ation with Imperfe t Verier

Inthisse tionweviewour onstru tioninamoregenerallight.

Suppose weare givensome lassi alrandomizedalgorithmA that su eeds

insolvingsomeproblemwithprobabilityp.Inaddition,wearegivenaBoolean

fun tionthattakesasinputanoutputfromalgorithmA,andoutputswhether

itisasolutionornot.Then,wemayndasolutiontoourproblembyrepetition.

Werst apply algorithmA, obtainingsome andidate solution,whi hwethen

giveasinputtotheverier.Ifoutputsthatthe andidateindeedisasolution,

weoutput it and stop, and otherwise we repeat the pro ess by reapplying A.

Theprobabilitythatthispro essterminatesbyoutputtingasolutionwithinthe

rst(

1

p

)iterationsoftheloop,islowerboundedbya onstant.

A quantum analogueof boosting the probability of su ess is to boost the

amplitude ofbeingina ertainsubspa eofaHilbertspa e.Thusfar,amplitude

ampli ation[6℄hasassumedthatwearegivenaperfe tverier: whenevera

andidatesolutionisfound,we andeterminewith ertaintywhetheritisasolu-

tionornot.Formally,wemodelthisbylettingbe omputedbyadeterministi

lassi alsubroutineoranexa tquantumsubroutine.

Themain resultofthispapermaybeviewedasanadaptationofamplitude

ampli ation to the situation where the verier is not perfe t, but sometimes

tum subroutine,wearegivenabounded-errorquantumsubroutine. Previously,

the onlyknown te hniqueforhandlingsu h aseshasbeenby straightforward

simulationofaperfe t verier: onstru tasubroutinefor omputing wither-

ror 1

2 k

byrepeatingagivenbounded-errorsubroutine oforder (k)times and

thenusemajorityvoting.Usingsu hdire tsimulations,wemay onstru tgood

but sub-optimal quantum algorithms, liketheO(

p

nlogn)queryalgorithm for

quantumsear hoftheintrodu tion.Here,wehaveintrodu edamodi ationof

the amplitude ampli ationpro ess that allows us to eÆ iently dealwith im-

perfe tveriers.Essentially,ourresultsaysthatimperfe t veriersareasgood

asperfe tveriers(up toa onstantmultipli ativefa torinthe omplexity).

A knowledgments

We thankRi hard Clevefor usefuldis ussions, aswellasfor hostingMM and

RdWattheUniversityofCalgary,wheremostofthisworkwasdone.

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