Improved Quantum Communication Complexity Bounds for Disjointness and Equality

PeterHyer 1;?

andRonalddeWolf 2;??

1

Dept.ofComp.S i.,Univ.ofCalgary,AB,Canada.hoyer ps .u algary. a

2

UCBerkeley.583SodaHall,BerkeleyCA94720,USA.rdewolf s.berkeley.edu

Abstra t. Weprovenewboundsonthequantum ommuni ation om-

plexity ofthe disjointnessand equality problems.Forthe ase of exa t

andnon-deterministi proto olsweshowthatthese omplexitiesareall

equalton+1,thepreviousbestlowerboundbeingn=2.Weshowthisby

improvingageneralboundfornon-deterministi proto olsofdeWolf.We

alsogiveanO(

p

n
log



n

)-qubitbounded-errorproto olfordisjointness,

modifyingandimprovingtheearlier O(

p

nlogn)proto olofBuhrman,

Cleve, andWigderson, andprovean( p

n )lowerboundfor a lassof

proto olsthatin ludestheBCW-proto olaswellasournewproto ol.

1 Introdu tion

The area of ommuni ation omplexity deals with abstra ted models of dis-

tributed omputing,whereoneonly aresaboutminimizingtheamountof om-

muni ationbetweenthepartiesandnotabouttheamountof omputationdone

by the individual parties. The standard setting is the following. Two parties,

Ali e and Bob,wantto omputesome fun tion f : f0;1g n

f0;1g n

!f0;1g.

Ali e re eives input x 2 f0;1g n

, Bob re eives y 2 f0;1g n

, and they want to

ompute f(x;y). Forexample,they maywant to nd outwhether x = y (the

equality problem) orwhether x and y are hara teristi ve torsof disjoint sets

(thedisjointnessproblem).A ommuni ationproto olisadistributedalgorithm

where Ali e rstdoessome omputation on herside, then sends amessage to

Bob, whodoes some omputation on his side, sends a messageba k, et . The

ostoftheproto olismeasuredbythenumberofbits(orqubits,inthequantum

ase) ommuni atedonaworst- aseinput(x;y).

Asinmanyotherbran hesof omplexitytheory,we andistinguishbetween

variousdierent\modes"of omputation.LettingP(x;y)denotethea eptan e

probability of the proto ol (the probability of outputting 1), we onsider four

dierentkindsofproto olsfor omputingf,

{ Anexa tproto olhasP(x;y)=f(x;y),forallx;y

?

SupportedinpartbyCanada'sNSERCandthePa i InstitutefortheMathema-

ti alS ien es.

??

SupportedbyTalentgrantS62{565fromtheNetherlandsOrganizationforS ienti

Resear h.Work ondu tedwhile at CWI, Amsterdam,partially supportedby EU

{ Anon-deterministi proto olhasP(x;y)>0if andonlyiff(x;y)=1,for

allx;y

{ Aone-sidederrorproto olhasP(x;y)1=2iff(x;y)=1,andP(x;y)=0

iff(x;y)=0

{ Atwo-sidederrorproto olhasjP(x;y) f(x;y)j1=3,forallx;y.

These four modes of omputation orrespond to those of the omputational

omplexity lassesP,NP,RP,andBPP,respe tively.

Proto olsmaybe lassi al(sendandpro ess lassi albits)orquantum(send

andpro essquantumbits).Classi al ommuni ation omplexitywasintrodu ed

byYao[35℄,andhasbeenstudiedextensively.Itiswellmotivatedbyitsintrinsi

interest as well asby its appli ations in lowerbounds on ir uits, VLSI, data

stru tures,et .WerefertothebookofKushilevitzandNisan[26℄fordenitions

andresults. WeuseD(f), N(f),R

1

(f),and R

2

(f)to denote theminimal ost

of lassi alproto olsforf in theexa t,non-deterministi ,one-sidederror,and

two-sidederrorsettings,respe tively. 1

NotethatR

2

(f)R

1

(f)D(f)n+1

andN(f)R

1

(f)D(f)n+1forallf.SimilarlywedeneQ

E

(f),NQ(f),

Q

1

(f),andQ

2

(f)forthequantumversionsofthese ommuni ation omplexities

(we will be abit morepre ise about the notionof a quantum proto ol in the

nextse tion).Forallof these omplexities, weassumeAli eand Bobstartout

withoutanysharedrandomnessorentanglement.

Quantum ommuni ation omplexity was introdu ed by (again) Yao [36℄

and the rst examples of fun tions where quantum ommuni ation omplex-

ity is less than lassi al ommuni ation omplexity were given in [14,10,11,

15,9℄. In parti ular, Buhrman, Cleve, and Wigderson [9℄ showed for a spe-

i promise version of the equality problem that Q

E

(f) 2 O(logn) while

D(f)2(n).Theyalsoshowedfortheinterse tion problem(thenegationofthe

disjointnessproblem)thatQ

1 (INT

n )2O(

p

nlogn),whereasR

2 (INT

n

)2(n)

is awell known and non-trivial result from lassi al ommuni ation omplex-

ity [20,31℄. Later, Raz [30℄ exhibited a promise problem with an exponential

quantum- lassi algapeveninthebounded-errorsetting:Q

2

(f)2O(logn)ver-

susR

2

(f)2(n 1=4

=logn).Otherresultsonquantum ommuni ation omplex-

itymaybefoundin [25,2,28,13,21,34,24,23℄.

Theaimof this paperis to sharpenthe bounds on thequantum ommuni-

ation omplexities ofthe equalityand disjointness(or interse tion) problems,

inthefourmodeswedistinguishedabove.Wesummarizewhatwasknownprior

to thispaper,

{ n=2Q

1 (EQ

n );Q

E (EQ

n

)n+1[25,13℄

n=2NQ(EQ

n

)n+1[34℄

Q

2 (EQ

n

)2(logn)[25℄

{ n=2Q

1 (DISJ

n );Q

E (DISJ

n

)n+1[25,13℄

n=2NQ(DISJ

n

)n+1[34℄

lognQ

1 (INT

n );Q

2 (DISJ

n )2O(

p

nlogn)[9℄.

1

KushilevitzandNisan [26℄useN 1

(f) forourN(f), R 1

(f)for ourR1(f)andR (f)

forourR (f).

InSe tion3werstsharpenthenon-deterministi bounds,byprovingageneral

algebrai hara terizationofNQ(f).In[34℄itwasshownforallfun tionsf that

lognrank(f)

2

NQ(f)log(nrank(f))+1;

where nrank(f) denotesthe rank ofa \non-deterministi matrix" for f (to be

dened more pre isely below). It is interesting to note that in many pla es

in quantum omputing one sees fa tors of 1

2

appearing that are essential, for

example in the query omplexity of parity [4,17℄, in the bounded-error query

omplexity of allfun tions [16℄, in superdense oding [5℄, and in lowerbounds

forentanglement-enhan edquantum ommuni ation omplexity[13,28℄.In on-

trast,weshowherethatthe 1

2

intheabovelowerbound anbedispensedwith,

andtheupperbound istight, 2

NQ(f)=log(nrank(f))+1:

Equality and disjointness both have non-deterministi rank 2 n

, so their non-

deterministi omplexities are maximal: NQ(EQ

n

) = NQ(DISJ

n

) = n+1.

(This ontrastswiththeir omplements:NQ(NEQ

n

)=2[27℄andNQ(INT

n )

N(INT

n

)=logn+1.)Sin eNQ(f)lowerboundsQ

1

(f)andQ

E

(f),wealsoob-

tainoptimalboundsfortheone-sidedandexa tquantum ommuni ation om-

plexities of equality and disjointness. In parti ular, Q

E (EQ

n

) = n+1, whi h

answersaquestion posed to one ofus (RdW) by GillesBrassardin De ember

2000.

The two-sided error bound Q

2 (EQ

n

) 2 (logn) is easy to show, whereas

thetwo-sidederror omplexityofdisjointnessisstillwideopen.InSe tion4we

give a one-sidederror proto ol for the interse tion problem that improvesthe

O(

p

nlogn)proto olofBuhrman,Cleve,andWigdersonbynearlyalog-fa tor,

Q

1 (INT

n )2O(

p

n
log

?

n

);

where is a (small) onstant. The fun tion log

?

n is dened as the minimum

number of iterated appli ations of the logarithm fun tion ne essary to obtain

a number less than or equalto 1: log

?

n = minfr  0 j log (r)

n  1g, where

log (0)

isthe identityfun tion and log (r)

=logÆlog (r 1)

. Eventhough log

?

n

is

exponentialin log

?

n,itisstillverysmallinn,in parti ular log

?

n

2o(log (r)

n)

forevery onstantr1.Itshouldbenotedthatourproto olisasymptoti ally

somewhat more eÆ ient than the BCW-proto ol ( p

n log

?

n

versus p

nlogn),

butisalsomore ompli atedtodes ribe;itisbasedonare ursivemodi ation

oftheBCW-proto ol,anideathat previouslyhasbeenusedfor law-ndingby

Buhrmanet al.[12,Se tion5℄.

Proving good lower bounds on the Q

2

- omplexity of the disjointness and

interse tion problems isoneof themain openproblems in quantum ommuni-

ation omplexity. Onlylogarithmi lowerbounds areknown so farforgeneral

2

Similarly we an improve the query omplexity result ndeg (f)=2  NQ

q (f) 

ndeg (f)of[34℄totheoptimalNQ (f)=ndeg (f).

proto ols [25,2,13℄. A lower bound of (n 1=k

) is shown in [24℄ for proto ols

ex hangingat mostk 2O(1) messages.InSe tion 4.1 we proveanearly tight

lowerboundof( p

n)qubitsof ommuni ationforallproto olsthatsatisfythe

onstraintthattheira eptan eprobabilityisafun tionofx^y(then-bitAND

ofAli e'sxandBob'sy),ratherthanofxandy\separately."Sin eDISJ

n itself

is also afun tion only of x^y, this does notseem to bean extremely strong

onstraint.The onstraintis satised by a lass of proto ols that in ludes the

BCW-proto olandournewproto ol.Itseemsplausiblethatthegeneralbound

isQ

2 (DISJ

n )2(

p

n)aswell,but wehavesofarnotbeenabletoweakenthe

onstraintthat thea eptan eprobabilityisafun tionof x^y.

2 Preliminaries

2.1 Quantum Computing

Herewebrie ysket hthesettingofquantum omputation,referringtothebook

ofNielsenandChuang[29℄formoredetails.Anm-qubitquantumstatejiisa

superpositionorlinear ombinationoverall lassi alm-bitstates,

ji= X

i2f0;1g m



i jii;

withthe onstraintthat P

i j

i j

2

=1.Equivalently,jiisaunit ve torinC 2

m

.

Quantum me hani s allows us to hange this state by means of unitary (i.e.,

norm-preserving) operations: j

new

i = Uji, where U is a 2 m

2 m

unitary

matrix. A measurement of ji produ es the out ome i with probability j

i j

2

,

andthenleavesthesystemin thestatejii.

Thetwomain examplesof quantum algorithmsso far,are Shor'salgorithm

for fa toring n-bit numbers using a polynomial number (in n) of elementary

unitarytransformations[32℄andGrover'salgorithmforsear hinganunordered

n-element spa e using O(

p

n) \look-ups" or queries in the spa e [18℄. Below

we use a te hnique alled amplitude ampli ation, whi h generalizes Grover's

algorithm.

Theorem 1 (Amplitude ampli ation [7℄). There exists a quantum algo-

rithm QSear h with the following property. Let A be any quantum algorithm

thatusesnomeasurements,andlet:f1;:::;ng!f0;1gbeanyBooleanfun -

tion. Let adenote the initial su ess probability of Aof nding asolution (i.e.,

the probability of outputting some i 2 f1;:::;ng so that (i) = 1). Algorithm

QSear hnds asolution using anexpe ted numberof O



1

p

a



appli ations of

A, A 1

,and ifa>0,anditruns forever ifa=0.

Considertheproblem of sear hing anunordered n-elementspa e. An algo-

rithmAthat reatesauniformsuperpositionoveralli2f1;:::;nghassu ess

probabilitya 1=n, soplugging this into theabovetheorem andterminating

afterO(

p

n)appli ationsgivesusanalgorithmthatndsasolutionwithprob-

2.2 Communi ation Complexity

For lassi al ommuni ation proto ols we refer to [26℄. Here we brie y dene

quantum ommuni ationproto ols,referringtothesurveys[33,8,22,6℄formore

details.Thespa einwhi hthequantumproto olworks, onsistsofthreeparts:

Ali e's part,the ommuni ation hannel, andBob's part(wedo notwrite the

dimensions of these spa es expli itly). Initially these three parts ontain only

0-qubits,

j0ij0ij0i:

WeassumeAli estartstheproto ol.SheappliesaunitarytransformationU A

1 (x)

to her partand the hannel. This orresponds to her initial omputation and

her rst message.The length of this message is the number of hannel qubits

ae ted. Thestateisnow

(U A

1 (x)I

B

)j0ij0ij0i;

wheredenotestensorprodu t,andI B

denotestheidentitytransformationon

Bob'spart. ThenBob applies aunitary transformationU B

2

(y) to his partand

the hannel.This operation orrespondsto Bobreading Ali e'smessage,doing

some omputation, and puttinga return-messageon the hannel. This pro ess

goesba k andforth for somek messages, so the nal stateof the proto ol on

input(x;y)willbe(in aseAli egoeslast)

(U A

k (x)I

B

)(I A

U B

k 1

(y))


(I A

U B

2 (y))(U

A

1 (x)I

B

)j0ij0ij0i:

Thetotal ostoftheproto olisthetotallengthofallmessagessent,onaworst-

ase input(x;y). Forte hni al onvenien e, weassume that at the end of the

proto ol theoutput bit is the rst qubit on the hannel. Thus thea eptan e

probabilityP(x;y)oftheproto olistheprobabilitythatameasurementofthe

nalstategivesa`1'intherst hannel-qubit.Notethatwedonotallowinter-

mediate measurements during the proto ol.This is without loss of generality:

itiswellknownthat su hmeasurements anbepostponeduntiltheendofthe

proto olatnoextra ommuni ation ost.Asmentionedin theintrodu tion,we

useQ

E

(f),NQ(f),Q

1

(f),andQ

2

(f)to denotethe ostofoptimalexa t,non-

deterministi ,one-sidederror,and two-sidederrorproto olsforf,respe tively.

Thefollowinglemma wasstatedsummarily withoutproofby Yao [36℄ and

in moredetailbyKremer[25℄.Itiskeyto manyoftheearlierlowerboundson

quantum ommuni ation omplexityaswellasto ours,andis easilyprovenby

indu tion on`.

Lemma2 (Yao [36℄;Kremer[25℄).Thenalstateofan `-qubitproto ol on

input(x;y) an bewrittenas

X

i2f0;1g

` jA

i (x)iji

` ijB

i (y)i ;

where the A

i (x);B

i

(y) are ve tors (not ne essarily of norm 1), andi

`

denotes

Thea eptan eprobabilityP(x;y)oftheproto olisthesquarednormofthe

partofthenalstatethathasi

`

=1.Lettinga

ij

bethe2 n

-dimensional omplex

olumnve torwiththeinnerprodu tshA

i (x)jA

j

(x)iasentries,andb

ij the2

n

-

dimensional olumn ve torwith entries hB

i (y)jB

j

(y)i , we anwrite P (viewed

as a 2 n

2 n

matrix) as the sum P

i;j:i

`

=j

`

=1 a

ij b

T

ij of 2

2` 2

rank 1 matri es,

so therank of P is at most 2 2` 2

. Forexample, for exa tproto ols this gives

immediatelythat`islowerboundedby 1

2

timesthelogarithmoftherankofthe

ommuni ationmatrix,and fornon-deterministi proto ols ` islowerbounded

by 1

2

timesthe logarithmof thenon-deterministi rank(denedbelow).Inthe

nextse tionweshowhowwe angetridofthefa tor 1

2

inthenon-deterministi

ase.

Weusex^ytodenotethebitwise-ANDofn-bitstringsxandy,andsimilarly

xy denotes thebitwise-XOR.LetORdenotethen-bitfun tion whi his1if

at leastoneofits ninputbits is1,and NORbeitsnegation.We onsiderthe

followingthree ommuni ation omplexityproblems,

{ Equality:EQ

n

(x;y)=NOR(xy)

{ Interse tion:INT

n

(x;y)=OR(x^y)

{ Disjointness:DISJ

n

(x;y)=NOR(x^y).

3 Optimal Non-Deterministi Bounds

Let f : f0;1g n

f0;1g n

! f0;1g. A 2 n

2 n

omplex matrix M is alled a

non-deterministi matrixforf ifithasthepropertythatM

xy

6=0ifandonlyif

f(x;y)=1(equivalently, M

xy

=0ifandonlyiff(x;y)=0).Weusenrank(f)

todenotethenon-deterministi rankoff,whi histheminimalrankamongall

non-deterministi matri esforf.In[34℄ itwasshownthat

lognrank(f)

2

NQ(f)log(nrank(f))+1:

Inthisse tionweshowthat theupperboundisthetruebound.Theproofuses

thefollowingte hni allemma.

Lemma3. If thereexisttwofamilies of ve torsfA

1

(x);:::;A

m

(x)gC d

and

fB

1

(y);:::;B

m

(y)gC d

su hthat for allx2f0;1g n

andy2f0;1g n

,we have

m

X

i=1 A

i

(x)B

i

(y)=0if andonlyif f(x;y)=0;

thennrank(f)m.

Proof. Assume there exist twosu h families of ve tors. LetA

i (x)

j

denote the

jthentryofve torA

i

(x),andletsimilarlyB

i (y)

k

denotethekthentryofve tor

B

i

(y). We use pairs (j;k) 2 f1;:::;dg 2

to index entries of ve torsin the d 2

-

iff(x;y)=0then m

i=1 A

i (x)

j B

i (y)

k

=0forall(j;k),

iff(x;y)=1then P

m

i=1 A

i (x)

j B

i (y)

k

6=0forsome(j;k).

Asarststep,wewanttorepla etheve torsA

i

(x)andB

i

(y)bynumbersa

i (x)

and b

i

(y) that have similar properties. Weuse the probabilisti method [1℄ to

showthat this anbedone.

LetI be an arbitraryset of 2 2n+1

numbers. Choose oeÆ ients 

1

;:::;

d

and

1

;:::;

d

,ea h oeÆ ientpi keduniformlyatrandomfromI.Foreveryx,

dene a

i (x) =

P

d

j=1 

j A

i (x)

j

, and for everyy dene b

i (y) =

P

d

k =1 

k B

i (y)

k .

Considerthenumber

v(x;y)= m

X

i=1 a

i (x)b

i (y)=

d

X

j;k =1 

j 

k m

X

i=1 A

i (x)

j B

i (y)

k

!

:

Iff(x;y)=0,thenv(x;y)=0forall hoi esofthe

j

;

k .

Now onsidersome(x;y)withf(x;y)=1.There isapair(j 0

;k 0

)forwhi h

P

m

i=1 A

i (x)

j 0

B

i (y)

k 0

6=0.Wewanttoprovethat v(x;y)=0happensonlywith

very small probability. In order to do this, x the random hoi es of all 

j ,

j 6= j 0

, and 

k

, k 6= k 0

, and view v(x;y) as a fun tion of the two remaining

not-yet- hosen oeÆ ients=

j 0

and =

k 0

,

v(x;y)=

0 +

1 +

2 +

3 :

Hereweknowthat

0

= P

m

i=1 A

i (x)

j 0B

i (y)

k

0 6=0.Thereisatmostonevalueof

forwhi h

0 +

2

=0.Allothervaluesofturnv(x;y)intoalinearequationin

,soforthosethereisatmostone hoi eof thatgivesv(x;y)=0.Hen eout

ofthe(2 2n+1

) 2

dierentwaysof hoosing(;),atmost2 2n+1

+(2 2n+1

1)
1<

2 2n+2

hoi esgivev(x;y)=0.Therefore

Pr[v(x;y)=0℄<

2 2n+2

(2 2n+1

) 2

=2 2n

:

Usingtheunionbound,wenowhave

Pr



there isan(x;y)2f 1

(1)forwhi hv(x;y)=0





X

(x;y)2f 1

(1)

Pr[v(x;y)=0℄<2 2n

2 2n

=1:

Thisprobabilityisstri tlylessthan1,sothereexistsetsfa

1

(x);:::;a

m

(x)gand

fb

1

(y);:::;b

m

(y)gthatmakev(x;y)6=0forevery(x;y)2f 1

(1).Wethushave

that

m

X

i=1 a

i (x)b

i

(y)=0ifandonlyiff(x;y)=0:

Viewthea

i andb

i as2

n

-dimensionalve tors,letAbethe2 n

mmatrixhaving

the a

i

as olumns, and B be the m2 n

matrix having the b

i

as rows. Then

(AB)

xy

= P

m

i=1 a

i (x)b

i

(y), whi h is 0ifand onlyiff(x;y)=0. Thus AB isa

Lemma 3 allows us to prove tight bounds for non-deterministi quantum

proto ols.

Theorem4. NQ(f)=log (nrank(f))+1.

Proof. TheupperboundNQ(f)log(nrank(f))+1wasshownin[34℄(a tually,

the upperbound shownthere waslog (nrank(f))for proto ols where onlyBob

hastoknowtheoutputvalue).Forthesakeof ompletenesswerepeatthatproof

here.Letr=nrank(f)and M bearank-r non-deterministi matrixfor f.Let

M T

=UV bethesingularvaluede ompositionofthetransposeofM [19℄,so

U andV areunitary,andisadiagonalmatrixwhoserstrdiagonalentriesare

positiverealnumbersandwhoseotherdiagonalentriesare0.Belowwedes ribe

aone-roundnon-deterministi proto olforf,usinglog(r)+1qubits.FirstAli e

preparesthe statej

x i=

x

Vjxi, where

x

>0isanormalizingrealnumber

that depends onx.Be ause onlytherstrdiagonalentries of arenon-zero,

onlytherstramplitudesofj

x

iarenon-zero,soj

x

i anbe ompressedinto

logr qubits. Ali esends these qubits to Bob. Bob then applies U to j

x i and

measures theresulting state.If heobservesjyi, then heputs 1 onthe hannel

andotherwiseheputs0there. Thea eptan eprobabilityof thisproto olis

P(x;y)=jhyjUj

x ij

2

= 2

x

jhyjUVjxi j 2

= 2

x jM

T

yx j

2

= 2

x jM

xy j

2

:

Sin e M

xy

isnon-zeroifand onlyiff(x;y)=1,P(x;y)will bepositiveifand

only iff(x;y) =1.Thus wehave anon-deterministi quantum proto ol for f

withlog(r)+1qubitsof ommuni ation.

Forthelowerbound, onsideranon-deterministi `-qubitproto olforf.By

Lemma2,itsnalstateoninput(x;y) anbewrittenas

X

i2f0;1g

` jA

i (x)iji

` ijB

i (y)i :

Without lossofgeneralityweassume theve torsA

i

(x) andB

i

(y)allhavethe

samedimension d.Let S =fi2 f0;1g

`

j i

`

=1g and onsider thepartof the

statethat orrespondstooutput1(wedropthei

`

=1andthej
i -notationhere),

(x;y)= X

i2S A

i

(x)B

i (y):

Be ausetheproto olhasa eptan eprobability0ifandonlyiff(x;y)=0,this

ve tor(x;y) will be the zero ve tor if and only if f(x;y) = 0.The previous

lemmagivesnrank(f)jSj=2

` 1

,andhen ethatlog(nrank(f))+1NQ(f).

u t

Note that any non-deterministi matrix for the equality fun tion has non-

zeroes on its diagonal and zeroes o-diagonal, and hen e has full rank. Thus

NQ(EQ

n

) = n+1, whi h ontrasts sharply with the non-deterministi om-

plexity of its omplement (inequality), whi h is only 2 [27℄. Similarly, a non-

deterministi matrixfordisjointnesshasfull rank,be ausereversingtheorder-

the diagonal. This gives tight bounds for the exa t, one-sided error, and non-

deterministi settings.

Corollary 5. We have that Q

E (EQ

n ) = Q

1 (EQ

n

) = NQ(EQ

n

) = n+1 and

that Q

E (DISJ

n )=Q

1 (DISJ

n

)=NQ(DISJ

n

)=n+1.

4 On the Bounded-Error Complexity of Disjointness

4.1 Improved Upper Bound

Hereweshowthatwe antakeomostofthelognfa torfromtheO(

p

nlogn)

proto olfortheinterse tionproblem(the omplementofdisjointness)thatwas

givenbyBuhrman,Cleve,andWigdersonin[9℄.

Theorem6. Thereexistsa onstant su hthat Q

1 (INT

n )2O(

p

n
log

?

n

).

Proof. Were ursivelybuildaone-sidederrorproto olthat anndanindex i

su h that x

i

= y

i

= 1, provided su h an i exists ( all su h an i a `solution').

Clearly this suÆ es for omputing INT

n

(x;y). Let C

n

denote the ost of our

proto olonn-bitinputs.

Ali eandBobdividethenindi esf1;:::;nginton=(logn) 2

blo ksof(logn) 2

indi esea h.Ali epi ksarandomnumberj 2f1;:::;n=(logn) 2

gandsendsthe

numberj toBob.Nowtheyre ursivelyrunourproto olonthejthblo k,ata

ost of C

(logn) 2

qubits of ommuni ation. Ali e then measures her partof the

state,andtheyverifywhetherthemeasurediis indeedasolution.Ifthereisa

solutioninthejthblo k,thenAli endsonewithprobabilityatleast1=2,sothe

overallprobabilityofndingasolution(ifthereisone)isatleast(logn) 2

=2n.By

usingasuperpositionoveralljwe anpushallintermediatemeasurementstothe

endwithoutae tingthesu essprobability.Therefore,applyingO(

p

n=logn)

rounds of amplitude ampli ation(Theorem 1) boosts this proto ol to having

errorprobabilityat most1=2.Wethushavethere urren e

C

n

O(1) p

n

logn C

(logn)

2+O(logn)

:

Sin e C

1

= 2, this re ursion unfolds to the bound C

n 2 O(

p

n
log

?

n

) for

some onstant .Carefulinspe tion oftheproto ol givesthat the onstant is

reasonablysmall. ut

4.2 LowerBound fora Spe i Class of Proto ols

We givea lowerbound for two-sidederror quantum proto ols for disjointness.

The lowerbound applies to allproto olswhose a eptan eprobabilityP(x;y)

isafun tionjustofx^y,ratherthanofxandy \separately."Inparti ular,the

proto olsof[9℄andofourpreviousse tion fallinthis lass.

Theorem7. Anytwo-sidederrorquantumproto olforDISJ

n

whosea eptan e

probability isafun tion of x^y,has to ommuni ate ( p

n)qubits.

Proof. Consideran`-qubitproto olwitherrorprobabilityatmost1=3.Bythe

ommentfollowingLemma2,we anwriteitsa eptan eprobabilityP(x;y)as

a2 n

2 n

matrixP ofrankr2 2` 2

.

WenowinvokearelationbetweentherankofthematrixP andpropertiesof

the2n-variatemultilinearpolynomialthatequalsP(x;y).

3

Thereisann-variate

fun tiong su hthatP(x;y)=g(x^y).Letg(z)= P

S a

S z

S

bethepolynomial

representationofg.ThenP(x;y)=g(x^y)= P

S a

S (x^y)

S

= P

S a

S x

S y

S ,so

the2n-variatemultilinearpolynomialP only ontainsmonomialsin whi h the

set of x-variables is thesame astheset of y-variables. Forpolynomialsof this

form( alled\even"),[13,Lemmas2and3℄implythatthenumberofmonomials

in P(x;y)equalstherankr ofthematrixP.

Settingy=xinP(x;y)givesapolynomialp(x)= P

S a

S x

S

thathasrmono-

mialsand thatapproximatesthen-bitfun tion NOR,sin ejp(x) NOR(x)j=

jP(x;x) DISJ (x;x)j  1=3. But [13, Theorem 8℄implies that every polyno-

mial that approximates NOR must have at least 2 p

n=12

monomials. Hen e

2 p

n=12

r2 2` 2

,whi h gives` p

n=48+1. ut

5 Open Problems

This paperts inasequen e ofpapersthat (slowly)extendwhat isknownfor

quantum ommuni ation omplexity, e.g. [9,2,30,13,21,34,24,23℄. The main

open question isstill the bounded-error omplexityof disjointness.Of interest

is whether itis possibleto proveanO(

p

n) upperbound fordisjointness,thus

gettingridofthefa torof log

?

n

inourupperboundofTheorem6,andwhether

it is possible to extend the lower bound of Theorem 7 to broader lasses of

proto ols. Sin e disjointness is oNP- omplete for ommuni ation omplexity

problems [3℄, stronglowerbounds onthe disjointness problem implya hostof

otherlowerbounds.

A se ond questionis whether qubit ommuni ation anbe signi antlyre-

du edin aseAli eandBob anmakeuseofpriorentanglement(shared EPR-

pairs).GivingAli eandBobnsharedEPR-pairstrivializesthenon-deterministi

omplexity(usetheEPR-pairsasapubli ointorandomlyguesssomen-bitz,

Ali ethensendsBob1bitindi atingwhetherx=z,ifx=zthenBob an om-

pute theanswerf(x;y)and sendit toAli e,if x6=z thentheyoutput 0),but

for theexa tand bounded-error models itis open whether priorentanglement

an makeasigni antdieren e.

3

ForS[n℄=f1;:::;ng,weusexSforthemonomial Q

i2S

xi.Ann-variatemultilin-

earpolynomialp(x)= P

S[n℄

aSxS,aS2R,isaweightedsumofsu hmonomials.

The number of monomials in p is the number of S for whi h aS 6= 0. One an

showthat foreveryfun tiong:f0;1g n

!Rthereisauniquen-variatemultilinear

n

A knowledgments

WethankHarryBuhrmanandHartmutKlau kforhelpful dis ussions on ern-

ing the proof of Lemma 3, and Ri hard Cleve for helpful dis ussions on the

proto olfordisjointness.

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