RintonPress
BOOK REVIEW
on
An Introdu tion to QuantumComputing Algorithms
byArthurO.Pittenger
Birkhauser, De ember1999
Hard over$44.95 (138pages) ISBN: 0817641270
and
QuantumComputing
byMikaHirvensalo
Springer,May 2001
Hard over$44.95 (190pages) ISBN: 3540667830
and
Classi aland Quantum Computation
byA.Yu.Kitaev,A. Shen,andM.N.Vyalyi
Ameri an Mathemati al So iety,July 2002
Hard over$59.00 (272 pages) ISBN: 082182161X (soft overshould be onsiderably heaper)
A qui k surveyon amazon. omshowsthat the number ofbooks onquantum omputing
( 20)ismorethan10timesashighasthenumberofquantumalgorithmsthat wehave
today(2: Shor'sandGrover's). Manypeopleinthe eld,in ludingthisreviewer,feelthat
Nielsenand Chuang'sQuantum Computation andQuantum Informationis probablythe
best andmost omprehensiveof these. Nevertheless, thereis roomforsomeotherbooks
with di erentperspe tives. Here wereview and ompare three books that fo usmainly
onthealgorithmi side ofthe eld, andhen e ana ordto besigni antlyshorterthan
NielsenandChuang's675-pagevolume.
1. Pittenger
ArthurPittenger'sAnIntrodu tiontoQuantumComputingAlgorithmsre e tsitsau-
thor'sownexperien einlearningthemathemati sandtheoreti alphysi srequiredforthe
subje t, as he writes in the a knowledgments. It is generally written in apleasant and
informalstyle, with mu h motivation in betweenthe mathemati s. Of the three books
reviewedhereitis probablythemostreadable.
It onsistsofthreepartsofabout40pagesea h. The rstpart(Chapters1and2) overs
the omputationalmodel: statesand operations,somequantum me hani alba kground,
and the ir uit model of quantum gates, generalizing lassi al reversible ir uits. The
hapter doesnot go into the issue of approximating arbitrary ir uitsby a nite set of
basis gates. The se ond part(Chapter 3) re e ts thetitle of the book. It explains the
mainquantumalgorithms: Deuts h-Jozsa,Simon,Grover,Shor,andthegeneralizationof
thelattertotheAbelianhiddensubgroupproblem. The hapterisself- ontained,ex ept
for some of the number theory needed for the lassi al post-pro essing and analysis of
Shor'salgorithm. The nalpart(Chapter4)dis ussesquantumerror- orre tionindetail.
It overs the 9-qubit, 7-qubit, and 5-qubit odes, as well as the general framework of
stabilizer odesandCalderbank-Shor-Steane(CSS) odes. Fault-tolerantimplementation
ofgates is notdealtwith. Still,in just 120pagesthis book managesto explain mu h of
the oreofquantum omputing,andto explainitwell.
2. Hirvensalo
MikaHirvensalo'sQuantumComputingismorere entbut oversquitesimilarground.
It toois mu h shorterthan Nielsen-Chuangandhasa lear omputers ien efo us;one
wonderswhether Hirvensalo wasawareof the earlierPittenger book, whi h he doesnot
referen e. Hisstyleofwritingismoreformalandmathemati althanPittenger's.
Thesix hapters ofhis book explainthe ir uitmodel(likePittenger,without adis-
ussionofuniversalgatesets), Shor'salgorithm,thehidden subgroupproblem, Grover's
algorithm, and thepolynomialmethod for query omplexity lowerbounds. As abonus,
thelast 80 pages of thebook onsist of twovery extensiveappendi es. The rst overs
themathemati alfoundationsofquantumme hani s. Thisin ludes quiteafewadvan ed
topi s thatare notusedin themain text, su hasanentropi versionofthe un ertainty
relations, Gleason's theorem that every well-behaved assignment of probabilities omes
fromadensitymatrix,anda hara terizationof ompletelypositivemapsondensityma-
tri es. These ondappendix explainssomemathemati alba kground: grouptheory,the
dis reteFouriertransform,linearalgebra,numbertheoryin ludingthe ontinuedfra tion
expansionusedin Shor's lassi alpost-pro essing,andsomeinformationtheory.
3. Kitaev,Shen, and Vyalyi
The very re ent Classi al and Quantum Computation by Kitaev, Shen, and Vyalyi
(KSV) grew out of a ourse on lassi al and quantum omputing given by Kitaev and
Shenin Mos ow in 1999. It is atranslated and expanded versionof an earlier Russian
book,whi hisstillavailableforfreeathttp://www.m me.ru/free-booksforthosewho
anreadRussian
. Fromaresear her'sperspe tiveitisbyfarthemostinterestingofthe
threebooks,andIwill orrespondinglybemoredetailed indis ussingit.
Thebookhasthreeparts: lassi al omputing,quantum omputing, andsolutionsto
exer ises. The lassi alpartis quiteex ellent. Ina learand intuitivestyleofwritingit
des ribestheessentialsof thetheoryof lassi alalgorithmsand omplexity. This overs
Turingma hines, ir uits,reversible omputing,NP- ompleteness,randomizedalgorithms,
andthepolynomialhierar hy. This30-pageexpositionalsoin ludessomemoreadvan ed
resultslikeBPP P=poly andBPP
2 .
Thequantumpartofthebook(Chapters6to15)devotesabouthalfofits120pagestoa
thoroughexpositionofthequantum ir uitmodel,in ludingrepresentingorapproximating
arbitraryunitariesbymeansofelementarygates,quantum omputationwithmixedstates,
and a detailed a ount of measurement. After all the details of the ir uit model are
in pla e, the book ontinues in Chapter 13 to des ribe the phase estimation te hnique
(originallyduetoKitaev)andthewayit anbeusedtosolvetheAbelianhiddensubgroup
problem, in ludingfa toring and dis retelogarithms. Chapter 14 dealswith aquantum
versionofthe omplexity lassNPanda ompletepromiseproblemforthis lass. Finally,
Chapter 15 des ribes quantum error- orre ting odes and ends with abrief des ription
of Kitaev's work on tori odes and anyons, where error orre tion would be a natural
propertyoftheunderlyingphysi alsystemitself.
Apart from its on iseness and rigor, one of the main strengths of this book is the
attention it givesto Kitaev's ontributionsto quantum omputing. These in lude ade-
tailed analysis of eÆ ient approximation of arbitrary ir uits using only gates from a
spe i nite basis, the Abelian hidden subgroup problem, quantum NP- ompleteness,
andtori odes. These topi sareexplainedinmu hdetailandwithmanysubtletiesand
insightsthat areoften glossed over in other presentations|and for things like quantum
NP- ompleteness there is no otherpresentation. Agood understanding ofthe quantum
partofthisbook (in ludingthe exer isesand theirsolutions)will provide theresear her
withinvaluableinsightsandtoolsfornewresear h.
At the same time, this bias towardsKitaev's quantum work may also be viewed as
aweakness of this book. Even if onerestri ts attention to omputer s ien e aspe ts of
quantum omputing,variousthingsaremissing. Theworkofanumberofkeyresear hers
is ompletelyignored,in ludingthatofAndrisAmbainis,HarryBuhrman,DanielGottes-
man, Peter H yer, Ri hard Jozsa,Mi hele Mos a,and Umesh Vazirani. Between them,
thesepeoplehave ontributedalargefra tionofthemainresultsonquantumalgorithms
and omplexity, yet none of their papersis even ited. The Deuts h-Jozsa algorithm is
absent;thereisnothingabouttheappli ationsofGrover'salgorithmin ounting, ollision-
nding et . The result that Grover's algorithm is optimal for quantum sear h is only
mentionedinpassingandthepaperthat rstprovedthis(Bennett, Bernstein,Brassard,
andVazirani \Strengths and weaknessesof quantum omputing") is not ited. There is
nothingelse on lowerbounds, virtuallynothing aboutquantum ommuni ationor om-
muni ation omplexity,noquantum ryptography,et . Thebook's oversuggestsusingit
asatextbookforagraduate ourseonquantum omputing,but Ifearthatsu ha ourse
wouldgiveasomewhatbiasedviewof the eld.
Ase ondproblemwithusingthisbookfora ourseisthedisparitybetweenits lassi al
andquantumparts. Theseareapparentlywrittenpredominantlybydi erentauthors. The
lassi alpartisgenerallyverywellandintuitivelywritten,anddoesnotpresupposemu h.
Ontheotherhand,thequantumpartisamu hlesssmoothread. Itissigni antlymore
demanding andnotquite self- ontained. Forinstan e, theproofthat thestandardbasis
of elementary gates an eÆ iently approximate ir uits over other bases (Se tion 8.3)
assumessomeknowledgeofLiegroupsandLiealgebras,andthelastse tionsabouterror-
orre ting odesassumesomea quaintan ewithhomologyofmanifolds. Again,thismay
beproblemati whenusingKSVasatextbookfora ourse,sin emoststudentswillnotbe
veryfamiliarwiththismaterial. Asanotherexample,theuseofquantumphaseestimation
in quantum algorithms is originally due to Kitaev, but the exposition of his method in
Mos aismu hmore learthantheexpositiongivenhere.
The abovepointsnotwithstanding, alot anbe learnedfrom this book; mu h more
thanfrom theother two,but itrequires agreatere ortbythereader. Thisis ne when
that readerisaresear her|andthat is probablywhere thebook willbeused themost:
asavaluableresour eforpeoplewhowanttolookuporlearntheintri a iesofthingslike
the ir uitmodel, quantum NP- ompleteness,et .
4. Comparisonand on lusion
Theoverlapbetweenthesethreebooksisquitelarge. Allthreeareexpli itlyoriented
towards omputer s ien e, devoting most of their pages to the quantum ir uit model
andthe main quantum algorithms. The ontentsof the Pittengerand Hirvensalobooks
in parti ular are very lose. The main di eren es are that Pittenger has a hapter on
error- orre tionand his style of writingis somewhat more informaland intuitive, while
Hirvensalohasa hapteronlowerboundsandsomemoremathemati alba kground(su h
as ontinuedfra tions).
The KSV book o ersmore than the other two books. This in ludes a su in t but
veryni eintrodu tionto alot of lassi al omplexity theory, amorein-depthdis ussion
ofthequantum ir uitmodel,and topi slikequantumNP- ompleteness andtori odes
thatare notreadilyavailable inanyotherbooks. Onthedownside, Ifound itsquantum
partmoredemandingandharderto readattimesthantheothertwobooks.
All three books are pre ise and reasonablysu in t introdu tions to the algorithmi
aspe tsofthe eldofquantum omputation. Assu htheywillbemostusefulto omputer
s ientists and mathemati ians who want to learn about the algorithms without being
bothered too mu h by the physi s. The books are suitable for a 1-semester ourse on
quantumalgorithms(omittingsomeofthemoreadvan edse tionsinthe aseofKSV).All
threebookshavemanyexer ises,butKSVistheonlyonethatgivesthesolutionsaswell.
Hirvensalodis ussessomelowerbounds butPittenger andKSV donot,and noneofthe
books treat ommuni ation-based topi s likequantum ryptography, hannel apa ities,
or ommuni ation omplexity,northevarious appli ationsofGrover'salgorithm.
So,whi hbookto hoose? Ifyouwantaverya essible rstintrodu tiontoquantum
algorithms,Iwouldre ommendPittenger'sbook. Ifyouprefermoreformalmathemati s,
Hirvensalo also gives agood rst introdu tion to roughly the same topi s. If you have
suÆ ient mathemati al maturity and/or are prepared to do some work while reading,
thengo forthe KSVbook. Foreveryonewith abroaderinterestin quantum omputing
(in luding quantum information theory), I would still re ommend Nielsen and Chuang
overthesebooks. It ontainsmu h morematerial, isveryreadable,andnotsigni antly
moreexpensivethanthethreebooksdis ussedhere.
Finally,toallthose urrentlyworkingonyetanotherbookaboutquantum omputing:
whatthis eld needsmostismorealgorithms,notmorebooks.
Ronaldde Wolf(rdewolf wi.nl)
CWI
Kruislaan413
1098SJAmsterdam
