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computationalcomplexitythenatleasttocomputationmoregenerally. Giventhesteadydecrease
inthesizeofcomputingcomponents,itisinevitablethatquantummechanicswillbecomeincreas-
ingly relevant to the construction of computers—for quantum mechanics provides a remarkably
accurate description of extremely small physical systems (on the scale of atoms) where classical
physical theories have failed completely. Indeed, an extrapolation of Moore’s Law predicts sub-
atomiccomputingcomponentswithinthenexttwodecades[83,78];apossibilityinconsistentwith
quantummechanicsasitiscurrentlyunderstood.
Thatquantummechanicsshouldhaveimplicationstocomputationalcomplexitytheory,how-
ever, is much less clear. It is only through the remarkable discoveries and ideas of several re-
searchers, including Richard Feynman [50], David Deutsch [41, 42], Ethan Bernstein and Umesh
Vazirani [30, 31], and Peter Shor [93, 94], that this potential has become evident. In particular,
Shor’s polynomial-time quantum factoring and discrete-logarithm algorithms [94] give strong
supporttotheconjecturethatquantumandclassicalcomputersyielddifferingnotionsofcompu-
tationalhardness. Otherquantumcomplexity-theoreticconcepts,suchastheefficientverification
ofquantumproofs,suggestawiderextenttowhichquantummechanicsinfluencescomputational
complexity.
It may be said that the principal aim of quantum computational complexity theory is to un-
derstandtheimplications ofquantumphysicstocomputationalcomplexitytheory. Tothisend,it
considersthe hardness of computational problems with respectto models of quantum computa-
tion,classificationsofproblemsbasedonthesemodels,andtheirrelationshipstoclassicalmodels
andcomplexityclasses.
II Introduction
This article surveys quantum computational complexity, with a focus on three fundamental no-
tions: polynomial-time quantum computations, the efficient verification of quantum proofs, and
quantum interactive proof systems. Based on these notions one defines quantum complexity
classes, such as BQP, QMA, and QIP, that contain computational problems of varying hardness.
2
Properties of these complexity classes, and the relationships among these classes and classical
complexity classes, are presented. As these notions and complexity classes are typically defined
within the quantum circuit model, this article includes a section that focuses on basic properties
ofquantumcircuits thatare importantinthesettingofquantumcomplexity. A selectionofother
topicsinquantumcomplexity,includingquantumadvice,space-boundedquantumcomputation,
andbounded-depthquantumcircuits,isalsopresented.
Twodifferentbutcloselyrelatedareasofstudyarenotdiscussedinthisarticle: quantumquery
complexityandquantumcommunicationcomplexity. Readersinterestedinlearningmoreaboutthese
interesting and active areas of research may find the surveys of Brassard [35], Cleve [36], and
deWolf[107]tobehelpfulstartingpoints.
It is appropriate that brief discussions of computational complexity theory and quantum in-
formation precede the main technical portion of the article. These discussions are intendedonly
tohighlighttheaspectsofthesetopicsthatarenon-standard,requireclarification,orareofpartic-
ular importanceinquantumcomputationalcomplexity. Inthesubsequentsectionsofthisarticle,
the reader is assumed to have basic familiarity with both topics, which are covered in depth by
severaltextbooks[14,44,61,68,84,87].
II.1 Computationalcomplexity
Throughout this article the binary alphabet 0,1 is denoted Σ, and all computational problems
{ }
are assumed to be encoded over this alphabet. As usual, a function f : Σ Σ is said to
∗ ∗
→
be polynomial-time computable if there exists a polynomial-time deterministic Turing machine that
outputs f(x) foreveryinput x Σ . Tworelatedpointsontheterminologyusedthroughoutthis
∗
∈
articleareasfollows.
1. A function of the form p : N N (where N = 0,1,2,... ) is said to be a polynomial-
→ { }
bounded function if and only if there exists a polynomial-time deterministic Turing machine
thatoutputs1f(n) oninput1n foreveryn N. Suchfunctionsareupper-boundedbysome
∈
polynomial,andareefficientlycomputable.
2. Afunctionoftheparticularforma : N [0,1]issaidtobepolynomial-timecomputableifand
→
only if there exists a polynomial-time deterministic Turing machine that outputs a binary
representation of a(n) on input 1n for each n N. References to functions of this form in
∈
this article typically concern bounds on probabilities that are functions of the length of an
inputstringtosomeproblem.
Thenotionofpromise problems [45,53]iscentraltoquantumcomputationalcomplexity. These
aredecisionproblemsforwhichtheinputisassumedtobedrawnfromsomesubsetofallpossible
inputstrings. Moreformally, apromiseproblemisapair A = (A ,A ), where A , A Σ
yes no yes no ∗
⊆
are sets of strings satisfying A A = ∅. The strings contained in the sets A and A
yes no yes no
∩
are called the yes-instances and no-instances of the problem, and have answers yes and no, re-
spectively. Languages may be viewed as promise problems that obey the additional constraint
A A = Σ . Althoughcomplexitytheoryhastraditionallyfocusedonlanguagesratherthan
yes no ∗
∪
promise problems, little is lost and much is gained in shifting one’s focus to promise problems.
Karp reductions (also called polynomial-time many-to-one reductions) and the notion of com-
pletenessaredefinedforpromiseproblemsinthesamewayasforlanguages.
Severalclassicalcomplexityclassesarereferredtointhisarticle,andcomparedwithquantum
complexity classes when relations are known. The following classical complexity classes, which
should hereafter be understood to be classes of promise problems and not just languages, are
amongthosediscussed.
3
P Apromiseproblem A = (A ,A )isinPifandonlyifthereexistsapolynomial-
yes no
timedeterministicTuringmachine Mthatacceptseverystringx A andrejects
yes
∈
everystring x A .
no
∈
NP A promise problem A = (A ,A ) is in NP if and only if there exists a
yes no
polynomial-bounded function p and a polynomial-time deterministic Turing ma-
chine M with the following properties. For every string x A , it holds that M