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initionisthefact,discussedinthenextsection,thatanyefficientalgorithm(quantumorclassical)
canbeconvertedtoaquantumcircuitthatcloselyapproximatestheactionofthesegates.
Finally, one may consider a more general situation in which the predicate A is replaced by
a function that outputs multiple bits. The definition of each gate R is adapted appropriately.
n
Alternately, one may restrict their attention to single-bit queries as discussed above, and use the
Bernstein–Vaziranialgorithm[31]tosimulateonemultiple-bit querywithonesingle-bitqueryas
illustratedinFigure6.
IV Polynomial-time quantum computations
Thissectionfocusesonpolynomial-timequantumcomputations. Thesearethecomputationsthatare
viewed,inanabstractandidealizedsense,tobeefficientlyimplementablebythemeansofaquan-
tumcomputer. Inparticular,thecomplexityclassBQP(shortforbounded-error quantumpolynomial
time)isdefined. Thisisthemostfundamentallyimportantofallquantumcomplexityclasses,asit
representsthecollection ofdecisionproblems thatcan beefficiently solvedby quantumcomput-
ers.
11
IV.1 Polynomial-time generated circuitfamiliesandBQP
TodefinetheclassBQP usingthequantumcircuit model,it isnecessarytobriefly discussencod-
ingsofcircuitsandthenotionofapolynomial-timegeneratedcircuitfamily.
It is clear that any quantum circuit formed from the gates described in the previous section
could be encoded as a binary string using any number of different encoding schemes. Such an
encoding scheme must be chosen, but its specifics are not important so long as the following
simplerestrictionsaresatisfied:
1. Theencodingissensible: everyquantumcircuitisencodedbyatleastonebinarystring,and
everybinarystringencodesatmostonequantumcircuit.
2. The encoding is efficient: there is a fixed polynomial-bounded function p such that every
circuit of size N has an encoding with length at most p(N). Specific information about the
structureofacircuitmustbecomputableinpolynomialtimefromanencodingofthecircuit.
3. Theencodingdisallowscompression: itisnotpossibletoworkwithencodingschemesthat
allowforextremelyshort(e.g.,polylogarithmic-length)encodingsofcircuits;soforsimplic-
ityitisassumedthatthelengthofeveryencodingofaquantumcircuit isatleastthesizeof
thecircuit.
Now,asanyquantumcircuitrepresentsafinitecomputationwithsomefixednumberofinput
andoutputqubits,quantumalgorithmsaremodelledbyfamiliesofquantumcircuits. Thetypical
assumption is that a quantum circuit family that describes an algorithm contains one circuit for
eachpossibleinputlength. Preciselythesamesituationariseshereasintheclassicalsetting,which
is that it should be possible to efficiently generate the circuits in a given family in order for that
family to represent an efficient, finitely specified algorithm. The following definition formalizes
thisnotion.
Definition 2. Let S Σ be any set of strings. Then a collection Q : x S of quantum cir-
∗ x
⊆ { ∈ }
cuits is said to be polynomial-time generated if there exists a polynomial-time deterministic Turing
machinethat,oneveryinput x S,outputsanencodingofQ .
x
∈
This definition is slightly more general than what is needed to define BQP, but is convenient
forotherpurposes. Forinstance,itallows onetoeasilyconsiderthesituationin whichtheinput,
or some part of the input, for some problem is hard-coded into a collection of circuits; or where
a computation for some input may be divided among several circuits. In the most typical case
that a polynomial-time generated family of the form Q : n N is referred to, it should
n
{ ∈ }
be interpreted that this is a shorthand for Q : n N . Notice that every polynomial-time
1n
{ ∈ }
generated family Q : x S has the property that each circuit Q has size polynomial in
x x
{ ∈ }
x . Intuitively speaking, the number of quantum and classical computation steps required to
| |
implementsuchacomputationispolynomial;andsooperationsinducedbythecircuitsinsucha
familyareviewedasrepresentingpolynomial-time quantumcomputations.
ThecomplexityclassBQP,whichcontainsthosepromiseproblemsabstractlyviewedtobeeffi-
cientlysolvableusingaquantumcomputer,maynowbedefined. Moreprecisely,BQPistheclass
ofpromiseproblemsthatcanbesolvedbypolynomial-timequantumcomputationsthatmayhave
somesmall probability tomake an error. Fordecisionproblems, thenotionofapolynomial-time
quantumcomputationisequatedwiththecomputationofapolynomial-timegeneratedquantum
circuit family Q = Q : n N , where each circuit Q takes n inputqubits, and producesone
n n
{ ∈ }
output qubit. The computation on a given input string x Σ is obtained by first applying the
∗
∈
12
circuit Q to the state x x , and then measuring the output qubit with respect to the standard
|x | | ih |
basis. Themeasurementresults0and1are interpretedasyesandno(or acceptandreject), respec-
tively. TheeventsthatQaccepts xand Qrejectsxareunderstoodtohaveassociatedprobabilities
determinedinthisway.
BQP Let A = (A ,A ) be a promise problem and let a,b : N [0,1] be functions.
yes no
→
Then A BQP(a,b) ifand only ifthereexistsapolynomial-time generatedfamily of
∈
quantumcircuits Q = Q : n N ,whereeachcircuit Q takesninputqubitsand
n n
{ ∈ }
producesoneoutputqubit,thatsatisfiesthefollowingproperties:
1. if x A thenPr[Q accepts x] a( x ),and
yes
∈ ≥ | |
2. if x A thenPr[Q accepts x] b( x ).
no
∈ ≤ | |
TheclassBQPisdefinedasBQP = BQP(2/3,1/3).
Similar toBPP,thereis nothingspecial about theparticular choice oferror probability 1/3, other
thanthatitisaconstantstrictlysmallerthan1/2. Thisismadeclearinthenextsection.
There are several problems known to be in BQP but not known (and generally not believed)
tobeinBPP.Decision-problemvariantsoftheintegerfactoringanddiscretelogarithmproblems,