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showntobeinBQPbyShor[94],areatpresentthemostimportantandwell-knownexamples.
IV.2 Errorreduction for BQP
When one speaksof the flexibility, or robustness, ofBQP with respectto error bounds, it is meant
thattheclassBQP(a,b) isinvariant underawiderangeof“reasonable”choicesofthefunctions a
andb. Thefollowingpropositionstatesthismoreprecisely.
Proposition 3 (Error reduction for BQP). Suppose that a,b : N [0,1] are polynomial-time com-
→
putable functions and p : N N is a polynomial-bounded function such that a(n) b(n) 1/p(n)
→ − ≥
for all but finitely many n N. Then for every choice of a polynomial-bounded function q : N N
∈ →
satisfying q(n) 2forallbutfinitelymanyn N,itholdsthat
≥ ∈
BQP(a,b) = BQP = BQP 1 2−q,2−q .
−
(cid:0) (cid:1)
The above proposition may be proved in the same standard way that similar statements are
provedforclassicalprobabilisticcomputations: byrepeatingagivencomputationsomelarge(but
stillpolynomial)numberoftimes,overwhelmingstatisticalevidenceisobtainedsoastogivethe
correctanswerwithanextremelysmallprobabilityoferror. Itisstraightforwardtorepresentthis
sort of repeated computation within the quantum circuit model in such a way that the require-
mentsofthedefinitionofBQParesatisfied.
IV.3 Simulatingclassical computationswith quantumcircuits
Itshouldnotbesurprisingthatquantumcomputerscanefficientlysimulateclassicalcomputers—
forquantuminformationgeneralizesclassicalinformation,anditwouldbeabsurdiftherewerea
lossofcomputationalpowerinmovingtoamoregeneralmodel. Thisintuitionmaybeconfirmed
byobservingthecontainmentBPP BQP. Hereanadvantageofworkingwiththegeneralquan-
⊆
tumcircuitmodelarises: forifonetrulybelievestheUniversalityTheorem,thereisalmostnothing
toprove.
13
NANDgate FANOUTgate
X
1 Tr X D Y 1
X
2 Tr 0 Y
2
| i
1 Y
| i
Randombit
0 H D Y
| i
Figure7: Quantumcircuit implementationsofaNANDgate,aFANOUTgate,andarandombit.
The phase-damping gates, denoted by D, are only included for aesthetic reasons: they force the
purely classical behavior that would be expected of classical gates, but are not required for the
quantumsimulationofBPP.
Observe first that the complexity class P may be defined in terms of Boolean circuit families
in asimilar manner toBQP.Inparticular, agivenpromiseproblem A = (A ,A ) isin Pifand
yes no
only if there exists a polynomial-time generated family C = C : n N of Boolean circuits,
n
{ ∈ }
whereeachcircuit C takesninputbitsandoutputs1bit,suchthat
n
1. C(x) = 1forall x A ,and
yes
∈
2. C(x) = 0forall x A .
no
∈
ABooleancircuit-baseddefinitionofBPPmaybegivenalongsimilarlines: agivenpromiseprob-
lem A = (A ,A ) is in BPP if and only if there exists a polynomial-bounded function r and a
yes no
polynomial-time generatedfamily C = C : n N of Boolean circuits, where each circuit C
n n
{ ∈ }
takesn+r(n)inputbitsandoutputs1bit,suchthat
1. Pr[C(x,y) = 1] 2/3forall x A ,and
yes
≥ ∈
2. Pr[C(x,y) = 1] 1/3forall x A ,
no
≤ ∈
wherey Σr( x ) ischosenuniformlyatrandominbothcases.
| |
∈
In both definitions, the circuit family C includes circuits composed of constant-size Boolean
logic gates—which for the sake of brevity may be assumed to be composed of NAND gates and
FANOUTgates. (FANOUToperationsmustbe modelledas gatesfor thesakeofthesimulation.)
Fortherandomizedcase,itmaybeviewedthattherandombitsy Σr( x ) areproducedbygates
| |
∈
that take no input and output a single uniform random bit. As NAND gates, FANOUT gates,
andrandombitsareeasilyimplementedwithquantumgates,asillustratedinFigure7,thecircuit
family Ccanbesimulatedgate-by-gatetoobtainaquantumcircuitfamily Q = Q : n N for
n
{ ∈ }
AthatsatisfiesthedefinitionofBQP.ItfollowsthatBPP BQP.
⊆
IV.4 The BQP subroutine theorem
There is an important issue regarding the above definition of BQP, which is that it is not an in-
herently“clean” definitionwith respecttothemodularization ofalgorithms. TheBQPsubroutine
theorem ofBennett,Brassard,BernsteinandVazirani[28]addressesthisissue.
14
Suppose that it is established that a particular promise problem A is in BQP, which by def-
inition means that there must exist an efficient quantum algorithm (represented by a family of
quantumcircuits)for A. Itisthennaturaltoconsidertheuseofthatalgorithmasasubroutinein
other quantum algorithms for more complicated problems, and one would like to be able to do
this without worrying about the specifics of the original algorithm. Ideally, the algorithm for A