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shouldfunctionasanoraclefor A,asdefinedinSectionIII.4.
A problem arises, however, when queries to an algorithm for A are made in superposition.
Whereas it is quite common and useful to consider quantum algorithms that query oracles in
superposition, a given BQP algorithm for A is only guaranteed to work correctly on classical
inputs. It could be, for instance, that some algorithm for A begins by applying phase-damping
gatestoallofitsinputqubits,orperhapsthishappensinadvertentlyasaresultofthecomputation.
Perhaps it is too much to ask that the existence of a BQP algorithm for A admits a subroutine
havingthecharacteristicsofanoraclefor A?
The BQP subroutine theorem establishes that, up to exponentially small error, this is not too
much toask: theexistenceofan arbitrary BQP algorithmfor A implies theexistenceofa“clean”
subroutinefor Awiththecharacteristicsofanoracle. Aprecisestatementofthetheoremfollows.
Theorem 4 (BQP subroutine theorem). Suppose A = (A ,A ) is a promise problem in BQP. Then
yes no
for any choice of a polynomial-bounded function p there exists a polynomial-bounded function q and a
polynomial-time generated family of unitary quantum circuits R : n N with the following proper-
n
{ ∈ }
ties:
1. Eachcircuit R implementsaunitaryoperation U onn+q(n)+1qubits.
n n
2. Forevery x A anda Σitholdsthat
yes
∈ ∈
x,0m,a 1 U x,0m,a 1 2−p(n)
n
h ⊕ | | i ≥ −
forn = x andm = q(n).
| |
3. Forevery x A anda Σitholdsthat
no
∈ ∈
x,0m,a U x,0m,a 1 2−p(n)
n
h | | i ≥ −
forn = x andm = q(n).
| |
The proof of this theorem is remarkably simple: given a BQP algorithm for A, one first uses
Proposition 3 to obtain a circuit family Q having exponentially small error for A. The circuit
illustrated in Figure 8 then implements a unitary operation with the desired properties. This is
essentially a bounded-error quantum adaptation of a classic construction that allows arbitrary
deterministiccomputationstobeperformedreversibly[27,97].
The following corollary expresses the main implication of the BQP subroutine theorem in
complexity-theoreticterms.
Corollary5. BQPBQP = BQP.
IV.5 Classicalupper boundson BQP
Thereisnoknownwaytoefficientlysimulatequantumcomputerswithclassicalcomputers—and
there would be little point in seeking to build quantum computers if there were. Nevertheless,
15
x
| i
Q Q 1
n −n
0m
| i
a
| i
Figure8: Aunitaryquantumcircuitapproximatinganoracle-gateimplementationofaBQPcom-
putation. Here Q is a unitary purification ofa circuit having exponentiallysmall error for some
n
probleminBQP.
some insight into the limitations of quantum computers may be gained by establishing contain-
mentsofBQPinthesmallestclassicalcomplexityclasseswherethisispossible.
Thestrongestcontainmentknownatthistimeis givenbycountingcomplexity. Countingcom-
plexity began with Valiant’s work [99] on the complexity of computing the permanent, and was
furtherdevelopedandappliedinmanypapers(including[22,48,96],amongmanyothers).
The basic notion of counting complexity that is relevant to this article is as follows. Given
a polynomial-time nondeterministic Turing machine M and input string x Σ , one denotesby
∗
∈
#M(x)thenumberofacceptingcomputationpathsofMonx,andby#M(x)thenumberofrejecting
computation paths of M on x. A function f : Σ Z is then said to be a GapP function if there
∗
→
exists a polynomial-time nondeterministic Turing machine M such that f(x) = #M(x) #M(x)
−
forall x Σ .
∗
∈
A variety of complexity classes can be specified in terms of GapP functions. For example, a
promise problem A = (A ,A ) is in PP if and only if there exists a function f GapP such
yes no
∈
that f(x) > 0for all x A and f(x) 0for all x A . Theremarkable closurepropertiesof
yes no
∈ ≤ ∈
GapP functions allows many interestingfacts to be proved about such classes. Fortnow’ssurvey
[51] on counting complexity explains many of these properties and gives several applications of
thetheoryofGapPfunctions. Thefollowingclosurepropertyisusedbelow.
GapP–multiplicationofmatrices. Let p,q : N N bepolynomial-boundedfunctions. Suppose
→
thatforeachn Nasequenceof p(n)complex-valuedmatricesisgiven:
∈
A , A , ..., A ,
n,1 n,2 n,p(n)
eachhaving rowsandcolumnsindexedbystringsin Σq(n). Supposefurtherthatthereexistfunc-
tions f,g GapPsuchthat
∈
f(1n,1k,x,y) = Re(A [x,y])
n,k
g(1n,1k,x,y) = Im(A [x,y])