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For most of the complexity classes listed above, there is a standard way to attach an oracle
to the machine model that defines the class, which provides a subroutine for solving instances
of a chosen problem B = (B ,B ). One then denotes the existence of such an oracle with a
yes no
superscript—for example, PB is the class of promise problems that can be solved in polynomial
timebyadeterministicTuringmachineequippedwithanoraclethatsolvesinstancesofB(atunit
cost). When classes of problems appear as superscripts, one takes the union, as in the following
example:
PNP = PB.
B ∈[NP
5
II.2 Quantuminformation
The standard general description of quantum information is used in this article: mixed states of
systems are represented by density matrices and operations are represented by completely pos-
itive trace-preserving linear maps. The choice to use this description of quantum information is
deserving of a brief discussion, for it will likely be less familiar to many non-experts than the
simplified picture of quantum information where states are represented by unit vectors and op-
erationsarerepresentedbyunitarymatrices. Thissimplifiedpictureisindeedcommonlyusedin
thestudyofbothquantumalgorithmsand quantumcomplexitytheory;and it is oftenadequate.
However,thegeneralpicturehasmajoradvantages: itunifiesquantuminformationwithclassical
probabilitytheory,bringswithitpowerfulmathematicaltools,andallowsforsimpleandintuitive
descriptionsinmanysituationswherethisisnotpossiblewiththesimplifiedpicture.
Classical simulations of quantum computations, which are discussed below in Section IV.5,
maybebetterunderstoodthroughafairly straightforwardrepresentationofquantumoperations
bymatrices. Thisrepresentationbeginswitharepresentationofdensitymatricesasvectorsbased
on the function defined as vec( x y ) = x y for each choice of n N and x,y Σn, and
| ih | | i| i ∈ ∈
extendedbylinearitytoallmatricesindexedbyΣn. Theeffectofthismappingistoformacolumn
vectorbyreadingtheentriesofamatrixinrowsfromlefttoright,startingatthetop. Forexample,
α
α β β
vec =  .
γ δ γ
(cid:18) (cid:19)
δ
 
 
Now,theeffectofageneralquantumoperationΦ,representedinthetypicalKrausformas
k
Φ(ρ) = ∑ A ρA ,
j ∗j
j=1
isexpressedasamatrixbymeansoftheequality
k
vec(Φ(ρ)) = ∑ A A vec(ρ).
j j
⊗
j=1 !
Thematrix
k
KΦ = ∑ A A
j j
⊗
j=1
issometimescalledthenaturalrepresentation (orlinearrepresentation) oftheoperationΦ. Although
thismatrixcouldhavenegativeorcomplexentries,onecanreasonablyviewitasbeinganalogous
toastochasticmatrixthatdescribesaprobabilisticcomputation.
Forexample,thecompletephase-dampingchannelforasinglequbitcanbewritten
D(ρ) = 0 0 ρ 0 0 + 1 1 ρ 1 1 .
| ih | | ih | | ih | | ih |
Theeffectofthismappingistozero-outtheoff-diagonalentriesofadensitymatrix:
α β α 0
D = .
γ δ 0 δ
(cid:18) (cid:19) (cid:18) (cid:19)
6
Φ
1
Y
X Φ Φ 1
1 2 4
X Y
2 2
Φ Φ
3 5
X Φ
3 6
X Y
4 3
X X
Figure 2: An example of a quantum circuit. The input qubits are labelled ,..., , the output
1 4
Y Y
qubits are labelled ,..., , and the gates are labelled by (hypothetical) quantum operations
1 3
Φ ,...,Φ .
1 6
Thenaturalrepresentationofthisoperationiseasilycomputed:
1 0 0 0
0 0 0 0
K =  .
D
0 0 0 0
 
0 0 0 1
 
 
Thenaturalmatrixrepresentationofquantumoperationsiswell-suitedtoperformingcompu-