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AnoperatorAon ,denotedbyA(u )for u ,
classical probability. The intrinsic stochastic nature
H | i | i∈H
is a function mapping from to that satisfies
of quantum theory indicates a deep connection be-
H H
A(a u +b v )=aA(u )+bA(v )forany u , v
tweenquantummechanicsandprobability.Sincethe
and| a,i C| .i can| rei present| ai operato| ti h| roi u∈ ghH
mainfocusofthispaperisonquantumcomputation b We n r
∈
and quantum information, we give a brief descrip- a matrix. Suppose that A is an operator on and
H
tion of quantum theory in this section to provide e 1,...,e k form an orthonormal basis in . Then
H
some quantum background for the purpose of re- each A(e j ) and there exists a unique k k
| i ∈H ×
viewing quantum computation and quantum sim- matrix (a ) such that A(e )= k e a , j =
jℓ | j i ℓ=1| ℓ i ℓj
ulation in subsequent sections. For further reading 1,...,k. We will identify operatorPA with matrix
on the subjects we recommend textbooks by Saku- (a ) and use A for both operator and matrix (a ).
jℓ jℓ
rai and Napolitano (2010) at the graduate level and An operator A on is said to be self-adjoint if
H
Griffiths(2004)attheundergraduatelevel forquan- its corresponding matrix A is Hermitian, that is,
4
Y.WANG
A=A . We also refer to self-adjoint operators as case that the quantum system is in one of states
†
Hermitian operators. An operator U is said to be ψ , j=1,...,J, with probability p being in state
j j
| i
unitary if its corresponding matrix U is unitary, ψ , and the corresponding density operator
j
| i
that is, UU =U U=I. We say an operator A is
† † J
semi-positive (or positive) definite if its correspond- (2) ρ= p ψ ψ .
j j j
ing matrix A is semi-positive (or positive) definite, | ih |
Xj=1
that is, uρ u 0 for u (or uρ u 0 for
h | | i≥ | i∈H h | | i≥ See Griffiths (2004), Sakurai and Napolitano (2010)
u with equality only for u =0). The trace
| i∈H | i and Shankar (1994).
of an operator A, denoted by Tr(A), is defined to
be the trace of its corresponding matrix A=(a ), 2.3 Quantum Probability
jℓ
that is, Tr(A)= k a .
j=1 jj We can test the theory of quantum mechanics by
2.2 Quantum SyP stem checking its predictions with experiments of per-
forming measurements on quantum systems in the
Quantummechanics depictsphenomenaatmicro-
laboratory. The usual quantum measurements are
scopiclevelsuchaspositionandmomentumofanin-
on observables such as position, momentum, spin,
dividual particle like an atom or electron, spin of an
andsoon,whereanobservable Xisdefinedasaself-
electron,detectionoflightphotons,andtheemission
adjoint operator on Hilbertspace .Theobservable
and absorption of light by atoms. Unlike classical H
definition is motivated from the fact that the eigen-
mechanics where physical entities like position and
valuesofself-adjointoperatorsarereal.Assumethat
momentum can bemeasured precisely, thetheory of an observable X has a discrete spectrum with the
quantum mechanics is intrinsically stochastic in a
following diagonal form
sensethatwecan onlymakeprobabilisticprediction
p
about the results of the measurements performed.
(3) X= x Q ,
a a
Quantum mechanics is mathematically described
Xa=1
byaHilbertspace andself-adjointoperatorson .
AquantumsystemH iscompletelycharacterizedbyiH ts where x a are real eigenvalues of X and Q a are the
corresponding one-dimensional projections onto the
state and the time evolution of the state. A state is
orthogonal eigenvectors of X. Consider such an ob-
defined to be a unit vector in . Let ψ(t) be the
H | i servablein thequantum system preparedinstate ρ.
state of the quantum system at time t, which is also
Measure space (Ω, ) is used to describe possible
referredtoasawavefunction.Thestates ψ(t ) and F
| 1 i measurement outcomes of the observable, and the
ψ(t ) at t and t are connected through ψ(t ) =
| 2 i 1 2 | 2 i result of the measurement is a random variable on
U(t ,t )ψ(t ) , where U(t ,t ) is a unitary opera-
1 2 | 1 i 1 2 (Ω, ) with probability distribution P ρ. We denote
tor depending only on time t and t . In fact, there F
1 2 by X theresultof themeasurement of observable X
exists a self-adjoint operator H, which is known as
given by (3). Then X is a random variable taking
the Hamiltonian of the quantum system, such that
valuesin x ,x ,..., ,andunderpurestate ψ ,the
1 2
U(t ,t )=exp[ iH(t t )]. With Hamiltonian H, { } | i
1 2 2 1 probability that measurement outcome x occurs is
− − a
we may describe the continuous time evolution of