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by gates or circuit elements.5
An importantclassof2m+1 2m+1 matricesarethe controlled gatesΛ (U)
m
×
defined by
3Weleave openthe physicalimplementation ofthe wires. Theyshouldnot bethought of
as classical wires, but rather in terms of some unspecified interaction between gate outputs
andinputs,orbygates sharingqubits.
4Otherindexrangesaresometimesconvenient.
5Afewwordsonterminology;agate(circuitelement)isrepresentedbyaunitaryoperator
whichisrealizedasaunitary matrixinsomebasis. Thesameholdsforthecompletecircuit,
itselfbuiltoutofgates.
126
u x ,...,x ,0 +u x ,...,x ,1
y0 1 m y1 1 m
| i | i
if m x =1
 ∧k=1 k
Λ m(U)( |x 1,...,x m,y i= (6.12)
x ,...,x ,y
1 m
| i
if m x =0
 ∧k=1 k
or in a different notation where x x x denotes the product of the bits
1 2 n
···
x ,x ,...,x
1 2 n
Λ m(U)x 1,...,x m,y = x 1,...,x m,y Ux1x2 ···xn y ,
| i | i | i
or, explicitly, in block-matrix form
I 0 0
2m
 0 u 00 u 01,
0 u u
10 11
 
where I denotes the 2m 2m identity matrix. The operator Λ (U) applies
2m m
×
the operation U to the (m+1)-th qubit conditioned on the first m qubits all
being equal to 1, otherwise nothing is done. The controlled operations are the
quantum analogs of the selection primitive of classical computation.
Another notation in common use for controlled gates, is Cm(G). The gate
G need not be a single qubit gate, though in most cases it is.
Figure 6.3 shows the diagrammatic representation of a Λ (G) gate. Note
2
that conditioning onthe controlbit being 1 is denotedby anfilled circle on the
corresponding wire.
G
Figure 6.3: A Λ (G) gate.
2
Thereisnothingspecialaboutconditioningon1. Itissometimesconvenient
to condition on 0, or on combinations of 0 and 1 for different control qubits.
No special notation will be introduced for this case, but I will refer to it as
a generalized Λ (G), and an example is given in figure 6.4 to exemplify the
n
concept.
127
G
Figure 6.4: A generalized Λ (G) gate with conditioning on 0 and 1.
2
6.2.3 Special discrete one-qubit gates
We first list a set of simple 1-qubit gates. In section 4.3.1, the spin-1/2 Pauli
matrices were introduced. With a change of notation they are
0 1 0 i 1 0
X = , Y = − , Z = . (6.13)
(cid:18)1 0(cid:19) (cid:18)i 0 (cid:19) (cid:18)0 1(cid:19)
−
For the algebraic identities and commutation relations satisfied by these
matrices, refer back to section 4.3.1.
Apart from these gates, the Hadamard gate H, the phase gate S, and the
π/8-gate T are given by the matrices
1 1 1 1 0 1 0
H = , S = , T = . (6.14)
√2(cid:18)1 1(cid:19) (cid:18)0 i(cid:19) (cid:18)0 exp(iπ/4)(cid:19)
−
These single qubit gates are important, as they can be used together with
the CNOT-gate to give universal sets of discrete quantum gates.
When simplifying circuits, the following identities are useful
HXH =Z, HYH = Y, HZH =X. (6.15)
−
TheHadamardgatecanbeusedtoproduceequallyweightedsuperpositions
as the following simple example shows
1
H 0 = (0 + 1 ), (6.16)
| i √2 | i | i
1
H 1 = (0 1 ). (6.17)
| i √2 | i−| i
6.2.4 One-qubit rotation operators
By formally exponentiating the Pauli matrices one obtains a set of continuous
rotation operators
128
θ θ cosθ isinθ
R x(θ)=e−iθX/2 =cos 2I −isin 2X = isin2 − cosθ2 (cid:19), (6.18)
(cid:18) θ
− 2 2
θ θ cosθ sinθ