Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
1
qubit gates and CNOT’s.
Let U be single qubit unitary gate. Then there exist single qubit operators
A,B and C such that
ABC =I (6.45)
eiαAXBXC =U. (6.46)
Proof
In terms of the rotation operators (6.18), (6.19) and (6.20), put
A=R (β)R (γ/2),
z y
B =R ( γ/2)R ( (δ+β)/2),
y z
− −
C =R ((δ β)/2).
z
−
Then
ABC =R (β)R (γ/2)R ( γ/2)R ( (δ+β)/2)R ((δ β)/2)=I.
z y y z z
− − −
Next, using the identities 6.24, 6.25 as well as X2 =I
XBX =XR ( γ/2)XXR ( (δ+β)/2)=R (γ/2)R ((δ+β)/2),
y z y z
− −
so that
AXBXC =R (β)R (γ/2)R (γ/2)R ((δ+β)/2)R ((δ β)/2)=
z y y z z
−
R (β)R (γ)R (δ).
x y z
Thus U can be decomposed as U =eiαAXBXC with ABC=I.
These equations allow a decomposition of a general controlled-U operation
asinfigure6.13. Whenthecontrolqubitis 0 ,theoperationABC isappliedto
| i
138
=
U P A B C
Figure 6.13: Simulation of a controlled-U operation in terms of single qubit
operations and CNOT’s.
thetargetqubit. Ontheotherhand,whenthecontrolqubitis 1 ,theoperation
| i
P(α)AXBXC is applied to the target qubit.
In order to obtain a somewhat more simple diagrammatic representation,
the circuit identity of figure 6.14 is useful. This identity can be derived using
the methods of section 6.3.7. Using the identity, we get the circuit of figure
6.15.
E(a)
=
P(a)
Figure 6.14: Identity between Λ (P(α)) and E(α) I.
1
⊗
139
E
=
U A B C
Figure 6.15: Simplified decomposition of a controlled-U operation in terms of
single qubit operations and CNOT’s.
Thus, the operation Λ (U) can be decomposed into six basic operations.
1
Decomposition of a general Λ (U) gate
2
Foranyunitary2 2operationU,aΛ (U)canbedecomposedasinfigure6.16
2
×
where V is a matrix that satisfies V2 =U.
c
1
c =
2
U V V+ V
Figure6.16: Simulationofa Λ (U) operationin terms ofΛ (V) operationsand
2 1
CNOT’s.
140
Proof
The gate construction is proved correct by an examination of the four cases of
combinations of basis states for the two target qubits c and c
1 2
| i | i
When c =0 and c =0, I is applied to the target.
1 2
•
When c =0 and c =1, VV =I is applied to the target.
1 2 †
•
When c =1 and c =0, V V =I is applied to the target.
1 2 †
•
Finally, when c =1 and c =1, VV =U is applied to the target.
1 2
•
The circuit identities developed so far allow for the decomposition of a gen-
eral Λ (U) gate in terms of 16 single qubit gates and CNOT gates. A naive
2
countingbasedonfigures6.15and6.16wouldimply the needto use20elemen-
tary gates, but a closer examinationshows that a few single qubit gates can be
merged and eliminated. The details are left to the reader.
Decomposition of a general Λ (U) gate
n 1