Page 186 - Principles and Applications of NanoMEMS Physics

174 Chapter 4

where the bar over X and Y denotes a rotation of -90 degrees with respect to
X or Y.
The NMR-based implementation of the C-NOT quantum gate involves a
series of two-qubit rotations, namely [201]: 00 → 00 , 01 → 01 ,
10 → 11 , and 11 → 10 . Addressing a particular qubit, without

affecting the neighboring one, is accomplished by exploiting the fact that
different atoms possess different resonance frequencies, ω , or that the
0
same type of atoms with a different chemical shift also possess different
ω . Taking two coupled spin-1/2 atoms with resonance frequencies ω and
1
0
ω , and coupling J , the C-NOT gate is implemented if applying a
2 12
narrowband 180-degree pulse at a frequency ω + J 2 / , causes spin 2 to
2 12
be inverted only if spin 1 is in state 1 . In this case, spin 1 is the control
qubit and spin 2 the target qubit. Pictorially, the C-NOT gate may be
visualized following the construction of Steffen, Vandersypen and Chuang
[201], see Fig. 4-18. The sequence of rotations is produced as follows: 1) An
RF pulse at a frequency ω , of a bandwidth such covering the frequency
2
range ω ± J , but that does not overlap with ω , rotates spin 2 from +Z to
2 12 1
–Y; 2) The spin system is allowed to evolve freely for a duration of /1 J 2
12
seconds; 3) During the free evolution period, the precession frequency of
spin 2 will be shifted by J± 2 / according to whether spin 1 is in the 1
12
or 0 state. This will result in the rotation of spin 2 to either +X or –X by
e
the end of this period, depending on the state of spin 1; 4) A 90-d gree pulse
applied to spin 2 about the –Y axis rotates spin 2 to +Z if spin 1 is in state
0 , or to –Z if spin 1 is in state 1 .

90
90 Delay(1/2J ab ) ) 90
90
Z Z X X -Y
Delay(1/2J ab
-Y
Y Y
X X

Figure 4-18. Left-to-right: Sequence of qubit rotations for implementing the C-NOT quantum
gate in NMR-based QC. The coordinate system rotates around the z ˆ axis at a frequency ω
2
when spin 1 is 0 (solid line), and 1 (dashed line). (After [201].)