Page 185 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 173

locus of the tip of a qubit vector as a function of ω − ω , for a given RF
RF 0
pulse duration and the parameter ω .
1
NMR-based quantum gates are generated by “tuning” the parameters in
the control Hamiltonian to achieve a desired qubit rotation. Since any
quantum gate may be constructed from single-qubit rotations and the C-NOT
gate, the problem of NMR-based quantum computing reduces to determining
the control Hamiltonian that will implement these. In this context, we note
that the most general qubit rotation is defined by [194],
G
ª ișˆ ⋅ı º
n
R = exp − , (25)
n ˆ « »
¬ 2 ¼
where n ˆ denotes the 3-dimensional axis of rotation, θ is the angle of
G
rotation, and ı = ı x ˆ + ı y ˆ + ı z ˆ is a vector of Pauli matrices.
x y z
Furthermore, it can be shown that any qubit transformation may be
implemented as a sequence of rotations about only two axes. In particular,
Bloch’s theorem stipulates such a transformation as [194],
U = e α i R () ( ) ( ) δγβ R R . (26)
x y x
Therefore, in terms of the control Hamiltonian parameters, implementing a
single-qubit gate may be accomplished in the rotating frame using RF
pulses. Specifically, if an RF field of amplitude ω and frequency is
1
ω = ω is applied to a single spin, this will evolve according to [194],
RF 0
U = exp [iω (cosφ I + sin I φ )t ], (27)
1 x y pulse
where the RF pulse duration is given by t . In the context of the Bloch
pulse
sphere, this transformation would rotate the qubit by an angle ~ ω t ,
θ
1 pulse
with respect to an axis in the x ˆ − y ˆ plane given by the phase φ . For
instance, the parameters: φ = π and ω t = π 2 effect the R ()
90
1 pulse x
rotation about x ˆ , whereas doubling the pulse duration implements
R (180 ), and changing the phase to φ = − π 2 effects the rotation about
x
y ˆ . In general, the phase of the RF pulse determines the nutation axis in the
rotating frame, so that to perform x ˆ and y ˆ rotations it is not necessary to
orient the RF field along these axes; changing the phase suffices. A rotation
about the z ˆ axis in terms of rotations about x ˆ and y ˆ is given by [194],
U = R () XR=θ ()X =θ YR ( )Yθ− , (28)
z y x