Ph.D. Thesis - Physics

The same effect as number 2 above is achieved if we regard ˆz rotations as merely
a shifting of the reference frame by the angle θ in each Rˆz(θ) rotation. The effect
of ˆz rotations is implemented by shifting the phases of each subsequent single-qubit
pulse to account for them. Choosing between this method and number 2 is a practical
matter; because it requires less modification to the pulse sequence, this method is
generally preferred.
2.1.4 Two-qubit operations
Now that we are armed with the ability to do arbitrary single-qubit rotations and with an in-
teraction Hamiltonian (Eq. 2.4) we can construct two-qubit gates such as the controlled-not
(CNOT). The CNOT, along with arbitrary single-qubit rotations, is sufficient to implement
universal quantum computation; all the operations needed for quantum simulation are con-
tained within this, and in practice less control is often required. In the computational basis
({|00〉 , |01〉,|10〉 , |11〉}), the CNOT has this matrix representation:
⎡ ⎤
1 0 0 0
⎢ ⎥
⎢
CNOT12 = ⎢
0 1 0 0 ⎥
⎢
⎣ 0 0 0 1
⎥ . (2.17)
⎦
0 0 1 0
Suppose there are two coupled spins. The basic pulse sequence rotates spin 2, the “target
qubit,” from a ˆz eigenstate into the ˆx-ˆy plane of the Bloch sphere. Then, during a period
of free evolution under H0 + HI, the direction of the rotation of this spin (in the rotating
frame) depends on the state of spin 1, called the “control qubit.” After a time equal to
1/(2J12), the spin is rotated back into a ˆz eigenstate. This effectively flips (or doesn’t) spin
2 based on the state of spin 1. Specifically, the pulse sequence looks like:
U ′ CNOT = Rˆx(π/2)UI(1/(2J12))Rˆy(π/2) =
⎡ ⎤
1 0 0 0
⎢ ⎥
⎢
0 i 0 0 ⎥
⎢
⎣ 0 0 0 −i
⎥ ,
⎦
(2.18)
0 0 1 0
where UI(t) = exp(−iHIt/) is the unitary evolution under HI. Fig. 2-1 shows the opera-
tion of the CNOT gate as described here.
This does not exactly produce the CNOT presented in Eq. 2.17, but the gate can be
exactly duplicated by applying appropriate single-qubit pulses to spins 1 and 2. Specifically,
a Rˆz(−π/2) is required on qubit 2 and Rˆz(π/2) on qubit 1.
2.1.5 Refocusing
We briefly present one more invaluable technique for NMR quantum control. In NMR, the
ZZ interaction between spins is “always on,” in the sense that it’s a part of the Hamilto-
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