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safely ignored. This step is called the “Rotating Wave Approximation”. R 1 and
R 2 can also be ignored if ω mn is sufficiently different from ω.
Since R 9 does not change the magnitude of the level populations, but only
their relative phases, a transition between levels can therefore only be driven if
ω ≈ ω mn . Thanks to the non-linearity in the qubit potential, this makes it possible
to operate the qubit solely in the lowest two states (the qubit states) of the shallow
minimum by ensuring that the interaction does not drive transitions between one
of the qubit states and undesired other states.
In our example case, if the qubit is initialized into state | 12 〉 and ω ≈ ω | 12 〉,| 14 〉 ,
the only states that will ever be populated are states | 12 〉 and | 14 〉, which we
will from now on call | 0 〉 and | 1 〉 to denote the two logical qubit states. The
drive ∆I dc (t) can then drive transitions between the qubit states via:
i ∂ t A(t) =
(
Tmn
2 (X(t)σ x + Y (t)σ y ) + T )
nn − T mm
Z(t)σ z A(t) (3.47)
2
3.3.2 Operations on a Single Qubit
The evolution of the qubit can be simulated numerically by approximating
∆I dc (t) with sections ∆I dc (t) → ∆I dc (t + ∆t) of temporarily constant drive amplitudes
X t→t+∆t , Y t→t+∆t , and Z t→t+∆t . For each section, Equation 3.47 can then
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