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• The x-range over which the potential is approximated needs to be large
enough for the wave-functions to “comfortably” go to zero on both sides.
• In a real qubit potential, the right minimum will most likely have hundreds
of states at energies below the ground-state of the operating minimum. Since
an n × n matrix will yield only the lowest n eigenstates, M therefore needs
to have several hundred rows and columns.
• The ground-state in the operating minimum will usually not be the level
with the lowest overall energy if the other minimum is deeper. In the figure
shown, states 9 and 11 (counting from 0) are localized mostly in the left
minimum, the states above level 11 span both minima, and all other states
are localized in the right minimum. Thus, it is necessary to sort the levels
into the correct minimum before subtracting their energies to find transition
frequencies.
3.2.2 Eigenstates of Coupled Qubit Systems
It is theoretically possible to extend this method to finding the eigenstates of
a system of coupled qubits. For this, the state of the second qubit is added to
the Schrödinger equation, making it two-dimensional (a function of δ 1 and δ 2 ).
ψn(δ r 1 , δ 2 ) and V (δ 1 , δ 2 ) would then be rewritten as vectors following a convention
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