# Conditional Phase Gate Superconducting Qubits

In the paper "Demonstration of two-qubit algorithms with a superconducting quantum processor" (L. DiCarlo et al., Nature 460, 240 (2009), arXiv) they demonstrate how to realize conditional phase gates with superconducting qubits.

Specifically, they use the $|{1,1}\rangle \leftrightarrow |0, 2\rangle$ to create a conditional phase gate. I quote "his method of realizing a C-Phase gate by adiabatically using the avoided crossing between computational and non-computational states is generally applicable to qubit implementations with finite anharmonicity, such as trans- mons or phase qubits".

My question is how this technique works, especially why it is a controlled gate.

Each of the two spins, $q\in\{L,R\}$, has a bunch of energy levels $\{|n\rangle_q\}$, each at energy $\omega_{n}^q$. In other words, the basic Hamiltonian of the spins is: $$H=\sum_{n=0}^{N}\omega_{n}^L|n\rangle\langle n|_L+\omega_{n}^R|n\rangle\langle n|_R$$ Written like this, the two spins are not interacting, so we won't get a two-qubit gatewithout doing something extra.
We we're talking about a qubit, we specifically focus on populating just the $|0\rangle$ and $|1\rangle$ levels of each spin. Nothing else is ever populated (hopefully). Under the Hamiltonian $H$, as basis element $|x\rangle$ for $x\in\{0,1\}^2$ acquires a phase $$e^{-i(\omega^L_{x_L}+\omega^R_{x_R})t}.$$
In addition to this basic Hamiltonian of the spins, there is a cavity, containing photons that interact with both spins. This is what will mediate the two-qubit interaction. In effect, by manipulating the interaction parameters, we can change the energy level of the $|1\rangle_L|1\rangle_R$ state independently of the $|10\rangle$ and $|01\rangle$ states. Thus, in principle, one creates a different phase pno all 4 basis states, and these can be combined to give a controlled-phase gate.
You'll notice I haven't mentioned the $|0,2\rangle$ level yet. In some ways this is irrelevant; everything I've said is (a sufficiently good approximation to) true. The issue is that usually, when you change the parameters, you don't get the independent control of the different energies. The place to go looking, if you want to find such independent control, is in the region of an 'avoided crossing', where the usual linear variation of energy with parameters would suggest that two energy levels should be the same (e.g. the $|11\rangle$ and $|02\rangle$ levels). The avoided crossing means that the energy takes on a quadratic form near the (not) crossing point, and it's that non-linearity that you're making use of. It also defines important constraints on the adiabatic evolution: since you do not want to populate the $|02\rangle$ level, you have to move slowly with respect to the energy gap between $|11\rangle$ and $|02\rangle$, which is comparatively small, and therefore the evolution time is quite slow.