Analysis of an all-to-all connected star array of transmon qubits

We analyzed quantum $XX$ and $ZZ$ coupling and state transfer in an all-to-all connected star array of capacitively coupled superconducting transmon qubits. It is shown that in a highly-connected system like this a variety of different $ZZ$ couplings arise that correspond to the different ways qubits can interact with each other, opening different channels for unwanted qubit crosstalk and thus qubit operation errors. We studied the dependence of both the $XX$ and the $ZZ$ coupling on qubit detuning that controls qubit-qubit interaction. The $XX$ coupling, quantified by the error state occupation probability, shows a $\Delta\omega^{-2}$ decay with qubit detuning $\Delta\omega$. On the other hand, all $ZZ$ coupling frequencies show spikes at values in the lower detuning region that correspond to resonances between qubit states and states out of the computational basis, after which all couplings quickly decay to zero as qubit detuning further increases. This allows to define an operational region where near-zero qubit coupling can be achieved. We derive equations for the couplings as a function of qubit detuning that agree with numerical results solving the Schr\"odinger equation.