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

Abstract

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 $Δω^{-2}$ decay with qubit detuning $Δω$. 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 Schrödinger equation.

Figures (6)

• Figure 1: (a) Circuit schematics of the system of three transmon qubits that are capacitively coupled in a star array, and (b) equivalent circuit, where the labeled gray dashed boxes contain the qubits represented as gray circles in (a). The current through each coupling capacitor $C_{xi}$ ($i=1,2,3$) is $I_{xi}$, and each qubit is biased by corresponding external magnetic flux $\Phi_{ei}$.
• Figure 2: Time evolution of the single-excitation qubit state occupation probabilities for initial state $|\Psi_0\rangle = |010\rangle$. System's parameters are $I_{c[1,2,3]}= 40\,\text{nA},\,C_{1,2,3} = C= 100\,\text{fF},\, C_{x[1,2,3]} = C_x = 1\,\text{fF},\, \omega_{1,2,3} = \omega_{qb} = 6\,\text{GHz}$.
• Figure 3: Error state occupation probability $\mathcal{P}_e(010)$ vs. qubit detuning $\Delta\omega$ in log-log scale. Black solid line: Numerical results. Red dashed line: Analytical results using Eq. (\ref{['eq:pe']}). Inset: Same plot in linear scale. Except for qubits 1 and 3 frequencies, which were detuned by $\mp\Delta\omega/(2\pi)$ from qubit 2 frequency of 6 GHz, all system's parameters are the same as in the degenerate case (table \ref{['tabledegenerate']}).
• Figure 4: (Top) Energy spectrum of the system vs. qubit detuning in the two-excitation region, where the energy levels are labeled by the corresponding state. (Bottom) Same as the top panel but with the energy levels having different colors and line styles to track the corresponding state as detuning varies. Parameters are the same as in Fig. \ref{['pe010']}.
• Figure 5: Pairwise $ZZ$ couplings vs. qubit detuning $\Delta\omega$.. Red dashed lines are analytical results using Eqs. (\ref{['eq:zeta110']}) and (\ref{['eq:zeta101']}-\ref{['eq:zeta011']}). The dotted line at zero coupling is a guide for the eye. Parameters are the same as in Fig. \ref{['pe010']}