Stabilizer Quantum Error Correction Toolbox for Superconducting Qubits

We present a general protocol for stabilizer operator measurements in a system of $N$ superconducting qubits. Using the dispersive coupling between the qubits and the field of a resonator as well as single qubit rotations, we show how to encode the parity of an arbitrary subset of $M\le N$ qubits, onto two quasiorthogonal coherent states of the resonator. Together with a fast cavity readout, this enables the efficient measurement of arbitrary stabilizer operators without locality constraints.

Figure 1

(a) 2D array of $N$ transmon qubits in a 3D cavity. The ancilla qubit and the upper cavity are used for readout or reset and manipulation purposes. Cavity (b) and qubit (c) spectra in the ultrastrong dispersive regime.

Figure 2

(a) Quantum circuit diagram for encoding the parity of qubits 2 and 4 [full (blue) lines]. $D_{\alpha }$ represents the displacement operation, ${\pi }_x$ a single-qubit $\pi$ pulse and $T_{1/2}$ a free dispersive evolution of duration $\pi /(4\chi )$. (b) Numerical simulation of the evolution of the $Q$ function of the cavity [31], with an initial qubit state $|\psi {⟩}_4=(|ggge⟩+|ggeg⟩+|eeeg⟩)/\sqrt{3}$. Dissipation from photon loss at a rate $\kappa /(2\pi )=10\mathrm{kHz}$ and qubit decoherence with $T_1=T_2=20\mu \mathrm{s}$ are included as well as a finite displacement and $\pi$-pulse duration of 1 ns. Other parameters are $\alpha =2$, $\chi /(2\pi )=5\mathrm{MHz}$ and $K/(2\pi )=80\mathrm{kHz}$. The self-Kerr term leads to an additional phase rotation $\Delta {\varphi }_{\mathrm{nl}}=2K\overline{n}\Delta t$, where $\Delta t=50.3\mathrm{ns}$ is the total duration of the encoding. Taking this rotation into account, we obtain a fidelity of $F=98\%$ to the ideal target state $|\Psi {⟩}_{\mathrm{ideal}}=|{\tilde{\alpha }}_2⟩{\mathit{P}}_{\{2,4\}}^{+}|\psi {⟩}_4+|-{\tilde{\alpha }}_2⟩{\mathit{P}}_{\{2,4\}}^{-}|\psi {⟩}_4$.

Figure 3

Fidelity of the prepared state to $\exp [-i(\pi /8)\overline{X}]|+⟩$. The (red) dashed curve represents the boundary of the inequality $K\ge {\chi }^2/(4{\alpha }_q)$ which relates the self-Kerr $K$ with the dispersive shift $\chi$ and the single-qubit anharmonicity ${\alpha }_q$ [28]. Here ${\alpha }_q/(2\pi )=200\mathrm{MHz}$.