Full-Permutation Dynamical Decoupling in Triple-Quantum-Dot Spin Qubits

Dynamical decoupling of spin qubits in silicon can increase fidelity and can be used to extract the frequency spectra of noise processes. We demonstrate a full-permutation dynamical decoupling technique that cyclically exchanges the spins in a triple-quantum-dot qubit. This sequence not only suppresses both low-frequency charge-noise-induced and magnetic-noise-induced errors; it also refocuses leakage errors to first order, which is particularly interesting for encoded exchange-only qubits. For a specific construction, which we call “NZ1y,” the qubit is isolated from error sources to such a degree that we measure a remarkable exchange pulse error of $2.8\times {10}^{-5}$. This sequence maintains a quantum state for roughly 18,000 exchange pulses, extending the qubit coherence from $T_2^{\ast }=2\text{μ}\mathrm{s}$ to $T_2=720\text{μ}\mathrm{s}$. We experimentally validate an error model that includes $1/f$ charge noise and $1/f$ magnetic noise in two ways: by direct exchange-qubit simulation and by integration of the assumed noise spectra with derived filter functions, both of which reproduce the measured error and leakage with respect to a change of the repetition rate.