Controlling decoherence due to nuclear spins in III-V compounds: Which price do we pay?

Nuclear spins of the host lattice are the dominant source of decoherence in semiconductor donor and quantum dot spin qubits. There are two channels for nuclear induced decoherence: (1) Loss of visibility arising from the non-secular hyperfine coupling; (2) Spectral diffusion arising from the combined effect of inter-nuclear dipolar coupling and the secular hyperfine term. We performed numerical calculations to show that application of a moderate static magnetic field ($\sim$ 2 Tesla) is enough to suppress mechanism (1) within the $10^{-4}$ criteria of quantum error correction. On the other hand a much greater overhead is required to control mechanism (2). We consider the Carr-Purcell-Meiboom-Gill sequence as a means to control (2) and provide a realistic assessment of the required overhead in number of qubit $\pi$-pulses. We show that the required rate of $\pi$-pulsing is proportional to the nuclear spin quantum number squared, showing that robust coherent manipulation in the large spin environments characteristic of the III-V compounds is possible without resorting to nuclear spin polarization.