Nonideality of quantum operations with the electron spin of a31Pdonor in a Si crystal due to interaction with a nuclear spin system

We examine a ${}^{31}\mathrm{P}$ donor electron spin in a Si crystal to be used for the purpose of quantum computation. The interaction with an uncontrolled system of ${}^{29}\mathrm{Si}$ nuclear spins influences the electron-spin dynamics appreciably. The hyperfine field at the ${}^{29}\mathrm{Si}$ nuclei positions is noncollinear with the external magnetic field. Quantum operations with the electron wave function, i.e., using magnetic-field pulses or electrical gates, change the orientation of hyperfine field and disturb the nuclear-spin system. This disturbance produces a deviation of the electron spin qubit from an ideal state, at a short-time scale in comparison with the nuclear-spin diffusion time. For H${}_{\mathrm{ext}}\approx 9$ T, the estimated error rate is comparable to the threshold value required by the quantum error correction algorithms. The rate is lower at higher external magnetic fields.