We theoretically consider the possibility of using solid state nuclear spins in a semiconductor nanostructure environment as long-lived, high-fidelity quantum memory. In particular, we calculate, in the limit of a strong applied magnetic field, the fidelity of P donor nuclear spins as a function of time in random bath environments of Si and GaAs, and the lifetime of excited intrinsic spins in polarized Si and GaAs environments. In the former situation, the nuclear spin dephases due to spectral diffusion induced by the dipolar interaction among nuclei in the bath; in the interest of the high fidelity requirements necessary for fault-tolerant quantum computing, we consider the initial quantum memory decay caused by a non-Markovian bath. We calculate this nuclear spin memory time in the context of Hahn and Carr-Purcell-Meiboom-Gill ͑CPMG͒ refocused spin echoes using a formally exact cluster expansion technique which has previously been successful in dealing with electron spin dephasing in a solid state nuclear spin bath. With decoherence dominated by transverse dephasing ͑T 2 ͒, we find it feasible to maintain high fidelity ͑losses of less than 10 −6 ͒ quantum memory on nuclear spins for times of the order of 100 s ͑GaAs:P͒ and 1 to 2 ms ͑natural Si:P͒ using CPMG pulse sequences of just a few ͑ϳ2-4͒ applied pulses. We also consider the complementary situation of a central flipped intrinsic nuclear spin in a bath of completely polarized nuclear spins where decoherence is caused by the direct flip-flop of the central spin with spins in the bath. Exact numerical calculations that include a sufficiently large neighborhood of surrounding nuclei show lifetimes on the order of 1 -5 ms for both GaAs and natural Si. Our calculated nuclear spin coherence times may have significance for solid state quantum computer architectures using localized electron spins in semiconductors where nuclear spins have been proposed for quantum memory storage.

We investigate pure dephasing decoherence (free induction decay and spin echo) of a spin qubit interacting with a nuclear spin bath. While for infinite magnetic field B the only decoherence mechanism is spectral diffusion due to dipolar flip-flops of nuclear spins, with decreasing B the hyperfine-mediated interactions between the nuclear spins become important. We give a theory of decoherence due to these interactions which takes advantage of their long-range nature. For a thermal uncorrelated bath we show that our theory is applicable down to B approximately 10 mT, allowing for comparison with recent experiments in GaAs quantum dots.

We investigate decoherence due to pure dephasing of a localized spin qubit interacting with a nuclear spin bath. Although in the limit of a very large magnetic field the only decoherence mechanism is spectral diffusion due to dipolar flip-flops of nuclear spins, with decreasing field the hyperfine-mediated interactions between the nuclear spins become important. We take advantage of their long-range nature, and resum the leading terms in an 1/N expansion of the decoherence time-evolution function (N , being the number of nuclear spins interacting appreciably with the electron spin, is large). For the case of the thermal uncorrelated bath we show that our theory is applicable down to low magnetic fields (∼ 10 mT for a large dot with N = 10 6 ) allowing for comparison with recent experiments in GaAs quantum dot spin qubits. Within this approach we calculate the free induction decay and spin echo decoherence in GaAs and InGaAs as a function of the number of the nuclei in the bath (i.e. the quantum dot size) and the magnetic field. Our theory for free induction decay in a narrowed nuclear bath is shown to agree with the exact solution for decoherence due to hyperfine-mediated interaction which can be obtained when all the nucleielectron coupling constants are identical. For the spin echo evolution we show that the dominant decoherence process at low fields is due to interactions between nuclei having significantly different Zeeman energies (i.e. nuclei of As and two isotopes of Ga in GaAs), and we compare our results with recent measurements of spin echo signal of a single spin confined in a GaAs quantum dot. For the same set of parameters we perform calculations of decoherence under various dynamical decoupling pulse sequences, and predict the effect of these sequences in low B regime in GaAs. arXiv:0903.2256v3 [cond-mat.mes-hall]