Donor-dot scheme
6.8 Decoherence
There are several menaces to the full coherence of the quantum computer sketched in Sec. 6.6, and a discussion of them all is well beyond the scope of this thesis. Some brief attention is now given to the main decoherence sources occurring at the stage that this chapter has focused upon, i.e. performing an addressable CNOT between the data and the measurement qubits of the surface code. The charge (orbital) degrees of freedom are af- fected by (i) interactions with phonons of the silicon lattice, whose spectral densities differ significantly between the bulk of the layer and the region close to the interface [GKL04]; (ii) voltage noise due to the gates confining the dot, that are used to tune the magnitude of the donor/dot exchange couplingJ [FTCL09, DSH+13]; (iii) electromagnetic influences caused by stray charges, particularly dangling bonds trapped at the interface with the ox- ide [AFS82]. While the influence of the phonons can, to some extent, be reduced by a regime of low temperatures, the latter two sources of decoherence are known to cause1/f noise. Since the scheme proposed in this chapter relies on spin degrees of freedom, the only effect of such interactions with the environment could be to modify uncontrollably the strength of the coupling that enables the addressable CNOT between the data and measurement qubits in the surface code. Qualitatively one may expect that such influence should not be particularly detrimental, since the CNOT operations have been devised to depend intrinsically little on the magnitude of the coupling, but further work is needed to set quantitative thresholds to this statement.
The donor spin states, both NMR- and ESR-like, addressed with the SWAP described in the previous section, have their coherence protected by the clock transitions, that effec- tively enhance theirT2to timescales at least 3 orders of magnitude larger than the∼10µs
total gating times required to complete the total CNOT sequence proposed previously. The most important source of dephasing of the qubits involved, thus, affects the dot electron spin, which still suffers from spectral diffusion due to hyperfine coupling with the bath of29Si nuclei; moreover, the vicinity of a Si/SiO2 interface enhances the influence from
dipolar coupling with paramagnetic impurities in the oxide [MDL+14], as extensively de- scribed theoretically in Ref. [dS07].
Several theoretical treatments of the complete central spin - nuclear bath dynamics have been developed in literature [YLS06, WDS06], even extended to isotope-enriched silicon [WCM+10]: the most relevant feature highlighted by those works is probably the non-
Markovian character of the bath fluctuations induced by the29Si flip-flops, that take place
over timescales long compared to the central spin evolution, due to the weak magnetic moment of the nuclei. However, since we are not interested in the nuclear dynamics here, but only on the collective effect those flip-flops have on the local dot electron magnetic field, we can treat the hyperfine couplings as classical Overhauser field variables [RPB10],
6.9. CONCLUSIONS
fixed over the short SWAP timescales considered here [RB09]. Their Hamiltonian is: Hnoise = 1 2geµB(2S z doth z+S+ doth − +Sdot−h+), whereh=X n AnIn≡ X n hn, (6.48)
with the index n running over the non-zero nuclear spins located within the spatial ex- tent of the dot electron wavefuntion, andAn the corresponding hyperfine coupling con-
stant. The operating temperatures make sure that such nuclei are completely unpolarized, thus [RPB10] we can assume that the classical stochastic variablehnfollows an isotropic
Gaussian distribution p(hx,y,z
n ) =
1 √
2πσhe
−h2
n/2σh2, where the standard deviation σh can
be extracted directly from experimental measurement of the inhomogeneous broadening of the ESR linewidths. The most recent estimates related to quantum dots engineered in silicon layers include σh ≈ 1 MHzmeasured for natSi/SiGe dots [KSW+14], and an
astonishingσh ≈ 2.4kHz [VHY+14] in isotopically enriched (800 ppm of29Si nuclei)
silicon. We will show how the SWAP proposed here preserves high fidelities even with a conservative estimate ofσh ≈25 MHz.
First, we have to evaluate the expectation value of the Hamiltonian in Eq. 6.48 on the logical states|1iA,|2iAdefined in Eq. 6.24:
Ah1|H noise|1iA=−Ah2|Hnoise|2iA=−geµB hdot z 2 , Ah1|H noise|2iA= 0,
thus we find, as expected, that hyperfine coupling to nuclei only results in dephasing, not relaxation. Second, we evaluate how the dynamics is changed by any fixed value hz, weight each individual calculated fidelity|hψ(t0, hz)|(1,0)i|2 with the corresponding
Gaussian probabilityp(hz), and integrate over all possiblehzto get the overall fidelity.
We report in Fig. 6.18 the effect of spectral diffusion-induced dephasing on the coherent SWAP fidelities calculated in Sec. 6.7.2, assumingσh = 25 MHz. This analysis is again
performed across two orders of magnitude of exchange couplings relative to the different donor/dot pairs.
6.9
Conclusions
In this chapter we have described a quantum computing scheme involving quantum dots and bismuth donors in silicon, within an automatically scalable architecture as provided by the implementation of a surface code, which moreover poses greatly relaxed constraints over the coherence times of the quantum information manipulated. Rotations on the Bloch sphere of the physical bismuth qubits, playing the role of measurement qubits, are easily performed within the large Hilbert space made available by the spin levels of the Si:Bi system, where global microwave fields are well established means of manipulation. The
CHAPTER6: DONOR-DOT SCHEME