Optimizing electrically controlled echo sequences for the exchange-only qubit

Recently, West and Fong [New J. Phys. 14, 083002 (2012)] introduced an echo scheme for an exchange-only qubit, which relies entirely on the exchange interaction. Here, we compare two different exchange-based sequences and two optimization strategies, Uhrig dynamical decoupling and optimized filter function dynamical decoupling, which were introduced for a single spin qubit and are applied in this paper to the three spin exchange-only qubit. The calculation shows that the adaption of the optimization concepts can be achieved by straightforward calculation. We consider two types of noise spectra, Lorentzian and Ohmic noise. For both spectra, the results reveal a slight dependence of the performance on the choice of the echo sequence.

Figure 1

Sketch of the system of three quantum dots each hosting an electron represented by a large red (gray) dot and arrow. The electron spin states in dots 1 and 2 (2 and 3) can be coupled via the exchange interaction $J_{12} \left(J_{23}\right)$, while there is no direct coupling between dots 1 and 3. The electron spins in each dot experience the influence of the nuclear spins, which are represented by small yellow (light gray) arrows, via the hyperfine coupling. We describe this influence of the nuclear spins by fluctuating Overhauser fields.

Figure 2

Infidelity $1-F$ for Ohmic noise (a) and Lorentzian noise (b) in dependence of dimensionless storage time $T^{\prime }$ for pulse times chosen according to the CPMG scheme. The number of applied pulses $n$ is $n=3$ (cyan, light gray), $n=4$ (green, dark gray), and $n=10$ (black). The applied sequences are the all cyclic permutations realized by applying $P={\mathrm{SWAP}}_{12}{\mathrm{SWAP}}_{23}$ at every time $T{\delta }_j$ (solid lines), the sequence of pairs of $P$ and $P^{-1}$ (dotted lines), and the single spin operations (dashed lines).

Figure 3

Infidelity $1-F$ in dependence of dimensionless storage time $T^{\prime }$ for Ohmic noise (a) and Lorentzian noise (b) for pulse times chosen according to the UDD optimization strategy. The number of applied pulses $n$ is $n=2$ (orange, light gray), $n=4$ (green, dark gray), and $n=10$ (black). The applied sequences are plotted in the same line styles as in Fig. 2. In (b) the results for the exchange-based pulses with the all cyclic permutations and the pairs of $P$ and $P^{-1}$ are very similar, thus the solid and the dotted lines are on top of each other. For $n=2$ these sequences are identical by definition. The better performance of the single spin pulses (dashed lines) compared to the SWAP-based sequences for the same number of pulses, $n$, is due to the fact that it allows for the filter function to be zero up to the order $n$ while it is only order $n/2$ for the SWAP-based pulse sequences.

Figure 4

Double logarithmic plot of ${\delta }_1$, which is the smallest pulse interval, in dependence of the order $m$ for UDD applied to the all cyclic sequence (black circles), to the West-Fong sequence [red (gray) circles, values from Ref. 6], and for comparison to a single spin problem (gray squares).

Figure 5

OFDD values of ${\delta }_1,...,{\delta }_n$ for $n=3$ (a) and $n=4$ (b) in dependence of the dimensionless storage time $T^{\prime }$ considering single spin echo (black dashed lines), all cyclic operations (red solid lines), and the West-Fong sequence (dotted lines) where applying $P$ is indicated by red color and $P^{-1}$ by blue color. For comparison the UDD values of ${\delta }_j \left(j=1,...,n\right)$ are shown with gray lines in the respective line style. Note that for $n=3$ the UDD solution exists only for the single spin operations.

Figure 6

Infidelity $1-F$ in dependence of dimensionless storage time $T^{\prime }$ for Ohmic noise (a) and Lorentzian noise (b) for pulse times chosen according to OFDD. The color codes and the line styles are identical to Fig. 2.

Figure 7

Comparison of UDD (gray) and OFDD (black) optimized values of infidelity $1-F$ in dependence of the dimensionless storage time $T^{\prime }$ for Ohmic (a) and Lorentzian (b) noise. Results are shown with dashed lines for single spin echoes, with solid lines for the all cyclic sequence, and with dotted lines for the West-Fong sequence. The number of pulses is $n=10$.