Second-order shaped pulses for solid-state quantum computation

We present the construction and detailed analysis of highly optimized self-refocusing pulse shapes for several rotation angles. We characterize the constructed pulses by the coefficients appearing in the Magnus expansion up to second order. This allows a semianalytical analysis of the performance of the constructed shapes in sequences and composite pulses by computing the corresponding leading-order error operators. Higher orders can be analyzed with the numerical technique suggested by us previously. We illustrate the technique by analyzing several composite pulses designed to protect against pulse amplitude errors, and on decoupling sequences for potentially long chains of qubits with on-site and nearest-neighbor couplings.

Figure 1

(Color online) A single-spin second-order inversion $\left(\pi \right)$ pulse $Q_1\left(\pi \right)$. (a) Pulse profile over a complete period. (b) Evolution of the spin beginning with $s_z=1$. (c) The power spectrum of the pulse. The vertical lines denote the location of the harmonics. As seen, the spectral weight is almost entirely confined to $\omega <5{\omega }_0$.

Figure 2

(Color online) As in Fig. 1, but for the single-spin second-order $\pi ∕2$ pulse $Q_1(\pi ∕2)$.

Figure 3

(Color online) Pulse shapes for ${\varphi }_0=\pi ∕2$. Solid lines represent $Q_L(\pi ∕2)$, dashed lines correspond to $S_L(\pi ∕2)$. Pulse shapes with $L=1$ are drawn with thin blue lines, while those with $L=2$ are drawn with thick red lines. The black dotted line shows the Gaussian shape $G_{010}(\pi ∕2)$.

Figure 4

(Color online) As in Fig. 3 but for the inversion pulses, ${\varphi }_0=\pi$. Note that the first-order pulses [$S_L\left(\pi \right)$, dashed lines] actually have a smaller power than the Gaussian pulse.

Figure 5

(Color online) As in Fig. 3 but for the pulses with ${\varphi }_0=2\pi$. The pulse shapes appear to resemble those of a pair of consecutive $\pi$ pulses. Both first- and second-order pulses have a smaller power than the Gaussian pulse.

Figure 6

(Color online) Illustration of decoupling accuracy with sequences 8 and 32 for chains of $n=4$ qubits with different couplings as indicated on the plots. The plots show average infidelity [see Eq. (A4)] computed at fixed time $t$ as the pulse duration ${\tau }_p$ was reduced to accommodate a different number of sequences. The values of model parameters were randomly chosen and remained the same for all simulations. Symbols are the data points, lines are the single-parameter fits of the mismatch $\delta$ [see Eq. (A3)] to $\delta =b{\tau }_p^K$, where the values of $K$ indicated on the plots correspond to those in Table . (a) Gaussian pulses; (b) first-order pulses $S_1$; (c) second-order pulses $Q_1$.

Figure 7

(Color online) Scaling of the infidelity $1-F$ at $t∕{\tau }_p=128$ with the chain length $n$ for a particular realization of an $XXZ$ chain with on-site disorder ${\Delta }_i^z$, decoupling sequence 8, pulse shapes as indicated. (Data for the pulse $G_{010}$ divided by 10 to fit with the other data.) While the unitary matrix mismatch ${\delta }^2$ [see Eq. (A3)] grows exponentially with the chain length $n$, ${\delta }^2\propto 2^n$, the leading-order contribution to the corresponding infidelity $(1-F)$ represents the probability of error in one of the clusters, and it scales only linearly, as also seen in the plots.

Figure 8

(Color online) Contour plots of the average fidelity for the composite pulse SCROFULOUS ${\pi }_{60}{\pi }_{300}{\pi }_{60}$ with (a) Gaussian pulses $G_{010}$; (b) first-order self-refocusing pulses $S_1$ (the plots for pulses $Q_1$ look similarly but symmetric with respect to horizontal axis); (c) first-order pulse with amplitude correction $\upsilon ={\upsilon }^{\prime }={\upsilon }^{″}=0$; (d) second-order pulse with amplitude correction, $\upsilon ={\upsilon }^{\prime }={\upsilon }^{″}=\alpha =0$. The axes are the relative frequency mismatch ${\tau }_p\Delta$ and the relative pulse amplitude $(1+f)$, see text.

Figure 9

(Color online) Contour plots of the average infidelity $1-F$ for the composite pulse ${\mathrm{BB}}_1^{\left(W\right)}$ $\left[{\pi }_0{\pi }_{\varphi }{\left(2\pi \right)}_{3\varphi }{\pi }_{\varphi }\right]$ with (a) first-order self-refocusing pulses $S_1$; (b) second-order pulses $Q_1$; (c) first-order pulses with amplitude correction; (d) second-order pulses with amplitude correction. The axes are the relative frequency mismatch ${\tau }_p\Delta$ and the relative pulse amplitude $(1+f)$, see text.

Figure 10

(Color online) As in Fig. 9 but for the symmetrized sequence ${\mathrm{BB}}_1^{\left(CLJ\right)}$ $\left[{(\pi ∕2)}_0{\pi }_{\varphi }{\left(2\pi \right)}_{3\varphi }{\pi }_{\varphi }{(\pi ∕2)}_0\right]$. The pulse shapes are (a) first-order pulses $S_1$ (the fidelity for the second-order pulses $Q_1$ is similar); (b) first-order pulses with amplitude correction; (c) second-order pulses with amplitude correction. Note how regular are the shapes of high-fidelity regions.

Figure 11

(Color online) As in Fig. 9 but for the sequence $\mathrm{B}{\mathrm{B}}_1^{W^{\prime }}$ $\left[{\pi }_0{\pi }_{\varphi }{\pi }_{3\varphi }{\pi }_{3\varphi }{\pi }_{\varphi }\right]$ using only $\pi$ pulses. (a) First-order pulses with amplitude correction; (b) Second-order pulses with amplitude correction. The absence of the error terms linear in $\alpha$ [see Eqs. (54, 55)] produces a much wider high-fidelity region already with first-order pulses.

Figure 12

(Color online) Decoupling errors as a function of relative pulse amplitude for a chain of $n=4$ qubits with different decoupling schemes as indicated. (a) Ising chain in the presence of individual chemical shifts ${\Delta }_i^z$. Eight-pulse sequence with $\mathrm{B}{\mathrm{B}}_1$ composite pulses offers the best accuracy, which remains essentially the same whether the sequence is used with regular first- or second-order pulses or with the pulses stabilized against amplitude errors (first-order pulses $L_1$ and second-order pulses $L_3$). However, the pulse $L_3$ works well enough even with regular eight-pulse sequence. While the details of the amplitude scaling differ, at the level of $1-F={10}^{-4}$, the first-order amplitude-protected pulses $L_1$ and regular second-order pulses $Q_1$ have comparable accuracy. The use of first-order pulses show relatively poor performance even on resonance. (b) $XXZ$ chain in the presence of individual chemical shifts ${\Delta }_i^z$. With $XXZ$ coupling, the $\mathrm{B}{\mathrm{B}}_1$ composite pulse is no longer accurate, as the errors appear already in the linear order in ${\tau }_p{J}^{\perp }$ (not shown). With the regular eight-pulse sequence 8, the best accuracy is obtained for the pulses with amplitude correction.