Figure 1

Averaged trace-norm distance $\overline{D}\left(T\right)$ between the CDD-protected final state and the desired ground state as a function of total run time $T$ for Grover's algorithm, in units of the inverse minimum gap. Cutoff frequency is $\beta ={\Delta }_{\min }/5$. The ideal (solid black) and faulty (empty squares) evolutions are included for reference. Inset shows a closeup for large $T$, using a log scale for the vertical axis. Performance improves monotonically with concatenation level $l$, with the corresponding sequence denoted CDD${}_l$. Error bars are due to averaging over 30 random realizations of $H_{\mathrm{err}}\left(t\right)$.Reuse & Permissions

Figure 2

As in Fig. 1, for the 2-SAT on a ring problem. The curves for ideal evolution and CDD${}_4$ overlap to within our numerical accuracy up to $T=25/{\Delta }_{\min }^{2\mathrm{SAT}}$. Note the minimum at $T\approx 12{\Delta }_{\min }^{2\mathrm{SAT}}$ for the faulty evolution, suggesting the existence of an optimal open-system evolution time. This optimal time increases with concatenation level, until it disappears in the ideal case and for CDD${}_4$. CDD boosts the deviation by a factor of 10 from $\overline{D}\approx 0.2$ (at $T\approx 12.5/{\Delta }_{\min }^{2\mathrm{SAT}}$) in the faulty case to $\overline{D}\approx 0.02$ (at $T\approx 34.5/{\Delta }_{\min }^{2\mathrm{SAT}}$) for CDD${}_4$.Reuse & Permissions

Figure 3

CDD${}_4$ vs QDD${}_{15}$ for the Grover problem as a function of the normalized bath correlation time $1/\left(\beta T\right)$ and normalized total run time $T/{\Delta }_{\min }^{\mathrm{G}}$. Increasing the bath correlation time $1/\beta$ at fixed $T$ generally results in improved performance for CDD${}_4$, whose performance is significantly better than QDD${}_{15}$ in in the large-$T$ regime. All results were averaged over 30 realizations of $H_{\mathrm{err}}\left(t\right)$.Reuse & Permissions

Figure 4

Performance of CDD${}_4$-protected evolution for the 2-SAT problem, as a function of pulse interval $\tau$, for different values of the frequency cutoff $\beta$ ($\tau$ in units of $1/\beta$ to separate the curves). Total time $T=4^4\tau$. The minimum in each fixed $\beta$ curve corresponds to an optimal pulse interval ${\tau }_{\mathrm{opt}}\left(\beta \right)$, and corresponding $T_{\mathrm{opt}}\left(\beta \right)$. Peak performance $\overline{D}\left(T_{\mathrm{opt}}\left(\beta \right)\right)$ improves as $\beta$ is decreased, except at $\beta ={\Delta }_{\min }^{2\text{SAT}}/0.005$ where self-averaging effects result from rapid fluctuations in ${\epsilon }_j^{\mu }\left(t\right)$ (“motional narrowing”). Results are averaged over 30 realizations of $H_{\mathrm{err}}\left(t\right)$.Reuse & Permissions

Figure 5

Trace-norm distance between the QDD-protected final state $\rho \left(T\right)$ and the desired ground state ${\rho }_M$ as a function of total run time $T$ for Grover's algorithm, in units of the inverse minimum gap. Specific sequence orders are considered: $M=1,3,4,6,7,14,15$. Odd-parity sequence orders contain an number of pulses equal to those of the CDD sequences shown in Fig. 1 (main text). In contrast to CDD, QDD-protected evolution does not more closely resemble closed-system evolution at large $T$ as the number of pulses grows.Reuse & Permissions

Figure 6

Same as in Fig. 5, for the 2-SAT on a ring problem. QDD protection extends ideal evolution as sequence order increases, ultimately saturating at $M=14$.Reuse & Permissions

Figure 7

Performance of CDD${}_4$ for Grover's search algorithm as a function of the pulse interval $\tau$ for various values of the frequency cutoff $\beta$. Short correlation times ($\beta =200{\Delta }_{\min }^G$) demand a large pulse interval to obtain minimum deviation from ideal evolution. As $\beta$ is decreased, reduced minimum deviation is achieved at pulse intervals much smaller than the correlation time ($1/\beta$). Results are averaged over 30 realizations of $H_{\mathrm{err}}$.Reuse & Permissions

Figure 8

As in Fig. 7, for QDD${}_{15}$. The pulse interval $\tau$ represents the minimum delay between pulses for QDD. In contrast to CDD${}_4$, minimum deviation between DD-protected and ideal evolution does not increase with correlation time for QDD. Note also that the minimum deviations are approximately twice as large as those obtained for CDD and require a minimum pulse interval 10 times smaller.Reuse & Permissions

Figure 9

As in Fig. 4 (main text), for QDD${}_{14}$. Here, the pulse interval $\tau$ represents the minimum delay between pulses. Minimum deviations from ideal evolution are approximately twice as large as those obtained for CDD${}_4$. Peak performance appears insensitive to the value of $\beta$.Reuse & Permissions

Figure 10

Comparison between CDD${}_4$ and QDD${}_{14}$ performance for 2-SAT on a ring as a function of the normalized correlation time ${\left(\beta T\right)}^{-1}$ and total run time $T$. As in Fig. 3 (main text), sequence performance is nearly the same for short correlation times. Increases in correlation time result in smaller deviations between DD-protected and ideal evolution for both schemes.Reuse & Permissions

Figure 11

Performance of CDD for a constant error Hamiltonian [${\epsilon }_j^{\mu }\left(s\right)\equiv {10}^{-2}$ $\forall \mu ,j$ in Eq. (2) (main text)]. The curves for ideal evolution, faulty evolution, and CDD${}_4$ overlap up to $T\approx 4/{\Delta }_{\min }^G$, after which CDD${}_4$ continues to track the ideal evolution essentially perfectly throughout the simulated range.Reuse & Permissions

Figure 12

As in Fig. 11, for QDD with $M=\{1,3,4,6,7,14,15\}$. QDD-protected evolution deviates minimally from ideal evolution for $M=15$ up to $T\approx 25/{\Delta }_{\min }^G$, thereafter $M=6,7$ are the optimal sequence orders. Unlike the CDD case, increasing the number of pulses does not result in closed-system-like behavior over the entire range of $T$ considered. $M=14,15$ evolution closely coincides with ideal evolution more so than any other $M$ values; however it also diverges faster than all order sequence orders at critical $T$ values.Reuse & Permissions