Designing short robust not gates for quantum computation

Composite pulses, originally developed in nuclear magnetic resonance (NMR), have found widespread use in experimental quantum information processing (QIP) to reduce the effects of systematic errors. Most pulses used so far have simply been adapted from existing NMR designs, and while techniques have been developed for designing composite pulses with arbitrary precision, the results have been quite complicated and have found little application. Here, I describe techniques for designing short but effective composite pulses to implement robust not gates, bringing together existing insights from NMR and QIP, and present some composite pulses.

Figure 1

Vector diagrams showing the cancellation of first-order pulse strength error terms in composite pulse sequences with (a) $n=3$ and (b) $n=5$. The equilateral polygons can be traced in either direction, depending on the precise relationship chosen between phases in the toggling frame, but the overall orientation is fixed by the constraint on $\Phi$.

Figure 2

Fidelity achieved by (a) a simple pulse and composite pulses with $n=3$ optimized to suppress first-order pulse strength errors (b) and off-resonance errors (c). Fidelity is plotted as a function of the fractional pulse strength error $\epsilon$ and the off-resonance fraction $f$. Contours are drawn at 90$\%$, 99$\%$, and 99.9$\%$ fidelity, that is, logarithmically spaced infidelities with the inmost contour at an infidelity of ${10}^{-3}$.

Figure 3

Fidelity achieved by some composite pulses designed to suppress pulse strength errors: (a) ${\text{F}}_1$, (b) symmetric BB1, (c) a new symmetric pulse with $n=5$. The inmost contour is drawn at an infidelity of ${10}^{-4}$.

Figure 4

Coefficient of the fourth-order infidelity terms for pulse strength error ($F_{\epsilon }$) and off-resonance error ($F_f$) in five pulse composite pulses with simultaneous compensation of both errors to first order.

Figure 5

Fidelity achieved by some composite pulses with $n=5$ designed to suppress both pulse strength and off-resonance errors: (a) the symmetric Knill-type pulse, (b) pulse with suppression of second-order pulse strength errors, (c) pulse with suppression of second-order off-resonance errors. The inmost contour is drawn at an infidelity of ${10}^{-4}$.

Figure 6

Fidelity achieved by various composite pulses with $n=7$: (a) second-order suppression of both pulse strength and off-resonance errors, (b) symmetric pulse optimized for pulse strength errors, (c) symmetric pulse optimized for off-resonance errors. The inmost contour is at an infidelity of ${10}^{-5}$.

Figure 7

Fidelity achieved by various composite pulses with $n=9$ giving simultaneous suppression of first- and second-order pulse strength and off-resonance errors: (a) ASBO-9(7A), (b) ASBO-9(7B), (c) a new symmetric pulse. The inmost contour is at an infidelity of ${10}^{-6}$.