Arbitrarily accurate composite pulse sequences

Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error $ϵ$. Independent of the initial state of the system, current techniques allow the error to be reduced to $O\left({ϵ}^3\right)$. Here, we extend the composite pulse technique to cancel errors to $O\left({ϵ}^n\right)$, for arbitrary $n$.

Figure 1

Comparison of the narrowband and broadband composite pulse sequences generated by the TS method. The Wimperis BB1, PB1, and NB1 sequences are included in this family, and are equivalent to B2, P2, and N2.

Figure 2

Composite pulse error $E$ as a function of base error $ϵ$ for a variety of composite pulse sequences. $\mathrm{P}n$, $\mathrm{B}n$, $\mathrm{SK}n$, and $\mathrm{SB}n$ are the $n\text{th}$ order passband, broadband, SK, and combined B4-SK sequences, respectively. The number in the brackets refers to the number of imperfect $2\pi$ rotations in the correction sequence. Note how pulses of the same order (such as P6, B6, SK6, SB6) have the same slope (asymptotic scaling) for low values of $ϵ$, but can have widely varying performance when $ϵ$ is large. The inset plots the scaling of this sequence length with order $n$ for $\mathrm{SK}n$ ($\mathrm{SB}n$ is very similar) for $n⩽30$ and compares it with the upper bound obtained with numerical methods.