Reducing sequencing complexity in dynamical quantum error suppression by Walsh modulation

We study dynamical error suppression from the perspective of reducing sequencing complexity, with an eye toward facilitating the development of efficient semiautonomous quantum-coherent systems. To this end, we focus on digital sequences where all interpulse time periods are integer multiples of a minimum clock period and compatibility with digital classical control circuitry is intrinsic. We use so-called Walsh functions as a unifying mathematical framework; the Walsh functions are an orthonormal set of basis functions which may be associated directly with the control propagator for a digital modulation scheme. Using this insight, we characterize the suite of resulting Walsh dynamical decoupling sequences—including both familiar and novel control sequences—and identify the number of periodic square-wave (Rademacher) functions required to generate the associated Walsh function as the key determinant of the error-suppressing features. We also show how Walsh modulation may be employed for the protection of certain nontrivial logic gates. Based on these insights, we identify Walsh modulation as a digital-efficient approach for physical-layer error suppression.

Figure 1

Walsh functions listed by Paley ordering. (a) The first 32 Walsh functions. (b) The normalized pulse locations ${\delta }_j$ corresponding to the digital transition points of Walsh functions. Sequences ${\text{WDD}}_{2^r-1}$ are highlighted in both panels (thick lines, left; open symbols, right).

Figure 2

Error suppression properties of WDD sequences. (a) Filter functions calculated for sequences derived from $W_{2^r-1}(t/\tau )$ and corresponding to CDD${}_r$ (WDD1 at top to WDD31 at bottom), with fixed total sequence duration $\tau$. The horizontal axis corresponds to angular frequency in units of $1/\tau$. For increasing values of $r$ (top to bottom), the number of fundamental Rademacher functions that are needed to produce the sequence increases, and the order of error suppression, demonstrated graphically by the low-frequency roll-off of the filter function, increases by 6 dB/octave. Dashed lines correspond to the UDD sequence providing the same order of error suppression. (b) Filter functions for all WDD sequences with $n<32$ and $r=4$. UDD4 is represented by a dashed line, and WDD15 by a solid line. Dashed lines correspond, left to right, to WDD23, WDD27, WDD29, and WDD30. (c) Decoupling error as a function of time in units of the inverse cutoff frequency, $1/{\omega }_c$ in $S_{\beta }\left(\omega \right)\propto \alpha {\omega }^{-2}\exp [-{(\omega /{\omega }_c)}^2]$. Here, $\alpha =5\times {10}^{16}$, and frequency ranges are set as in Ref. [58]. Sequences are the same as in (a) and increase in $n$ from left to right.

Figure 3

Noise suppression at fixed frequency. (a) Filter function as a function of frequency $\omega /2\pi$ in units of $1/\tau$ on a semilog scale for various WDD sequences, resulting in a notch at frequency $\delta$ where noise is suppressed. The notch bandwidth increases with sequency and the order of error suppression about $\delta$ increases with $r$. Inset: The same filter functions plotted on a linear scale. (b) Filter function normalized by ${\omega }^2$ for WDD15 and WDD31. The solid thick line (bottom trace) shows WDD15 at $\delta /2\pi =2^5$, where the horizontal axis represents detuning from $\delta$. Here the order of error suppression as a function of detuning has increased by one order and is comparable to that for WDD31 near zero frequency. See text for details.