Optimally band-limited spectroscopy of control noise using a qubit sensor

Classical control noise is ubiquitous in qubit devices, making its accurate spectral characterization essential for designing optimized error suppression strategies at the physical level. Here, we focus on multiplicative Gaussian amplitude control noise on a driven qubit sensor and show that sensing protocols using optimally band-limited Slepian modulation offer substantial benefit in realistic scenarios. Special emphasis is given to laying out the theoretical framework necessary for extending nonparametric multitaper spectral estimation to the quantum setting by highlighting key points of contact and differences with respect to the classical formulation. In particular, we introduce and analyze two approaches (adaptive vs single setting) to quantum multitaper estimation, and show how they provide a practical means to both identify fine spectral features not otherwise detectable by existing protocols and to obtain reliable prior estimates for use in subsequent parametric estimation, including high-resolution Bayesian techniques. We quantitatively characterize the performance of both single- and multitaper Slepian estimation protocols by numerically reconstructing representative spectral densities, and demonstrate their advantage over dynamical-decoupling noise spectroscopy approaches in reducing bias from spectral leakage as well as in compensating for aliasing effects while maintaining a desired sampling resolution.

Figure 1

Effect of spectral leakage and bias on PSD estimation. The FF generated by a 12-pulse CPMG control sequence (blue solid line) is depicted in two spectral estimation scenarios, with the shaded area corresponding to the passband centered at ${\omega }_s$. Leakage is manifest in the presence of a high-frequency lobe, marked with an arrow. In (a), the PSD being estimated (black dashed line) has minimal spectral overlap with the high-frequency lobe. Consequently, the leakage has little effect on the expected value of the estimate in Eq. (25). In (b), the PSD (black dashed line) and the high-frequency lobe have significant overlap. This causes the high-frequency lobe to contribute to the overlap integral in Eq. (25), resulting in a biased, overestimated PSD at ${\omega }_s$.

Figure 2

DPSSs and DPSWFs for $N=500$ and $W=4/N$. The DPSSs $\left\{v_n^{\left(k\right)}(N,W)\right\}$ (top row) are plotted vs $n$ for orders $k=0,1,2,3$ (left to right). The squared magnitudes of DPSWFs $|U^{\left(k\right)}{(N,W;\omega )|}^2$ (bottom row) are plotted vs angular frequency scaled by $\mathrm{\Delta }t/\left(2\pi \right)$ for orders $k=0,1,2,3$ (left to right). The boundaries of the passbands at $\omega \mathrm{\Delta }t/\left(2\pi \right)=\pm W$ are marked by dashed vertical lines.

Figure 3

Convergence to an ideal bandpass filter with increasing $N$. The average squared magnitude of the first $K$ DPSWFs, denoted by ${\rho }_K(N,W;\omega )$ [Eq. (30)], for $W=0.008$ and (a) $N=200$ and $K=3$, (b) $N=400$ and $K=6$, (c) $N=800$ and $K=12$, and (d) $N=1600$ and $K=25$. Boundaries of the passband at $\omega \mathrm{\Delta }t/\left(2\pi \right)=\pm W$ are marked with vertical dashed lines. With increasing $N,{\rho }_K(N,W;\omega )$ approximates an ideal (square, band-limited) filter with increasing accuracy.

Figure 4

Shifted DPSS amplitude filters. The unshifted DPSS FF of order $k=1$ with $N=500$ and $W=10/N$ (a) has a passband centered at $\omega \mathrm{\Delta }t/\left(2\pi \right)=0$. In (b), the $k=1$ DPSS filter is shifted by ${\omega }_s\mathrm{\Delta }t/\left(2\pi \right)=0.013$ with COS modulation, producing a filter with two duplicate passbands centered at $\omega \mathrm{\Delta }t/\left(2\pi \right)=\pm 0.013$. In (c), SSB modulation with ${\omega }_s\mathrm{\Delta }t/\left(2\pi \right)=0.013$ halves the passband of the $k=1$ DPSS filter, shifting the lower half by $-0.013$ and the upper half by 0.013. (d) The $k=0$ DPSS filter with $N=500$ and $W=4/N$ (inset) is shifted by ${\omega }_s\mathrm{\Delta }t/\left(2\pi \right)=0.003$ with SIN modulation (green dashed line), COS modulation (purple dash-dotted line), and CS modulation (black solid line).

Figure 5

Aliasing of filters with spectral concentration near the Nyquist frequency ${\omega }_{\text{N}}$. The unshifted $k=0$ DPSS filter, centered at ${\omega }_1=0$ (green), is plotted along with $k=0$ DPSS filters COS-shifted by ${\omega }_2=5\pi /\left(8\mathrm{\Delta }t\right)$ (blue) and ${\omega }_3=7\pi /\left(8\mathrm{\Delta }t\right)$ (purple). The $k=0$ DPSS filters shifted by ${\omega }_2$ and ${\omega }_3$ have alias peaks beyond ${\omega }_{\text{N}}$ (shaded area), centered at ${\omega }_2^{\prime }=11\pi /\left(8\mathrm{\Delta }t\right)$ and ${\omega }_3^{\prime }=9\pi /\left(8\mathrm{\Delta }t\right)$. The magnitude of the peaks decreases with $\omega$ according to the envelope function $s(\omega ,\mathrm{\Delta }t)$ (black dashed line).

Figure 6

Filters produced by the SSQM optimization procedure for $N=200$ (a), $N=400$ (b), $N=800$ (c), and $N=1600$ (d). In the time domain, the filters are linear combinations of the first $K-1$ DPSSs with $W=0.008$ and $\mathrm{\Delta }t=1\mu \mathrm{s}$. The boundaries of the filter passband at $\omega \mathrm{\Delta }t/\left(2\pi \right)=\pm W$ are marked with vertical dashed lines.

Figure 7

Sample FFs used to reconstruct the Lorentzian PSDs. Filters corresponding to the first four RSE sequences are plotted in (a), which are (from left to right) a constant amplitude control waveform and CPMG with $n=2,3,4$ sign changes. Plotted in (b) are the first four DPSS filters used in the reconstruction, which are generated by the $k=0$ DPSS shifted by ${\omega }_s=0,2\pi /T,3\pi /T,4\pi /T$ (left to right) with $N=500,\mathrm{\Delta }t=4\mu \mathrm{s}$, and $W=1/N$. For both the RSE and DPSS, the control waveforms have total duration $T=2$ ms and are normalized so that ${\int }_0^{T}dt\mathrm{\Omega }{\left(t\right)}^2=900{\mathrm{rad}}^2$ ${\mathrm{s}}^{-1}$, ensuring that the areas enclosed by the filters are equal by Eq. (22). The leakage of the RSE filter for CPMG with $n=7$ is shown in (c) with arrows highlighting the largest lobes outside the main passband. For comparison, the $k=0$ DPSS shifted by ${\omega }_s=7\pi /T$ is plotted in (d).

Figure 8

Effects of leakage bias on spectral reconstructions. Two Lorentzian PSDs as in Eq. (64), with $A=4\times {10}^{-4}{\mathrm{Hz}}^{-1},w_p/2\pi =1.11$ kHz, and peaks centered at $p/2\pi =0$ kHz (a) and $p/2\pi =4.62$ kHz (c), are estimated using the RSE and $k=0$ DPSS FFs depicted in Fig. 7. Panels (a) and (c) compare numerically simulated RSE (purple asterisks) and $k=0$ DPSS eigenestimates (green circles) of the target PSD (black solid line). The numerical estimates are produced by $M=2000$ simulated qubit measurements in the $z$ basis per control sequence. Also shown are the expected values of RSE (purple dashed line) vs DPSS eigenestimates (green dash-dotted line). For $p/2\pi =0$ kHz, the uncertainty and relative error $e\left(\omega \right)\equiv [\langle {\widehat{S}}_{\mathrm{\Omega }}\left(\omega \right)\rangle -S_{\mathrm{\Omega }}\left(\omega \right)]/S_{\mathrm{\Omega }}\left(\omega \right)$ of the RSE estimates (purple dashed line) and the DPSS eigenestimates (green dash-dotted line) are plotted in (b). The same quantities for $p/2\pi =4.62$ kHz are shown in (d).

Figure 9

Aliasing and sampling resolution for DPSS vs comb-based approaches. The shaded region is the area underneath the target PSD, a multipeaked Gaussian spectrum given by Eq. (65), with $C_1=0.5$ ms, ${\omega }_1/2\pi =0$ kHz, ${\sigma }_1/2\pi =3.50$ kHz, $C_2=0.35$ ms, ${\omega }_2/2\pi =23.9$ kHz, ${\sigma }_1/2\pi =6.21$ kHz. Both (a) and (b) show the expected $k=0$ eigenestimates (blue crosses) and the expected frequency comb estimates (red hollow circles). In (a), the parameters of the sequences used are chosen so that ${\omega }_{\text{N}}/2\pi ={\omega }_{\text{N}}^{\text{eff}}/2\pi =12.7$ kHz (dashed vertical line), whereas ${\omega }_{\text{N}}/2\pi ={\omega }_{\text{N}}^{\text{eff}}/2\pi =49.0$ kHz (dashed vertical line) in (b). In the comb approach, extending the Nyquist frequency comes at the cost of degrading the sampling resolution, impacting the visibility of the first peak.

Figure 10

Detection of fine spectral features by different Slepian-based estimation protocols. The inset of (a) shows the target PSD, $S_{\mathrm{\Omega }}\left(\omega \right)=C/\left\{\right[\left(\right|\omega |-p)/w_p{]}^2+1\}+s_0$, which consists of a narrow Lorentzian centered at $p/2\pi =7.96$ kHz, with amplitude $C=4$ ms and width $w_p/2\pi =0.08$ kHz, along with an additive white noise component of magnitude $s_0=2\times {10}^{-4}{\text{Hz}}^{-1}$ and a cutoff frequency of ${\omega }_c/2\pi =17.5$ kHz. Three different estimates used to detect the line component are plotted in (a): the $k=0$ eigenestimate ${\widehat{S}}_{\mathrm{\Omega }}^{\left(0\right)}\left(\omega \right)$ (green hollow circles), the SSQM estimate ${\widehat{S}}_{\mathrm{\Omega }}^{\text{ss}}\left(\omega \right)$ (blue asterisks), and the AQM estimate ${\widehat{S}}_{\mathrm{\Omega }}^{\text{m}}\left(\omega \right)$ (purple hollow squares). The filters corresponding to the $k=0$ eigenestimates and SSQM estimates at ${\omega }_{s,3}/2\pi =5.25$ kHz, ${\omega }_{s,4}/2\pi =7.00$ kHz, and ${\omega }_{s,5}/2\pi =8.75$ kHz are plotted in (b) and (c), respectively, whereas the effective filters of the AQM at the same frequencies are shown in (d). The location of the peak is represented by the black dashed line. As before, all DPSS amplitude control waveforms used are normalized so that ${\int }_0^{T}dt\mathrm{\Omega }{\left(t\right)}^2=900{\mathrm{rad}}^2$ ${\mathrm{s}}^{-1}$. For each shift frequency, $M=2600$ simulated qubit measurements along ${\sigma }_z$ are used for estimation. In the AQM protocol, these measurements are divided between 13 measurement settings, corresponding to each DPSS.

Figure 11

Reconstruction of a narrow Lorentzian peak via Bayesian estimation with multitaper prior. In (a), the prior mean of the discretized PSD, given by the renormalized interpolated estimate ${\widehat{S}}_q^{\mathcal{I}}$, is plotted (blue line) along with the 0.95 credible interval (shaded in gray). To obtain a high-resolution estimate of the fine spectral features in the 5.4–10.3 kHz region, additional measurements are performed using $k=0$ DPSS control waveforms with $N=500,\mathrm{\Delta }t=20\mu \text{s},W=1/N$ and the same time-domain normalization used previously. Plotted in (b) is a representative “high-resolution” filter centered at 7.5 kHz (blue line) with black dashed lines at 5.4 and 10.3 kHz shown for scale. After $M=2600$ simulated qubit measurements for each control sequence, we obtain the posterior distribution of the PSD shown in (c) with mean (blue dashed line) and 0.95 credible interval (shaded in gray). The mean shows excellent agreement with the actual PSD (black line). The width of the 0.95 credible interval in the 5.4–10.3 kHz region is substantially reduced due to the additional measurements.