Ubiquitous problem of learning system parameters for dissipative two-level quantum systems: Fourier analysis versus Bayesian estimation

We compare the accuracy, precision, and reliability of different methods for estimating key system parameters for two-level systems subject to Hamiltonian evolution and decoherence. It is demonstrated that the use of Bayesian modeling and maximum likelihood estimation is superior to common techniques based on Fourier analysis. Even for simple two-parameter estimation problems, the Bayesian approach yields higher accuracy and precision for the parameter estimates obtained. It requires less data, is more flexible in dealing with different model systems, can deal better with uncertainty in initial conditions and measurements, and enables adaptive refinement of the estimates. The comparison results show that this holds for measurements of large ensembles of spins and atoms limited by Gaussian noise as well as projection noise limited data from repeated single-shot measurements of a single quantum device.

Figure 1

Example of ideal measurement signal and data from simulated experiments with Gaussian noise with $\sigma =0.05$ on the left and projection noise on the right where each data point is the average of $N_e=100$ binary-outcome single-shot measurements.

Figure 2

Minimum, maximum, and median of relative error of $\omega$ on the left and $\gamma$ estimates on the right as a function of the magnitude of the Gaussian noise for the 10 model systems in Table 1, averaged over 1000 runs for each system and noise level.

Figure 3

Distribution of $\omega$ estimates in (a), (b), (c) and $\gamma$ estimates in (d), (e), (f) for 1000 runs for Model 1 with 1% Gaussian noise for Strategy 1 in (a) and (b), Strategy 2 in (b) and (e), and Strategy 3 in (c) and (f).

Figure 4

Limits of Fourier resolution and difficulty in estimating peak width for short, noisy signals with (a) 1% Gaussian noise, (b) 5% Gaussian noise, (c) 10% Gaussian noise.

Figure 5

Minimum, maximum and median of relative error of $\omega$ on the left and $\gamma$ estimates on the right as a function of the number of single-shot measurement repetitions per data point, $N_e$, for the 10 model systems in Table 1.

Figure 6

Maximum likelihood for 10 model systems, averaged over 100 runs each, obtained from Strategy 3.

Figure 7

Estimation of width of likelihood peak with regard to $\omega$ and $\gamma$.

Figure 8

Uncertainties of $\omega$ estimates (left plot) and $\gamma$ estimates (right plot) as a function of the Gaussian noise level for the 10 model systems in Table 1 shows that the uncertainty increases linearly with the noise level but for some models the slope is steeper than for others.

Figure 9

Maximum of log likelihood, Strategy 3, for the 10 model systems in Table 1 for different noise levels. The dips in the maximum likelihood for models 4, 5, and 10 correlate with steeper slopes in the uncertainty vs noise level plots in Fig. 8.

Figure 10

Uncertainties of the $\omega$ estimates on the left and $\gamma$ estimates on the right as a function of projection noise level for our 10 model systems show a linear increase with respect to the noise level, with the slope for models 4, 5, and 10 being greater than for the others, as was observed for Gaussian noise.

Figure 11

Estimates for parameters ${\alpha }_1$ and ${\alpha }_2$ including uncertainty as a function of the noise level $\sigma$ for 10 model systems.

Figure 12

Estimates for parameters ${\alpha }_1$ and ${\alpha }_2$ as a function of the number of single-shot repetitions $N_e$ for 10 model systems of the type described by Eq. (5), averaged over 100 runs each.

Figure 13

Plot of the minimum eigenvalue of the covariance matrix of the estimator minus the inverse Fisher information for various models as a function of $N_e$.

Figure 14

Iterative refinement by averaging of signal traces. The power spectra on the left were obtained from averaging the signals over multiple runs effectively coincide. Increasing the number of averages smooths out the spectrum but does not reduce the width of the peak. For Strategy 2 neither the $\omega$ estimates in the center nor the gamma estimates on the right converge to the true values (shown as dash-dot line), while for Strategy 3 both estimates appear to converge to the true values. For Strategy 1 the $\omega$ estimates improve and converge to the correct value, but the $\gamma$ estimates remain biased, here converging to a value larger than the true value.

Figure 15

Prior likelihood after 25 time samples ($t_n=1.2n$) for model 1 on the left with $\{{\omega }_j,{\gamma }_j\}$ samples shown as white dots, and corresponding predicted measurement traces $p_j\left(t\right)=p(t,\{{\omega }_j,{\gamma }_j\})$ and variance of $p_j\left(t\right)$ as function of $t$ on the right.

Figure 16

Selection of measurement times for iterative low-discrepancy sampling. The new measurement times in each iteration as chosen such as to fill in the largest existing gaps.

Figure 17

Median error of $\omega$ (open symbols) and $\gamma$ (filled symbols) parameter estimates for iterative LD sampling (solid lines) and adaptive sampling based on trace variance (dashed lines) for model system 4 with measurements subject to 5% Gaussian noise (circles) and projection noise $\sigma =N_e^{-1/2}$ (squares), respectively. The solid lines are below the dashed lines, suggesting that low-discrepancy sampling is actually preferable to our simple adaptive sampling strategy.

Figure 18

Ideal signal as continuous blue curve and sparsely sampled noisy data as green stars $*$ for $N_t=100$, $t\in [0,25]$ with $N_e=100$ single-shot experiments per data point for a system described by Eq. (30) with $\omega =1$, $\gamma =0.1$ on the left and corresponding log likelihood on the right.

Figure 19

Minimum, maximum and median of relative error (averaged over 100 runs for each system and noise level) of $\omega$ and $\gamma$ estimates as a function of noise level $\sigma$ on the left and estimates for the initial state and measurement angles ${\theta }_I$ and ${\theta }_M$ on the right for 10 model systems described by Eq. (30) with model parameters given in Table 1 for two experimental conditions: 2a maximum visibility: ${\theta }_I={\theta }_M=0$ and 2b: ${\theta }_I=\frac{\pi }{3}$, ${\theta }_M=\frac{\pi }{4}$.