Rapid estimation of drifting parameters in continuously measured quantum systems

We investigate the determination of a Hamiltonian parameter in a quantum system undergoing continuous measurement. We demonstrate a computationally rapid method to estimate an unknown and possibly time-dependent parameter, where we maximize the likelihood of the observed stochastic readout. By dealing directly with the raw measurement record rather than the quantum-state trajectories, the estimation can be performed while the data are being acquired, permitting continuous tracking of the parameter during slow drifts in real time. Furthermore, we incorporate realistic nonidealities, such as decoherence processes and measurement inefficiency. As an example, we focus on estimating the value of the Rabi frequency of a continuously measured qubit and compare maximum likelihood estimation to a simpler fast Fourier transform. Using this example, we discuss how the quality of the estimation depends on both the strength and the duration of the measurement; we also discuss the trade-off between the accuracy of the estimate and the sensitivity to drift as the estimation duration is varied.

Figure 1

Log-likelihood $\mathcal{L}\left(\mathrm{\Omega }\right)$ as a function of possible Rabi frequencies $\mathrm{\Omega }$. A long noisy $z$-measurement record $\left\{r_j\right\}$ of duration $T=1$ ms was simulated with $N={10}^5$ discrete time steps $\delta t=10$ ns, a characteristic measurement time ${\tau }_m=1\mu \mathrm{s}$, and a true Rabi oscillation frequency ${\mathrm{\Omega }}_T/2\pi =1$ MHz (solid green vertical line). The maximum likelihood estimator (dotted red vertical line) is the peak at ${\mathrm{\Omega }}_{\mathrm{ML}}/2\pi =1.0025$ MHz, with a $0.25\%$ error. The quadratic fit $\mathcal{L}\approx -{(\mathrm{\Omega }-{\mathrm{\Omega }}_{\mathrm{ML}})}^2/2{\sigma }^2$ to the peak yields the precision $\sigma /2\pi =0.0026$ MHz.

Figure 2

RMS error of maximum likelihood estimation (MLE) of frequency versus measurement time ${\tau }_m$ and total signal duration $T$, with $N_E=600$ trajectory realizations. (a) Error of the MLE, indicating that longer signals yield more accurate estimates, for a range of optimal measurement times ${\tau }_m$. (b) Slice of (a) with constant $T=40\mu \mathrm{s}$, showing the “sweet spot” where the RMS error does not strongly depend on ${\tau }_m$. (c) Slice of (a) with constant ${\tau }_m=0.65\mu \mathrm{s}$, showing the reduction in the error with increased collection time.

Figure 3

Power spectral density for $T=50\mu \mathrm{s}, {\tau }_m=1\mu \mathrm{s}$, and $\delta t=0.01\mu \mathrm{s}$, shown for a single realization of the measurement process (gray line). After applying a triangular moving-average filter (black line) with a width of roughly $T/\left(2\pi {\tau }_m\right)$ points of minimum frequency resolution $\delta \mathrm{\Omega }/2\pi =1/T$, the fluctuations in the power spectral density are reduced. With sufficient nearest-neighbor averaging, the filtered data will approach a Lorentzian profile (green line) with increasing $T$.

Figure 4

Frequency estimation via fast Fourier transform. The estimation of the peak frequency can be optimized by filtering the spectral density, as shown in Fig. 3. For a total run time of $T=50\mu \mathrm{s}$ and time step $\delta t=0.01\mu \mathrm{s}$ the estimated frequency approaches the real frequency ${\mathrm{\Omega }}_T/2\pi =1$ MHz. The RMS error averaged over $N_E=200$ realizations is of the order of $10\%$ for measurement times ${\tau }_m\sim 0.3\mu \mathrm{s}$ to ${\tau }_m\sim 0.8\mu \mathrm{s}$. However, this error increases for shorter measurement times, due to zero-frequency Zeno pinning, or for longer measurement times, where the measurement is too weak for the output signal to accurately estimate the frequency.

Figure 5

RMS error of fast Fourier transform (FFT) frequency estimate versus measurement time ${\tau }_m$ and total signal duration $T$, with $N_E=600$ trajectory realizations, and a bandpass filter to a window $\mathrm{\Omega }\in [0,2{\mathrm{\Omega }}_T]$, as well as a triangular moving average over the nearest spectral points of width $\delta \mathrm{\Omega }/2\pi =1/T$. (a) Error of the FFT, showing uniformly worse performance than with the MLE. (b) Slice of (a) with the same constant $T=40\mu \mathrm{s}$ as in Fig. 2. (c) Slice of (a) with the same constant, ${\tau }_m=0.65\mu \mathrm{s}$. These crude estimates may be used to accelerate the MLE.

Figure 6

The time-dependent frequency tracking, shown by two examples: (a) an ideal case with unit efficiency $\eta =1$ and no extra dephasing and (b) a realistic case with $50\%$ efficiency ($\eta =0.5$), a phase relaxation time $T_2=30\mu s$, and an energy relaxation time $T_1=50\mu s$. The true drifting frequency in both plots is represented by dotted gray lines. (a) Brown X's represent the estimation from the FFT and solid red circles represent the estimation from the MLE. (b) Brown X's, solid red circles, and open blue circles represent estimations from the FFT, MLE, and MLE ignoring nonidealities in the simulation, respectively. The moving window duration is $T_w=40\mu s$ for all estimation methods, stepped in 10-$\mu s$ intervals. Estimated frequencies are plotted at the midpoint of each estimation window.