Maximal Adaptive-Decision Speedups in Quantum-State Readout

The average time $T$ required for high-fidelity readout of quantum states can be significantly reduced via a real-time adaptive decision rule. An adaptive decision rule stops the readout as soon as a desired level of confidence has been achieved, as opposed to setting a fixed readout time $t_f$. The performance of the adaptive decision is characterized by the “adaptive-decision speedup,” $t_f/T$. In this work, we reformulate this readout problem in terms of the first-passage time of a particle undergoing stochastic motion. This formalism allows us to theoretically establish the maximum achievable adaptive-decision speedups for several physical two-state readout implementations. We show that for two common readout schemes (the Gaussian latching readout and a readout relying on state-dependent decay), the speedup is bounded by 4 and 2, respectively, in the limit of high single-shot readout fidelity. We experimentally study the achievable speedup in a real-world scenario by applying the adaptive decision rule to a readout of the nitrogen-vacancy-center (NV-center) charge state. We find a speedup of $\approx 2$ with our experimental parameters. In addition, we propose a simple readout scheme for which the speedup can, in principle, be increased without bound as the fidelity is increased. Our results should lead to immediate improvements in nanoscale magnetometry based on spin-to-charge conversion of the NV-center spin, and provide a theoretical framework for further optimization of the bandwidth of quantum measurements.

Popular Summary

It is desirable to accurately measure the state of a quantum system in the shortest possible amount of time in many quantum-computation and information-processing applications. One way to speed up such measurements is to adaptively stop each measurement as soon as its outcome is reasonably certain, instead of using a fixed measurement time. On average, this strategy reduces the measurement time without affecting the measurement accuracy. It is therefore of great importance to establish how far this approach can be taken, in principle. Here, we obtain the theoretically achievable reduction in average measurement time, i.e., the “adaptive-decision speedup,” for several realistic physical models of the discrimination between two quantum states.

We first provide bounds for the speedup of two idealized but commonly encountered physical models of a two-state measurement. These models help us to understand how to most rapidly differentiate between two charge states of a single nitrogen-vacancy (NV) center in diamond (i.e., the negatively charged ${\mathrm{NV}}^{-}$ and the neutral ${\mathrm{NV}}^0$) using fluorescence-based detection. We use both Monte Carlo simulations and experimental data to demonstrate that a significant speedup (approximately a factor of 2) is achievable in practice. This speedup in charge detection may be used, in combination with a conversion of spin to charge, to significantly improve the detection of magnetic fields on a nanometer scale. In addition, we show that the speedup can, in principle, be improved arbitrarily via careful design of the measurement scheme.

Our results provide a theoretical framework for optimizing the measurement time of commonly encountered measurement schemes. We expect that our findings can be applied to both atomic and quantum-dot systems.

Figure 1

(a) Readout trajectories for the Gaussian latching readout. The trajectory for state $|+⟩$ (solid blue) has an average of $+1$, while the trajectory for state $|-⟩$ (dashed red) has an average of $-1$. Both trajectories have the same signal-to-noise ratio per unit time $r$. The identification of the state is made based on the entire trajectory acquired during the readout time $t$ (dotted black). (b) Schematic representation of the adaptive decision rule for the Gaussian latching readout. As soon as the log-likelihood ratio ${\lambda }_t$ (solid magenta) satisfies the stopping condition ${\lambda }_t\ge \overline{\lambda }$, ${\lambda }_t\le -\overline{\lambda }$ or $t\ge t_M$, data acquisition is stopped. The sign of ${\lambda }_t$ is then used to choose the qubit state (dashed blue for $|+⟩$ and dotted red for $|-⟩$). Here, we choose $|+⟩$.

Figure 2

Adaptive decision rule for the state-dependent decay readout. The readout is stopped as soon as the likelihood ratio ${\lambda }_t$ (solid magenta) satisfies ${\lambda }_t<-\overline{\lambda }$ or ${\lambda }_t>\overline{\lambda }$. In this case, this pair of conditions is equivalent to stopping the readout as soon as an event is detected before a maximum time $t_M=\overline{\lambda }\tau$ (dot-dashed black). We choose the state according to the sign of ${\lambda }_t$ at the stopping time (dashed blue for $|+⟩$ and dotted red for $|-⟩$). In this example, we choose $|-⟩$. For the magenta trajectory shown above, this choice results in an error since an event occurs after time $t_M$, as indicated by the sudden divergence in ${\lambda }_t$.

Figure 3

Schematic representation of (a) the NV-center charge dynamics described in Sec. 4a, (b) the experimental apparatus, and (c) the experimental sequence.

Figure 4

(a) Experimental photon-count trajectory illustrating the state-dependent fluorescence rate and two-level switching of the NV-center charge state for an illumination power of $0.55\mu \mathrm{W}$. Here, the photon detection rate fluctuates between ${\gamma }_{+}\approx 720\mathrm{Hz}$ for ${\mathrm{NV}}^{-}$ and ${\gamma }_{-}\approx 50\mathrm{Hz}$ for ${\mathrm{NV}}^0$, as discussed in Sec. 4b. The corresponding average counts per 10-ms bin are indicated by the horizontal dotted black lines. The NV-center ionization and recombination rates are found to be ${\mathrm{\Gamma }}_{+}\approx 3.6\mathrm{Hz}$ and ${\mathrm{\Gamma }}_{-}\approx 0.98\mathrm{Hz}$, respectively. (b) Log-likelihood ratio ${\lambda }_t$ (solid magenta) and counts per 100-$\mu \mathrm{s}$ bin (dashed red) for the first 60 ms of the data shown in (a). The log-likelihood ratio was calculated using Eq. (8) and the experimental parameters given in Sec. 4b. The discontinuities in ${\lambda }_t$ indicate the detection of a photon.

Figure 5

Error rate as a function of average readout time for the experimental parameters of Sec. 4b. The lines give the Monte Carlo simulated error rate, and the symbols give the experimental values corrected for preparation errors using Eq. (e7). We display the error rate for the photon-counting model discussed in Appendix pp4 (solid blue and blue circles), for the nonadaptive MLE (dashed orange and orange squares), and for the adaptive MLE (dotted green and green triangles). The horizontal dotted black lines indicate the minimum error rate of the photon-counting (1.9%) and MLE methods (1.5%). As indicated by the vertical dotted black lines, the speedup of the adaptive MLE compared to the nonadaptive MLE is $t_f/T\approx 1.9$ when the target error rate is the minimum error rate of the photon-counting method.

Figure 6

Monte Carlo simulated error rate for an unbalanced initial charge distribution with $P(+)=0.25$ when (a) the value of $P(+)$ is unknown (MLE) and (b) the value of $P(+)$ is known (MAP). In both cases, we show the error rate for the photon-counting method of Ref. [12] (solid blue), the nonadaptive readout (dashed orange), and the adaptive readout (dotted green). The horizontal dotted black lines indicate the minimum error rates of the photon-counting method (1.6% in both cases) and of the optimal methods (1.4% for the MLE and 1.3% for the MAP). The dotted vertical lines show that significant adaptive-decision speedups of $t_f/T\approx 1.6$ and $t_f/T\approx 1.8$ are obtained when using the MLE and MAP, respectively. The discontinuous features in (a) occur because the MLE uses a nonoptimal discrimination threshold when $P(\pm )\ne 0.5$.

Figure 7

(a) Readout scheme based on state-dependent polarization of an emitted photon. When the state is $|+⟩=|e,R⟩$ ($|-⟩=|e,L⟩$) in the excited subspace, a right (left) circularly polarized photon is emitted. Both states decay to the ground state $|0⟩=|g⟩$ on a time scale $\tau$. (b) Similar scheme for readout of the charge state of a double quantum dot. When the electron is in the right (left) island $|R⟩$ ($|L⟩$), it is detected after tunneling into the right (left) lead. Both states tunnel into the leads on a time scale $\tau$, leaving the double quantum dot in the empty state $|0⟩$.

Figure 8

Unraveling of the NV-center charge readout dynamics. The state $|+⟩$ ($|-⟩$) represents ${\mathrm{NV}}^{-}$ (${\mathrm{NV}}^0$). Photons are detected at state-dependent rates ${\gamma }_{\pm }$. Each detection increases the number of observed photons $n$. The rates ${\mathrm{\Gamma }}_{\pm }$ describe ionization or recombination processes that are not associated with the detection of a photon.

Figure 9

Evolution of the ${\mathrm{NV}}^{-}$ probability $p_t$ for a trajectory generated assuming $|+⟩$ (solid blue) and $|-⟩$ (dashed red) for the same parameters as in Fig. 5. In this particular case, we assumed that $P(+)=P(-)$. The discontinuities in $p_t$ correspond to the detection of a photon. Data acquisition is stopped when $p_t$ first crosses the stopping thresholds $p_{+}$ and $p_{-}$ (dotted black) or when $t$ reaches the maximum acquisition time $t_M=25\mathrm{ms}$.

Figure 10

(a) Histogram of photon counts per 10-ms time bin (black dots). The solid green curve is a fit to a mixture of Poisson distributions, Eq. (e1). The dashed blue and dotted red curves are the Poisson distributions conditioned on states $|+⟩$ and $|-⟩$, respectively. We extract ${\gamma }_{+}\approx 720\mathrm{Hz}$ and ${\gamma }_{-}\approx 50\mathrm{Hz}$ from the means of the conditional distributions. (b) Average photon counts per 10 ms, ${\varrho }_{\pm }(t)$, for trajectories postselected on the initial states $|+⟩$ (blue circles) and $|-⟩$ (red squares). The solid black lines are a simultaneous fit of ${\varrho }_{+}(t)$ and ${\varrho }_{-}(t)$ to Eq. (e3). From the fit, we extract ${\mathrm{\Gamma }}_{+}\approx 3.6\mathrm{Hz}$ and ${\mathrm{\Gamma }}_{-}\approx 0.98\mathrm{Hz}$. The horizontal dotted black line is the steady-state average $B$ obtained from the fit to Eq. (e3).

Figure 11

(a) Experimentally measured probability density $P(\lambda )$ of the log-likelihood ratio $\lambda$ (black circles) for subtrajectories of duration 25 ms. The theoretical prediction $P_{\mathrm{theo}}(\lambda )$ (dotted black) and the corresponding weighted conditional distributions $P_{\mathrm{theo}}(\lambda |+)P_{\mathrm{theo}}(+)$ (solid blue) and $P_{\mathrm{theo}}(\lambda |-)P_{\mathrm{theo}}(-)$ (dashed red) are shown for comparison. The theoretical prior probabilities $P_{\mathrm{theo}}(\pm )$ are taken to be the stationary values corresponding to the extracted rates, Eq. (e6). (b) Experimental error rate $\tilde{\epsilon }$ of the photon-counting method (blue circles), the nonadaptive MLE (orange squares), and the adaptive MLE (green triangles). The theoretical error rate $\epsilon$ of Fig. 5 is reproduced for comparison (solid blue for the photon-counting method, dashed orange for the nonadaptive MLE, and dotted green for the adaptive MLE). The discrepancy between experiment and theory can be attributed to preparation errors [see Eq. (e7)].