B. D’Anjou    L. Kuret    L. Childress Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada    W.A. Coish Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada Quantum Information Science Program, Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
July 26, 2019
###### Abstract

Quantum Information, Quantum Physics

## I Introduction

Efficient discrimination of quantum states is important for, e.g., fast qubit readout Hume et al. (2007); Myerson et al. (2008); Burrell et al. (2010); Noek et al. (2013), rapid feedback and steering Sayrin et al. (2011); Blok et al. (2014); Murch et al. (2015), preparation of nonclassical states of light Brune et al. (2008); Deléglise et al. (2008); Sayrin et al. (2011); Peaudecerf et al. (2014), and nanoscale magnetometry Waldherr et al. (2012); Shields et al. (2015). Given sufficient information about the statistics and dynamics of a physical readout apparatus, it is possible to speed up a readout through a real-time adaptive decision rule (described below) Hume et al. (2007); Myerson et al. (2008); Burrell et al. (2010); Sayrin et al. (2011); Noek et al. (2013); Blok et al. (2014); Peaudecerf et al. (2014). An adaptive decision rule can result in a significant reduction in the average time per measurement without any significant deterioration of the readout fidelity. Adaptive decisions can be used to improve any readout scheme, in principle, given the ability to continuously monitor the state on a time scale that is short compared to the time required for high-fidelity readout Elzerman et al. (2004); Hume et al. (2007); Myerson et al. (2008); Barthel et al. (2009); Jiang et al. (2009); Morello et al. (2010); Neumann et al. (2010); Robledo et al. (2011a); Pla et al. (2013); Noek et al. (2013); Aslam et al. (2013); Harty et al. (2014); Jeffrey et al. (2014); Shields et al. (2015).

Adaptive-decision problems have a long history in probability theory. Their mathematical root can be traced back as far as the famous “Gambler’s ruin” problem introduced by Pascal in the 17th century Edwards (1983). A general mathematical theory of adaptive decision rules, known as sequential analysis, was developed during the World War II Wald (1947) and has since become a well-established part of statistical decision theory Poor (1994). Implementing an adaptive decision rule requires the ability to update a stochastically varying measure of confidence in the state, typically a likelihood function, in real time. A measurement outcome is then chosen when the confidence measure first achieves a desired value. Adaptive-decision problems are thus closely related to first-passage time analysis Risken (1989); Klebaner (2005); the stochastically varying confidence measure can be treated as the coordinate of a diffusing particle, which crosses a boundary when the desired confidence is reached. An adaptive decision allows for a reduction in the average measurement time, associated with a corresponding “adaptive-decision speedup.” Here, we formulate the adaptive decision for a two-state readout in terms of a first-passage time problem. We use this formalism to theoretically establish the maximum achievable speedups for several physical readout models. Moreover, we demonstrate that significant speedups can be achieved in practice by experimentally characterizing the adaptive-decision speedup for the detection of a NV-center charge state, effectively doubling the bandwidth of such a measurement.

To experimentally study the achievable speedup, we implement an adaptive decision rule for a readout that discriminates between two charge states of a single NV center in diamond Aslam et al. (2013); Shields et al. (2015). The NV-center charge readout relies on the ability to distinguish strong from weak fluorescence upon illumination of the NV-center impurity. Remarkably, the NV-center charge readout can approach either of the readout models described above in distinct limits. However, for realistic experimental parameters, the dynamics of the NV-center charge readout will be intermediate between these two extremes. We therefore analyze and experimentally quantify the associated speedup. We update our level of confidence in the charge state using a quantum trajectory formalism that can easily be generalized to more complex readout schemes. We find an adaptive-decision speedup both for our experimental parameters and the experimental parameters of Ref. Shields et al. (2015), in which a similar charge-state measurement has recently been performed. Since a shorter average measurement time increases the number of measurements that can be performed on a NV-center spin in a given time, this result should directly improve the sensitivity of magnetometry based on spin-to-charge conversion of the NV-center spin Shields et al. (2015). A similar approach can, in principle, be applied to magnetometry using the standard spin readout of the NV center Degen (2008); Maze et al. (2008); Taylor et al. (2008); Balasubramanian et al. (2008).

An adaptive decision can only speed up the aforementioned readout schemes by a factor of order unity. However, we show that the speedup can become parametrically large for other schemes. More precisely, we propose a simple readout, based on the discrimination of two distinct decay channels, for which the speedup becomes unbounded as the fidelity is increased. Such a readout could be realized in either atomic or quantum-dot systems.

The goal of readout is to discriminate between two particular states of a system of interest, and . To achieve this, we typically let the system interact with a measurement device for a finite amount of time , the readout time. During the time , the measurement device records a time-resolved trajectory in the form of, e.g., an electrical or photonic signal. As illustrated in Fig. 1(a), a distinct trajectory is recorded when the state is either (blue) or (red), making it possible to discriminate between the two states. In the presence of noise, perfect discrimination is not possible. We then decide whether the state was most likely or based on the entire trajectory acquired during the readout time. For an optimal readout, this decision maximizes the readout fidelity , namely, the probability that the state is identified correctly.

We now formalize the ideas introduced above. For a given readout time , the noisy state-dependent trajectory can be represented by a function defined in the interval , as depicted in Fig. 1(a). Given the observed trajectory, we wish to decide whether the initial state was most likely or in a way that maximizes the fidelity  111Other measures of state uncertainty, such as the entropy of the state probability distribution, may also be used when discriminating between more than two states Peaudecerf et al. (2014).. Alternatively, we wish to minimize the error rate . To do so, we calculate the likelihood ratio , or equivalently the log-likelihood ratio . Here, is the probability of observing the trajectory given that the state is initially . The log-likelihood ratio expresses a level of confidence in the state. When the prior probability for is , the decision rule that minimizes the error rate is to choose the maximum a posteriori estimate (MAP) of the state; if (), we decide that the initial state was most likely (). Here, is the optimal decision threshold. When the prior probabilities are assumed to be equal, and , the decision reduces to the maximum-likelihood estimate (MLE). In most of this article, we assume equal prior probabilities so that the MLE is optimal. We briefly discuss unequal prior probabilities and the MAP in Sec. IV.3.3.

When using a nonadaptive decision rule, we fix for each measurement and base our decision on the value of (nonadaptive MLE or MAP). When using an adaptive decision rule, in contrast, we stop each measurement as soon as , or , as illustrated in Fig. 1(b). Here, are stopping thresholds and is a maximum readout time. We then base our decision on the value of at the stopping time (adaptive MLE or MAP). The average value of from measurement to measurement gives the average readout time . For each of the three readout schemes listed in Table 1 , we have proven that a symmetric choice of stopping thresholds, , minimizes for a given error rate when the states are equally likely. For the readout discussed in Sec. IV, we optimize the stopping thresholds numerically.

## Iii Bounded speedup

In this section, we apply the ideas of Sec. II to two commonly encountered readout schemes, the Gaussian latching readout and the readout based on state-dependent decay. Although these schemes are usually not perfectly realized in practice, they are relevant to understand the limitations of a wide variety of readouts. In particular, both models can be mapped to extreme limits of the NV-center charge readout discussed in Sec. IV.

The Gaussian latching readout is the simplest, most widespread and most tractable model of state discrimination Kay (1998). In this scheme, each state gives rise to a state-dependent trajectory with constant average and subject to Gaussian white noise, as illustrated in Fig. 1(a). A latching readout is characterized by the absence of state relaxation at all times. Even though most readouts are limited by state relaxation Gambetta et al. (2007); D’Anjou and Coish (2014); Khezri et al. (2015), many implementations will approximately become latching readouts as the error rate is decreased Barthel et al. (2009); Neumann et al. (2010); Pla et al. (2013); Jeffrey et al. (2014); Shields et al. (2015).

Formally, the probability distributions for conditioned on the state are

 P(ψt|±)=Aexp{−r2∫t0dt′[ψt(t′)∓1]2}. (1)

Here, is the power signal-to-noise ratio per unit time and is a normalization constant independent of the state. From Eq. (1), it is straightforward to show that the log-likelihood ratio is . Thus, in the case of the Gaussian latching readout, a maximum-likelihood decision can be made by integrating the trajectory up to time .

If the trajectory is acquired over the fixed interval , the error rate takes the form Gambetta et al. (2007):

 ϵ=12erfc⎛⎝√rtf2⎞⎠, (2)

where is the complementary error function Gradshteyn and Ryzhik (2007). A derivation of Eq. (2) is included as part of Appendix A. Alternatively, we can implement an adaptive decision rule. Because of the symmetry of the problem under interchange of and , it is optimal to choose symmetric stopping thresholds. Therefore, we stop the readout as soon as , or . As in the nonadaptive case, we choose the state according to the sign of at the end of the sequence. The adaptive decision rule is illustrated in Fig. 1(b). The error rate and average readout time are then defined parametrically as a function of both and . The calculation of and can be recast as a first-passage time problem Risken (1989); Klebaner (2005), as detailed in Appendix A. For clarity, here we only give the result for the case when the probability of reaching is negligible, (see Appendix A). The corresponding error rate and average readout time are given by

 ϵ=11+e¯λ,T=¯λ2rtanh(¯λ2). (3)

By varying the stopping threshold , we can map the functional relation . As , Eq. (3) implies that . In the nonadaptive case, Eq. (2), we instead have . Therefore, in the limit of small , the adaptive decision rule asymptotically achieves the same error rate at an average time 4 times shorter than the nonadaptive decision rule Bechhofer (1960); Poor (1994). Thus, the maximal speedup is given by

 tfT∼4,(ϵ→0). (4)

### iii.2 State-dependent decay

As a second example, we consider the class of readouts relying on state-dependent decay. In these schemes, the state decays to with probability per unit time . The detection of a decay event, e.g., the detection of an emitted photon or of a tunneling electron, indicates that the initial state was . In contrast, the absence of a decay event indicates that the state was . This situation is approximately realized in, e.g., trapped ion qubits Noek et al. (2013), NV-center spin qubits Robledo et al. (2011a); Blok et al. (2014), and semiconductor spin qubits Elzerman et al. (2004); Morello et al. (2010). To facilitate the discussion, we assume that the events are detected with perfect efficiency. Moreover, we assume that a detected event is always the result of a transition; i.e., there are no “dark” counts. In this case, the log-likelihood ratio is when no event has been detected after a readout time because the detection probability of a decay event decreases exponentially with time. When an event is detected, suddenly becomes infinite because an event is guaranteed to have resulted from the initial state being . This scenario is illustrated in Fig. 2.

When using a nonadaptive decision rule, we wait for a time and base the decision on whether a decay event is detected in the interval , as indicated by the sign of . The error rate is given by the probability that the event occurs after time assuming an initial state , , multiplied by the probability that the state was initially :

 ϵ=12e−tf/τ. (5)

In contrast with the nonadaptive case, the adaptive decision rule stops the readout as soon as or , as shown in Fig. 2. Note that, for this particularly simple readout scheme, the use of a maximum readout time is redundant since the above stopping condition is the same as stopping the readout as soon as an event is detected before a maximum time . Variations of this adaptive decision rule have been implemented in Refs. Noek et al. (2013); Blok et al. (2014). The error rate is now given by the probability of an event occurring after time . To achieve the same error rate as the nonadaptive decision rule, Eq. (5), we thus set . However, we find that the readout time is now given, on average, by when the state is . Thus, the average readout time is because, as noted in Ref. Noek et al. (2013), we must always wait for the full duration when the state is . Thus, once again, we find a bounded speedup in the limit of vanishing error rate, :

 tfT∼2,(ϵ→0). (6)

## Iv Experiment: NV-center charge readout

The analysis of Sec. III provides limits on the achievable speedup for two idealized readout mechanisms. Many experimental realizations of a two-state readout can approach one or the other model in different limits, but under typical experimental conditions neither limit can be strictly realized. We must therefore use an algorithm for the adaptive decision rule suited to the underlying readout mechanism.

Here, we provide an explicit implementation of an adaptive decision rule for experimental two-state readout in a regime that lies intermediately between the two idealized examples of Sec. III. Specifically, we examine fluorescence-based charge-state detection of the NV center in diamond. As explained below, this system approaches a Gaussian latching readout in the limit of high fluorescence rates and long charge-state relaxation times, while it can begin to resemble state-dependent decay for very low excitation powers. In the following, we describe the main features of the NV charge readout system and use experimental data to extract the underlying system parameters. We then both simulate and experimentally implement the adaptive decision rule with the help of a simple algorithm that uses our knowledge of the dynamics to update the likelihood ratio in real time. For our experimental parameters, we find a speedup of . The resulting improvement in detection bandwidth can improve the sensitivity of an NV-center magnetometer using spin-to-charge conversion Shields et al. (2015).

### iv.1 Charge dynamics of the NV center in diamond

The NV center in diamond is typically observed in two charge states, namely, the negatively charged NV and the neutral NV. In keeping with our notation, we label NV and NV by and , respectively. When the NV impurity is illuminated with yellow light (), transitions between the NV ground and excited states scatter photons which are detected at a rate . The frequency of the yellow light is below the minimum threshold required to resonantly excite optical transitions of NV, and thus only a residual detection rate is observed in the neutral state. Note that the rates include the effect of dark counts and imperfect detection efficiency. This difference in fluorescence rates enables direct readout of the NV-center charge state Aslam et al. (2013); Shields et al. (2015). In addition, the incident light can cause ionization (recombination) of the NV (NV) state with probability per unit time (). As illustrated in Fig. 3(a), this leads to an alternating process of ionization and recombination that limits readout fidelity 222A similar process limits the readout fidelity of fluorescence-based readouts of trapped atoms and ions Gehr et al. (2010); Noek et al. (2013); Wölk et al. (2015). Typically, the condition is satisfied, so many transitions occur between ionization and recombination events. The trajectory can therefore be modeled as a hidden two-state Markov process subject to state-dependent Poissonian noise of intensity  Aslam et al. (2013); Shields et al. (2015). Because of detector bandwidth limitations, the maximum readout time is usually separated into bins of duration , so we observe photons in bins . Thus, the measured trajectory can be represented as an -component vector .

Note that the charge readout of the NV center approaches the two cases discussed in Sec. III in different limits. When , the noise statistics of the NV-center charge readout are approximately Gaussian, albeit with asymmetric signal-to-noise ratios per unit time:

 r±=(γ+−γ−)24γ±. (7)

To perform a high-fidelity readout, the rate of information gain must be faster than state relaxation, . For an adaptive advantage, the rate of information gain must also be slower than the sampling rate, . Combined with , the last inequality implies the necessary condition or . We conclude that when state relaxation is negligible and when and are both large and comparable in magnitude, the NV-center charge readout is approximately described by the model of Sec. III.1 with signal-to-noise ratio . In contrast, when , the NV-center charge readout can be approximately modeled by the state-dependent decay of Sec. III.2 with . Indeed, in this limit the presence or absence of a single photon on a time scale of a few is sufficient to choose the state with high confidence. Therefore, the detection of additional photons adds little information, and the state-dependent decay model is a good approximation. For our parameters, the experiments described below do not fall strictly into either of these limits. It is therefore interesting to characterize the speedup in this intermediate regime.

### iv.2 Experimental setup and data

We characterize the performance of the adaptive decision rule using experimental data. The data were acquired using a home-built confocal microscope with -nm excitation. The -nm wavelength lies in between the zero-phonon line of the NV () and NV (), so it only efficiently excites the negatively charged state. The excitation light is focused through a high numerical aperture objective (NA 1.35) onto a cut chemical-vapor-deposition-grown diamond in which single defects can be resolved. Photons emitted in the wavelength range are collected and detected with a single-photon counter. This detection range overlaps strongly with the NV fluorescence spectrum and only weakly with the NV fluorescence, while rejecting Raman scattering from the diamond. The experimental setup is illustrated schematically in Fig. 3(b).

To acquire fluorescence trajectories, -nm excitation is applied continuously and photon counts are recorded for in bins of , with a small separation of necessary for our field-programmable gate array (FPGA) card to process the counts. These time scales are chosen such that the total duration () is much greater than the time scale for ionization and recombination, and each time bin () is much shorter. A subset of such a time trace is illustrated in Fig. 4(a) with data rebinned to . The experiment is repeated to obtain statistics on fluorescence and ionization rates, interleaved with a tracking step (-nm green laser) that ensures the sample does not drift relative to the focal point of the microscope. The experimental sequence is illustrated in Fig. 3(c). Such data were acquired for excitation powers ranging from to (as measured going into the objective). For comparison, the saturation intensity at is estimated to be . We verified that and scale as the laser intensity and that and scale as the laser intensity squared, as shown in Ref. Aslam et al. (2013). Data sets were measured for two oriented NV centers, which yielded similar results. Very weak excitation is used to ensure that and are much larger than and .

For definiteness, we restrict the following analysis to a single power setting. Specifically, the data sets for the lowest laser power setting of were used to extract the parameters and , as detailed in Appendix E.1. We obtain the rates , , and , which are required to construct an algorithm for an adaptive decision rule.

In Refs. Aslam et al. (2013); Shields et al. (2015), single-shot readout of the charge state was performed by counting the total number of photons detected in the interval (see Appendix D). In contrast, implementing the adaptive decision rule requires an efficient algorithm to update the likelihood ratio after each time bin Gambetta et al. (2007); Myerson et al. (2008); Gehr et al. (2010); D’Anjou and Coish (2014); Ng and Tsang (2014); Gammelmark et al. (2014); Wölk et al. (2015). A hidden-Markov-model algorithm suitable to the NV-center charge readout has been developed in, e.g., Refs. Ng and Tsang (2014); Wölk et al. (2015). Here, we use an equivalent quantum trajectory approach, which is more easily generalized to quantum systems with multiple levels and coherent internal dynamics. Moreover, we find an update rule that is valid for an arbitrary value of the bin size . In particular, more than one ionization or recombination event may occur during one time bin, an occurrence that becomes more likely at high illumination power Aslam et al. (2013). The update rule of Ref. Wölk et al. (2015) is recovered when .

As shown in Appendix B, the likelihood ratio after bins is given by

 ΛN=Tr[M(δnN−1)…M(δn0)ℓ(+)0]% Tr[M(δnN−1)…M(δn0)ℓ(−)0]. (8)

In Eq. (8), and are initial state vectors representing and , respectively. Here, the trace Tr of a vector is defined as the sum of its elements. The matrix is a update matrix for the detection of photons in a bin. The exact form of the update matrices is given in Appendix B. They can be stored for all , where is the maximum number of photons with a non-negligible probability to occur. The likelihood ratio, Eq. (8), can then be updated in real time through simple matrix multiplication. An example of the application of Eq. (8) to experimental data is given in Fig. 4(b). Note that is nothing but the measurement superoperator for a classical Markov process subject to Poissonian noise. The approach can thus be generalized to any Markovian quantum trajectory by substituting with the appropriate measurement superoperator for direct detection and the vectors by any pair of initial states to be discriminated. In particular, the standard spin detection of the NV center modeled in, e.g., Refs. Manson et al. (2006); Robledo et al. (2011b); Doherty et al. (2013) can be processed in this way.

In the following, we first use Monte Carlo simulations to assess the ideal theoretical performance of the adaptive decision rule. We then apply the adaptive decision rule to experimental data to demonstrate a speedup under real experimental conditions. In all cases, we choose the experimental time bin and a maximum acquisition time . We use this value of and the rates given in Sec. IV.2 to calculate and store the update matrices for . Here, is chosen to match the maximum number of photons observed in any one time bin of the experimental data sets. The residual probability of observing more than photons in one time bin is .

#### iv.3.1 Monte Carlo simulations

We note that the parameters given in Sec. IV.2 satisfy the condition for which the NV-center charge readout reduces to the state-dependent decay readout of Sec. III.2, . It is therefore plausible that the speedup obtained in Fig. 5 has at least partially the same origin as that given in Eq. (6). Indeed, the speedup observed in Ref. Noek et al. (2013) was essentially explained by the state-dependent decay model for a readout whose dynamics are formally the same as the NV-center charge readout dynamics. However, the model of Sec. III.2 fails when the maximum readout time becomes comparable to , resulting in a much higher saturation error rate in Fig. 5 than that predicted by Eq. (5) with and . In our experiment, it is therefore necessary to use the update rule of Eq. (8) to obtain an accurate estimate of the error rate. In a context where it may be favorable or necessary to move towards the regime of the high-fidelity Gaussian latching readout of Sec. III.1, speedups larger than should be possible. Entering this regime may not be advantageous for the NV-center charge readout since the increase in excitation intensity required to reach the Gaussian regime results in a detrimental increase in the ionization and recombination rates. However, such an advantage may be possible in other systems with similar dynamics.

#### iv.3.2 Experimental verification

In the simulations of Sec. IV.3.1, the statistics of the simulated fluorescence trajectories are perfectly described by the two-state model assumed for readout. In real experimental conditions, however, the charge dynamics may deviate from this model. In this section, we show that the adaptive-decision speedup is robust to imperfections in our modeling by applying the adaptive decision rule to experimental data. To independently verify our model of charge dynamics, we split our data sets into a calibration set and a testing set. The calibration set is used solely to extract the experimental values of the rates and given in Sec. IV.2. The extracted parameters and Eq. (8) are then used on the testing set to verify the model. The details are given in Appendix E.2.

We first perform a preliminary verification by comparing the experimental distribution of log-likelihood ratios to the distribution predicted by Monte Carlo simulations [see Appendix E.2]. We find a close agreement between experiment and theory, showing that the model of charge dynamics discussed in Sec. IV.1 provides a good description of the statistics of the experimental trajectories. We can therefore use our model to perform approximate preparation of the charge state in postselection. We use this preparation to verify that the adaptive-decision speedup exists when the adaptive decision rule is applied to experimental data. From the testing set, we prepare () trajectories of with initial states (). We then read out the state for each trajectory using the photon-counting, nonadaptive MLE, and adaptive MLE methods (i.e. assuming equal prior probabilities). Comparing the result of the readout to the preparation allows us to estimate the experimental error rate , where are the error rates conditioned on the preparation in state . To better compare the experimental results with theory, we then fit the experimental error-rate curves to the theoretical prediction, accounting for an additional preparation error [see Eq. (48)]. This procedure gives as a single fit parameter. We emphasize that this adjustment is a uniform transformation of the error rate for all times and hence does not affect the measured speedup. The experimental error rate obtained after compensating for preparation error is shown along with the error rate determined by Monte Carlo simulation in Fig. 5 [the uncompensated error rate is shown in Appendix E.2 for comparison]. The experimentally measured error rate thus fits the theoretical prediction very well. Therefore, we conclude that the adaptive-decision speedup discussed in Sec. IV.3.1 persists when the adaptive decision rule is applied to real experimental data in spite of possible systematic errors and imperfect modeling.

#### iv.3.3 Unequal prior probabilities

A balanced probability distribution for the initial state, , is often desirable because it enables the extraction of one bit of information per measurement. However, in applications such as the magnetometry protocol of Ref. Shields et al. (2015), the probability distribution of the initial state is, in reality, unbalanced, . In general, the dependence of the adaptive-decision speedup on the prior probabilities is nontrivial and may depend on the particularities of the readout dynamics. A general analysis lies beyond the scope of this work. Here, we nevertheless discuss how to account for unequal prior probabilities. Moreover, we show that the adaptive-decision speedup of the experimentally relevant NV-center charge readout discussed in Sec. IV.3.1 persists even for unequal prior probabilities. We distinguish the case where are unknown from the case where are known.

When the prior probabilities are unknown, the best strategy is to calibrate the readout thresholds and by assuming that the prior probabilities are equal. This leads to the MLE readout discussed in Secs. IV.3.1 and  IV.3.2. Under this assumption, we obtain the error rates and average times conditioned on the state . When the MLE readout is used on an unbalanced sample of charge states, the error rate and average readout times are then given by and , respectively. The simulated error rate as a function of average readout time in this scenario is shown in Fig. 6(a) for the experimental parameters of Sec. IV.2 and . We also show the corresponding error-rate curve for the photon-counting method (see Appendix D). Using the minimum error rate of the photon-counting method as a reference, we find that a substantial speedup persists even when the prior probabilities are unknown.

When the prior probabilities are known, they can be used to improve the MLE readout by adjusting the thresholds to directly minimize for constant . This leads to the MAP readout mentioned in Sec. II (see Appendix C for details). We note that in a given application, the prior probabilities can easily be determined by using the MLE to estimate the relative proportion of and . The simulated error rate as a function of average readout time when using the MAP is shown in Fig. 6(b) for the same experimental parameters and . We find an adaptive-decision speedup using the minimum error rate of the photon-counting method as a reference, similar to the value obtained in Sec. IV.3.1 for . Therefore, we conclude that the adaptive-decision speedup persists for significant deviations from a balanced initial-state distribution.

## V Parametric improvement in speedup

From the examples of Secs. III and IV, we might be tempted to conclude that the speedup is fundamentally bounded by a constant of order unity. Here, we show that there exist readout schemes where the speedup can become arbitrarily large in the limit of low error rate.

To show this, we consider a simple variation of the state-dependent decay readout analyzed in Sec. III.2. We now suppose that the states and both decay to a third state through different decay channels and , respectively. The two channels may be associated with, e.g., different polarizations of emitted photons or two different leads into which the electron of a double-quantum-dot charge qubit can tunnel (see Fig. 7). If the two decay channels can be discriminated, e.g., with the help of a polarization analyzer, the state can be read out. For simplicity we assume that both states decay with the same probability per unit time .

When using a nonadaptive decision rule, we wait a time and base the decision on whether a or decay event has been detected. If no event is detected, we choose the state at random with equal probability. Assuming, for simplicity, that the channels and can be perfectly discriminated, the error rate is given by the probability that no decay event has occurred up to time , , multiplied by the probability that the random decision fails. Thus, the error rate is still given by Eq. (5).

The adaptive decision rule, in contrast, stops the readout as soon as either a decay event is detected or a maximum time is reached. Following the same argument as for the nonadaptive decision rule, the error rate is . To achieve the same error rate as the nonadaptive decision rule, we must therefore choose . However, we find that the average readout time is now . As the error rate decreases, , the average readout time tends to a constant so that

 tfT∼ln(12ϵ)→∞,(ϵ→0). (9)

Therefore, the speedup increases without bound as the error rate is decreased. This parametric improvement in speedup is achieved by transferring all the information about the state to the channel degree of freedom, or . This means that the readout no longer relies on discriminating decay from the absence of decay. We can thus arrange the readout so that both and decay on a time scale . Therefore, it is no longer necessary to wait for the entire duration when the state is .

Such a scheme could be applied in a variety of scenarios. For example, suppose that two atomic excited states, and , both decay to the ground state by emitting photons with right and left circular polarizations, respectively. This situation is depicted in Fig. 7(a). A polarization analyzer could then identify both states in a time , leading to the parametric improvement of Eq. (9). In another example, suppose that we want to discriminate between an electron being in the rightmost () and leftmost () islands of a double quantum dot. We imagine that, as shown in Fig. 7(b), the potential barriers between the dots and leads are lowered at the beginning of the readout phase. An electron in the right (left) dot will then tunnel out into the right (left) lead and be detected, with the left and right detectors playing the role of the polarization analyzers in the previous example. In perfect analogy with Fig. 7(a), stopping readout as soon as an electron is detected in either lead instead of waiting for a fixed time then leads to a parametric improvement in speedup, Eq. (9). Note that while the assumption of perfect detection efficiency is difficult to realize for photon detection, it is usually not a limitation for the detection of electron charges in quantum dots.

## Vi Conclusion

Our results are already applicable to a wide variety of systems, and they provide a direction for the optimization of measurement bandwidth in experimentally relevant readouts of quantum states. In particular, we have shown that magnetometry based on the NV-center spin-to-charge-conversion readout can be improved with currently achievable experimental parameters. Moreover, our results show that apparent limitations to the adaptive speedup can be overcome by careful redesign of the readout dynamics. In this work, our goal was to analyze the fundamental limitations of the adaptive-decision speedup for extreme limits of several physical readout schemes. We expect direct extensions of the first-passage time formalism developed here to allow for the derivation of analytical speedup bounds in the presence of, e.g., unwanted state relaxation, dark counts, or imperfect photon collection efficiency. On a more fundamental level, our analysis provides the necessary framework to study how the addition of measurement feedback and coherent readout dynamics can modify the maximal speedups discussed here. Moreover, this formalism may allow for the direct characterization of the achievable speedups for parameter estimation, or multiple-state discrimination in general, subject to experimentally relevant noise.

## Acknowledgments

We acknowledge financial support from NSERC, CIFAR, INTRIQ, and the Canada Research Chairs Program.

In this section, we sketch the derivation of the error rate and average readout time for both the nonadaptive and the adaptive decision rule applied to the Gaussian latching readout. According to Eq. (1), the expectation and covariance of the trajectory conditioned on the state are

 E[ψt(t′)|±]=±1,Cov[ψt(t′),ψt(t′′)|±]=r−1δ(t′−t′′), (10)

for and . Because the noise is Gaussian, all higher cumulants vanish. Correspondingly, the expectation and variance of the log-likelihood ratio are and . Therefore, is a simple drift-diffusion process distributed according to

 P(λt|±)≡G(±)(λt,t)=1√8πrtexp[−(λt∓2rt)28rt]. (11)

Here, is the Green’s function that solves the associated drift-diffusion equation for .

For a fixed readout time , the state is chosen according to the sign of . The error rates conditioned on the initial state being , , are thus

 ϵ(+)=∫0−∞dλtfP(λtf|+),ϵ(−)=∫∞0dλtfP(λtf|−). (12)

By symmetry of the problem, . Using Eq. (11), we can then calculate the error rate :

 ϵ=12erfc⎛⎝√rtf2⎞⎠. (13)

To assess the performance of the adaptive decision rule, we must calculate both the error rate and the average readout time , where is the average readout time conditioned on the state being . Using the symmetry of the problem again, we have and . We may therefore assume that the state is without loss of generality. We now reformulate the calculation of and as a first-passage time problem Risken (1989); Klebaner (2005) (an alternative formalism is given in Ref. Tantaratana and Poor (1982)). More precisely, let and be the error rate and average readout time conditioned on knowing that the log-likelihood ratio is at time . We wish to obtain and . Because drift diffusion is a Markov process, we can use Eq. (11) to condition the values of and on their possible values at time :

 ε(λ,t)=∫∞−∞dxG(+)(x,δt)ε(λ+x,t+δt),T(λ,t)=δt+∫∞−∞dxG(+)(x,δt)T(λ+x,t+δt). (14)

We then expand and around and on both sides and keep all terms of order . In the limit , Eqs. (14) then reduce to Kolmogorov backward partial differential equations Risken (1989); Klebaner (2005) of the drift-diffusion type 333We note that backward equations can also be obtained for the case of Poissonian noise. In this case, each partial differential equation is replaced by a discrete set of coupled rate equations. See, for example, Ref. Feller (1968).:

 −12r∂ε(λ,t)∂t=∂ε(λ,t)∂λ+∂2ε(λ,t)∂λ2,−12r∂T(λ,t)∂t=∂T(λ,t)∂λ+∂2T(λ,t)∂λ2+12r. (15)

The adaptive decision rule is implemented by setting appropriate boundary conditions. Since we have assumed that the state is , an error occurs only when we stop with . We must therefore have and . Similarly, the remaining readout time after stopping must vanish, and . Here, is the Heaviside step function.

To solve Eqs. (15) analytically, we first take . The corresponding solutions and are independent of and satisfy the ordinary differential equations:

 ∂ε∞(λ)∂λ+∂2ε∞(λ)∂λ2=0,∂T∞(λ)∂λ+∂2T∞(λ)∂λ2+12r=0, (16)

subject to the boundary conditions and . Solving Eqs. (16) gives Eq. (3):

 ϵ∞=ε∞(0)=11+e¯λT∞=T∞(0)=¯λ2rtanh(¯λ2). (17)

To obtain the solution for finite , we write and . Substituting these expressions in Eqs. (15), we find that and satisfy the homogeneous equations

 −12r∂η(λ,t)∂t=∂η(λ,t)∂λ+∂2η(λ,t)∂λ2,−12r∂ζ(λ,t)∂t=∂ζ(λ,t)∂λ+∂2ζ(λ,t)∂λ2, (18)

subject to the boundary conditions , and , , respectively. For each of the Eqs. (18), we decompose the boundary condition at time in terms of the basis of right eigenfunctions of the drift-diffusion operator . The eigenfunctions are chosen to vanish for . We then propagate each component backwards in time to find and . This gives the exact analytical expressions:

 ϵ=ϵ∞+∞∑m=0Ame−αmrtM,rT=rT∞+∞∑m=0Bme−αmrtM. (19)

Here, we define

 Am=2¯λ4π2(m+1/2)2+¯λ2,Bm=−16π¯λ2cosh(¯λ2)(−1)m(m+1/2)[4π2(m+1/2)2+¯λ2]2,αm=12+2π2(m+1/2)2¯λ2. (20)

Equation (19) reduces to Eq. (17) when .

## Appendix B Derivation of the update matrices

In this section, we derive the form of the update matrices using a number-resolved quantum trajectory formalism Cook (1981); Emary et al. (2007). The NV-center charge state at any given time can be described by a state vector , where is the probability of finding the charge state . The state vector obeys the equation of motion of a Markovian two-level fluctuator:

 ˙ρ(t)=Lρ(t). (21)

The state at time is . Here, is the Lindblad superoperator of the two-level fluctuator (in the basis ):

 L=(−Γ+Γ−Γ+−Γ−). (22)

To analyze the photon emission statistics, we resolve (or unravel) the state vector in the photon number:

 ρ(t)=∞∑n=0ℓ(n,t), (23)

where is the unnormalized state vector after measurement of photons in the interval . More precisely, the probability of detecting photons after time is

 P(n,t)=Trℓ(n,t), (24)

where the trace Tr is defined as the sum of all elements in the vector. Our goal is to find the measurement superoperator so that can be expressed as

 P(n,t)=Tr[M(n,t)ρ0]. (25)

We must first obtain the equations of motion for . The dynamical evolution of is illustrated in Fig. 8. At each time step, there are two possible types of transition. Either an ionization or recombination occurs ( is unchanged) or a photon is detected ( is increased by ). Thus, obeys the following set of coupled rate equations:

 ˙ℓ(n,t)=Lℓ(n,t)−K[ℓ(n,t)−ℓ(n−1,t)]. (26)

The initial condition is . Here, encodes the state-dependent photon detection:

 K=(γ+00γ−). (27)

Note that summing Eq. (26) over all recovers Eq. (21) after applying Eq. (23), as required by conservation of probability. The effect of dark counts and imperfect detection efficiency is simply to modify the rates .

Equation (26) can be solved by introducing the characteristic function (or Fourier transform) Emary et al. (2007)

 ℓ(χ,t)=∑nℓ(n,t)einχ,ℓ(n,t)=12π∫2π0dχℓ(χ,t)e−inχ, (28)

where is a counting field. Substituting Eq. (28) into Eq. (26), we have

 ˙ℓ(χ,t)=R(χ)ℓ(χ,t), (29)

where

 R(χ)=L−K+eiχK. (30)

Solving Eq. (29) and inverting the characteristic function then gives

 ℓ(n,t)=M(n,t)ℓ0, (31)

where the measurement superoperator is

 M(n,t)=12π∫2π0dχeR(χ)te−inχ,M(n,t)=1n!dndzneR(z)t∣∣∣z=0. (32)

In the last line, we expressed the result in terms of the -transform variable,  Cook (1981). We see that the measurement superoperators are generated by the matrix function .

We can now calculate the probability of the trajectory given the initial state as

 P(ψN|ℓ0)=Tr[M(δnN−1)…M(δn0)ℓ0], (33)

where is the desired update matrix. Equation (33) can be used to compare the likelihood function for two arbitrary initial states , in particular the two charge states and . If we choose a single time bin, , generates the photon number distributions given in Ref. Shields et al. (2015) (see Appendix D). Moreover, expanding for small yields the continuous-time measurement superoperator for direct detection Gambetta and Wiseman (2001). Note also that Eq. (33) is an exact solution of the continuous-time filtering equations discussed in Ref. Ng and Tsang (2014).

## Appendix C Details of Monte Carlo simulations

In this section, we describe the Monte Carlo simulations of the error rates in more detail. For a given initial state , we generate a trajectory iteratively using Eq. (33) as follows:

1. Knowing the state in bin , calculate the corresponding photon probability distribution for .

2. Sample a value at random from the distributions .

3. Find the postmeasurement state .

4. Go back to step with initial state .

We use this procedure to simulate trajectories using , , , and the rates extracted from experimental data (see Appendix E.1). Each trajectory is then processed using both the nonadaptive and adaptive MLE or MAP, depending on the assumed prior probabilities .

We first simulate the nonadaptive decision rule. We choose times ranging from to . For each trajectory, we use Eq. (8) to calculate . An error occurs if () when the initial state is (). Here, is the optimal decision threshold. Averaging the errors over all trajectories for each state gives the conditional error rate for each time . The error rate is then given by .

Next, we simulate the adaptive decision rule. We reformulate the stopping condition in terms of the stopping probabilities . Here,

 pt=Λt/Λth1+Λt/Λth=eλt−λth1+eλt−λth (34)

is the probability of the initial state being when the likelihood ratio is . To vary the average readout time, we choose stopping probabilities ranging from to and from to . For each trajectory, we then use Eq. (8) to find the first time for which , or . An error occurs if () when the initial state is (). Averaging the stopping times and errors over all trajectories for each state gives the conditional average readout time and the conditional error rate for each pair of stopping probabilities . The average readout time and the error rate are calculated as and , respectively. We then numerically choose the pairs of stopping probabilities that minimize the error rate under the constraint of a constant average readout time. An example of the application of the adaptive decision rule is illustrated in Fig. 9 for both initial states.

## Appendix D Photon-counting method

Here we summarize the analytical photon-counting readout model used in Ref. Shields et al. (2015). In this method, the number of photons detected during the interval is compared to a threshold . If , we choose , while if , we choose . To calculate the error rate, we need the conditional photon distributions . These can be obtained within our formalism via Eq. (25). However, from a computational point of view, it is more efficient to use the expressions given in Ref. Shields et al. (2015). These are

 P(n|±)=P(n,E|±)+P(n,O|±). (35)

Here, () corresponds to an even (odd) number of ionizations or recombinations having occurred after time . Each term in Eq. (35) is given by

 P(n,E|±)=P(n,μ(±)tf)e−Γ±tf+∫tf0dtP(n,μ(±)t)xtI1(2xt)tf−