Maximal Adaptive-Decision Speedups in Quantum-State Readout

B. D’Anjou,1 L. Kuret,1 L. Childress,1 and W. A. Coish1,21Department of Physics, McGill University, Montreal, Quebec, H3A 2T8 Canada

2Quantum Information Science Program, Canadian Institute for Advanced Research,Toronto, Ontario, M5G 1Z8 Canada

(Received 24 July 2015; revised manuscript received 13 November 2015; published 23 February 2016)

The average time T required for high-fidelity readout of quantum states can be significantly reduced viaa real-time adaptive decision rule. An adaptive decision rule stops the readout as soon as a desired level ofconfidence has been achieved, as opposed to setting a fixed readout time tf . The performance of theadaptive decision is characterized by the “adaptive-decision speedup,” tf=T. In this work, we reformulatethis readout problem in terms of the first-passage time of a particle undergoing stochastic motion. Thisformalism allows us to theoretically establish the maximum achievable adaptive-decision speedups forseveral 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 by4 and 2, respectively, in the limit of high single-shot readout fidelity. We experimentally study theachievable speedup in a real-world scenario by applying the adaptive decision rule to a readout of thenitrogen-vacancy-center (NV-center) charge state. We find a speedup of ≈2 with our experimentalparameters. In addition, we propose a simple readout scheme for which the speedup can, in principle, beincreased without bound as the fidelity is increased. Our results should lead to immediate improvements innanoscale magnetometry based on spin-to-charge conversion of the NV-center spin, and provide atheoretical framework for further optimization of the bandwidth of quantum measurements.

DOI: 10.1103/PhysRevX.6.011017 Subject Areas: Quantum Physics, Quantum Information

I. INTRODUCTION

Efficient discrimination of quantum states is importantfor, e.g., fast qubit readout [1–4], rapid feedback andsteering [5–7], preparation of nonclassical states of light[5,8–10], and nanoscale magnetometry [11,12]. Givensufficient information about the statistics and dynamicsof a physical readout apparatus, it is possible to speed up areadout through a real-time adaptive decision rule(described below) [1–6,10]. An adaptive decision rulecan result in a significant reduction in the average timeper measurement without any significant deterioration ofthe readout fidelity. Adaptive decisions can be used toimprove any readout scheme, in principle, given the abilityto continuously monitor the state on a time scale that isshort compared to the time required for high-fidelityreadout [1,2,4,12–22].Adaptive-decision problems have a long history in

probability theory. Their mathematical root can be tracedback as far as the famous “Gambler’s ruin” problemintroduced by Pascal in the 17th century [23]. A generalmathematical theory of adaptive decision rules, known assequential analysis, was developed during the World War II

[24] and has since become a well-established part ofstatistical decision theory [25]. Implementing an adaptivedecision rule requires the ability to update a stochasticallyvarying measure of confidence in the state, typically alikelihood function, in real time. A measurement outcomeis then chosen when the confidence measure first achieves adesired value. Adaptive-decision problems are thus closelyrelated to first-passage time analysis [26,27]; the stochas-tically varying confidence measure can be treated as thecoordinate of a diffusing particle, which crosses a boundarywhen the desired confidence is reached. An adaptivedecision allows for a reduction in the average measurementtime, associated with a corresponding “adaptive-decisionspeedup.” Here, we formulate the adaptive decision for atwo-state readout in terms of a first-passage time problem.We use this formalism to theoretically establish the maxi-mum achievable speedups for several physical readoutmodels. Moreover, we demonstrate that significant speed-ups can be achieved in practice by experimentally charac-terizing the adaptive-decision speedup for the detection of aNV-center charge state, effectively doubling the bandwidthof such a measurement.The maximum achievable speedup is ultimately deter-

mined by the specific dynamics of a given readoutapparatus. Some readouts may allow for a very largespeedup, while others are fundamentally limited. Here,we obtain upper bounds for the adaptive-decision speedupof two commonly encountered readout schemes. The first

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model we consider is the paradigmatic Gaussian latchingreadout, in which the two states are distinguished by twonoisy signals of constant but distinct intensity [28,29]. Thismodel can be a good approximation of, e.g., a readout forthe singlet and triplet states of two electron spins in adouble quantum dot [14], readouts of electron spin statesbased on repetitive measurement of nuclear spin ancillae[17,19], or a readout of superconducting qubits coupled to amicrowave cavity [22]. The second model we consider isthe readout based on state-dependent decay, in which thestate is identified conditioned on a measured decay event.Readouts of, e.g., trapped ion qubits [4], NV-center spinqubits [6,18], and semiconductor spin qubits [13,16] can beapproximated by this model. In the case of the Gaussianlatching readout, we find that the speedup is bounded by afactor of 4 as the fidelity is increased, while for the case ofstate-dependent decay, the speedup is bounded by a factorof 2.To experimentally study the achievable speedup, we

implement an adaptive decision rule for a readout thatdiscriminates between two charge states of a single NVcenter in diamond [12,20]. The NV-center charge readoutrelies on the ability to distinguish strong from weakfluorescence upon illumination of the NV-center impurity.Remarkably, the NV-center charge readout can approacheither of the readout models described above in distinctlimits. However, for realistic experimental parameters, thedynamics of the NV-center charge readout will be inter-mediate between these two extremes. We therefore analyzeand experimentally quantify the associated speedup. Weupdate our level of confidence in the charge state using aquantum trajectory formalism that can easily be generalizedto more complex readout schemes. We find an adaptive-decision speedup ≈2 both for our experimental parametersand the experimental parameters of Ref. [12], in which asimilar charge-state measurement has recently been per-formed. Since a shorter average measurement timeincreases the number of measurements that can be per-formed on a NV-center spin in a given time, this resultshould directly improve the sensitivity of magnetometrybased on spin-to-charge conversion of the NV-center spin[12]. A similar approach can, in principle, be applied tomagnetometry using the standard spin readout of the NVcenter [30–33].An adaptive decision can only speed up the aforemen-

tioned readout schemes by a factor of order unity. However,we show that the speedup can become parametrically largefor other schemes. More precisely, we propose a simplereadout, based on the discrimination of two distinct decaychannels, for which the speedup becomes unbounded as thefidelity is increased. Such a readout could be realized ineither atomic or quantum-dot systems.This article is organized as follows. In Sec. II, we

describe the general features of an adaptive decision rulethat uses the likelihood ratio as a measure of confidence. In

Sec. III, we discuss the maximal speedup for the Gaussianlatching readout (Sec. III A) and for the readout based onstate-dependent decay (Sec. III B). We show that thespeedup is bounded by a factor of 4 for the Gaussianlatching readout and by a factor of 2 for the state-dependentdecay readout. In Sec. IV, we introduce the charge readoutof the NV center (Sec. IVA). We describe our experimentalsetup as well as how the parameters of the charge dynamicswere extracted from experimental data (Sec. IV B). We thendiscuss how to apply the adaptive decision rule to the NV-center charge readout (Sec. IV C). We assess the perfor-mance of the adaptive decision rule both by performingMonte Carlo simulations based on experimental parametersand by applying the adaptive decision rule directly toexperimental data. We demonstrate a factor ≈2 speedup forthe readout of the NV-center charge state. In Sec. V, wediscuss a readout scheme where the speedup increaseswithout bound as the readout fidelity is increased. Wesummarize in Sec. VI. Technical details are given inAppendixes A–E.

II. ADAPTIVE DECISION RULE

The goal of readout is to discriminate between twoparticular states of a system of interest, jþi and j−i. Toachieve this, we typically let the system interact with ameasurement device for a finite amount of time t, thereadout time. During the time t, the measurement devicerecords a time-resolved trajectory ψ t in the form of, e.g., anelectrical or photonic signal. As illustrated in Fig. 1(a), adistinct trajectory is recorded when the state is either jþi(blue) or j−i (red), making it possible to discriminatebetween the two states. In the presence of noise, perfectdiscrimination is not possible. We then decide whether thestate was most likely jþi or j−i based on the entiretrajectory acquired during the readout time. For an optimalreadout, this decision maximizes the readout fidelity F,namely, the probability that the state is identified correctly.There is a tradeoff between readout fidelity and readout

time; a larger fidelity is typically achieved at the cost of alonger readout time. To achieve a given fidelity target, thesimplest approach is to choose a fixed readout time tf frommeasurement to measurement. The time tf is then selectedto achieve the desired fidelity. An alternative strategy is toimplement an adaptive decision rule. In this approach, eachindividual measurement is stopped as soon as a stoppingcondition is satisfied. The stopping condition is associatedwith reaching a minimum level of confidence in the state. Atypical example of a stopping condition is illustrated inFig. 1(b). When the fidelity is high, a large subset of thepossible trajectories will achieve a high level of confidencein a short amount of time, while others will requiresubstantially longer times. Therefore, the average time Trequired for high-fidelity adaptive readout of the state maybe significantly shorter than the fixed time tf required toachieve the same fidelity with a nonadaptive readout. This

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is the main idea underlying the field of sequential detectiontheory, which aims to discriminate between competinghypotheses with the shortest possible average samplingtime [24,25,34,35]. The relative performance of the adap-tive decision rule is characterized by the adaptive-decisionspeedup, tf=T. In this work, we analyze the maximumachievable speedup for three different readout schemeslisted in Table I. Each scheme will be described in detail inthe following sections. Note that the adaptive decision rulediscussed in this work does not require measurementfeedback. However, it can be combined with measurementfeedback to further reduce the average readout time [10].We now formalize the ideas introduced above. For a

given readout time t, the noisy state-dependent trajectorycan be represented by a function ψ tðt0Þ defined in the

interval t0 ∈ ½0; t�, as depicted in Fig. 1(a). Given theobserved trajectory, we wish to decide whether the initialstate was most likely jþi or j−i in a way that maximizes thefidelity F [36]. Alternatively, we wish to minimize the errorrate ϵ ¼ 1 − F. To do so, we calculate the likelihood ratioΛt ¼ Pðψ tjþÞ=Pðψ tj−Þ, or equivalently the log-likelihoodratio λt ¼ lnΛt. Here, Pðψ tj�Þ is the probability ofobserving the trajectory ψ t given that the state is initiallyj�i. The log-likelihood ratio expresses a level of con-fidence in the state. When the prior probability for j�i isPð�Þ, the decision rule that minimizes the error rate is tochoose the maximum a posteriori estimate (MAP) of thestate; if λt > λth (λt < λth), we decide that the initial statewas most likely jþi (j−i). Here, λth ¼ ln ½Pð−Þ=PðþÞ� isthe optimal decision threshold. When the prior probabilitiesare assumed to be equal, PðþÞ ¼ Pð−Þ and λth ¼ 0, thedecision reduces to the maximum-likelihood estimate(MLE). In most of this article, we assume equal priorprobabilities so that the MLE is optimal. We briefly discussunequal prior probabilities and the MAP in Sec. IV C 3.When using a nonadaptive decision rule, we fix t ¼ tf

for each measurement and base our decision on the value ofλtf (nonadaptive MLE or MAP). When using an adaptivedecision rule, in contrast, we stop each measurement assoon as λt ≥ λþ, λt ≤ λ−, or t ≥ tM, as illustrated inFig. 1(b). Here, λ� are stopping thresholds and tM is amaximum readout time. We then base our decision on thevalue of λt at the stopping time (adaptive MLE or MAP).The average value of t from measurement to measurementgives the average readout time T. For each of the threereadout schemes listed in Table I, we have proven that asymmetric choice of stopping thresholds, λþ ¼ −λ− ¼ λ̄,minimizes T for a given error rate ϵ when the states areequally likely. For the readout discussed in Sec. IV, weoptimize the stopping thresholds numerically.

III. BOUNDED SPEEDUP

In this section, we apply the ideas of Sec. II to twocommonly encountered readout schemes, the Gaussianlatching readout and the readout based on state-dependentdecay. Although these schemes are usually not perfectlyrealized in practice, they are relevant to understand thelimitations of a wide variety of readouts. In particular, bothmodels can be mapped to extreme limits of the NV-centercharge readout discussed in Sec. IV.

FIG. 1. (a) Readout trajectories for the Gaussian latchingreadout. The trajectory for state jþi (solid blue) has an averageof þ1, while the trajectory for state j−i (dashed red) has anaverage of −1. Both trajectories have the same signal-to-noise ratio per unit time r. The identification of the state ismade based on the entire trajectory acquired during the readouttime t (dotted black). (b) Schematic representation of the adaptivedecision rule for the Gaussian latching readout. As soon as thelog-likelihood ratio λt (solid magenta) satisfies the stoppingcondition λt ≥ λ̄, λt ≤ −λ̄ or t ≥ tM, data acquisition is stopped.The sign of λt is then used to choose the qubitstate (dashed blue for jþi and dotted red for j−i). Here, wechoose jþi.

TABLE I. Summary of the maximal speedups for the threereadout schemes discussed in this work.

Readout scheme Maximal tf=T

Gaussian latching readout 4State-dependent decay 2Decay-channel discrimination ∞

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A. Gaussian latching readout

The Gaussian latching readout is the simplest, mostwidespread and most tractable model of state discrimina-tion [28]. In this scheme, each state j�i gives rise to astate-dependent trajectory with constant average �1 andsubject to Gaussian white noise, as illustrated in Fig. 1(a).A latching readout is characterized by the absence of staterelaxation at all times. Even though most readouts arelimited by state relaxation [29,37,38], many implementa-tions will approximately become latching readouts as theerror rate is decreased [12,14,17,19,22].Formally, the probability distributions for ψ tðt0Þ condi-

tioned on the state are

Pðψ tj�Þ ¼ A exp

�− r2

Zt

0

dt0½ψ tðt0Þ∓1�2�: ð1Þ

Here, r is the power signal-to-noise ratio per unit time andA is a normalization constant independent of the state.From Eq. (1), it is straightforward to show that the log-likelihood ratio is λt ¼ 2r

Rt0 dt

0ψ tðt0Þ. Thus, in the case ofthe Gaussian latching readout, a maximum-likelihooddecision can be made by integrating the trajectory up totime t.If the trajectory is acquired over the fixed interval ½0; tf�,

the error rate takes the form [29]

ϵ ¼ 1

2erfc

� ffiffiffiffiffiffirtf2

r �; ð2Þ

where erfc is the complementary error function [39]. Aderivation 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 interchangeof jþi and j−i, it is optimal to choose symmetric stoppingthresholds. Therefore, we stop the readout as soon asλt ≥ λ̄, λt ≤ −λ̄ or t ≥ tM. As in the nonadaptive case, wechoose the state according to the sign of λt at the end of thesequence. The adaptive decision rule is illustrated inFig. 1(b). The error rate ϵ and average readout time Tare then defined parametrically as a function of both λ̄ andtM. The calculation of ϵ and T can be recast as a first-passage time problem [26,27], as detailed in Appendix A.For clarity, here we only give the result for the case whenthe probability of reaching t ≥ tM is negligible, rtM ≫min ð1; λ̄2Þ (see Appendix A). The corresponding error rateand average readout time are given by

ϵ ¼ 1

1þ eλ̄; T ¼ λ̄

2rtanh

�λ̄

2

�: ð3Þ

By varying the stopping threshold λ̄, we can map thefunctional relation ϵðTÞ. As ϵ → 0, Eq. (3) implies thatϵ ≈ exp ð−2rTÞ. In the nonadaptive case, Eq. (2), we

instead have ϵ ≈ expð−rtf=2Þ= ffiffiffiffiffiffiffiffiffiffiffi2πrtf

p. Therefore, in the

limit of small ϵ, the adaptive decision rule asymptoticallyachieves the same error rate at an average time 4 timesshorter than the nonadaptive decision rule [25,34]. Thus,the maximal speedup is given by

tfT∼ 4; ðϵ → 0Þ: ð4Þ

Here and throughout this article, “∼” indicates an asymp-totic equality.

B. State-dependent decay

As a second example, we consider the class of readoutsrelying on state-dependent decay. In these schemes, the statejþi decays to j−i with probability per unit time τ−1. Thedetection of a decay event, e.g., the detection of an emittedphoton or of a tunneling electron, indicates that the initialstate was jþi. In contrast, the absence of a decay eventindicates that the state was j−i. This situation is approx-imately realized in, e.g., trapped ion qubits [4], NV-centerspin qubits [6,18], and semiconductor spin qubits [13,16].To facilitate the discussion, we assume that the events aredetectedwith perfect efficiency.Moreover, we assume that adetected event is always the result of a transition; i.e., thereare no “dark” counts. In this case, the log-likelihood ratio isλt ¼ −t=τ when no event has been detected after a readouttime t because the detection probability of a decay eventdecreases exponentially with time. When an event isdetected, λt suddenly becomes infinite because an eventis guaranteed to have resulted from the initial state being jþi.This scenario is illustrated in Fig. 2.When using a nonadaptive decision rule, we wait for a

time tf ≳ τ and base the decision onwhether a decay event isdetected in the interval ½0; tf�, as indicated by the sign of λt.The error rate is given by the probability that the event occursafter time tf assuming an initial state jþi, e−tf=τ, multipliedby the probability 1=2 that the state was initially jþi:

ϵ ¼ 1

2e−tf=τ: ð5Þ

In contrast with the nonadaptive case, the adaptivedecision rule stops the readout as soon as λt < −λ̄ orλt > λ̄, as shown in Fig. 2. Note that, for this particularlysimple readout scheme, the use of a maximum readout timetM is redundant since the above stopping condition is thesame as stopping the readout as soon as an event is detectedbefore a maximum time tM ¼ λ̄τ. Variations of this adaptivedecision rule have been implemented in Refs. [4,6]. Theerror rate is now given by the probability of an eventoccurring after time tM. To achieve the same error rate asthe nonadaptive decision rule, Eq. (5), we thus set tM ¼ tf.However, we find that the readout time is now given, onaverage, by ð1 − e−tf=τÞτ when the state is jþi. Thus, theaverage readout time is T ¼ ½ð1 − e−tf=τÞτ þ tf�=2

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because, as noted in Ref. [4], we must always wait for thefull duration tf when the state is j−i. Thus, once again, wefind a bounded speedup in the limit of vanishing error rate,tf → ∞:

tfT∼ 2; ðϵ → 0Þ: ð6Þ

IV. EXPERIMENT: NV-CENTERCHARGE READOUT

The analysis of Sec. III provides limits on the achievablespeedup for two idealized readout mechanisms. Manyexperimental realizations of a two-state readout canapproach one or the other model in different limits, butunder typical experimental conditions, neither limit can bestrictly realized. We must therefore use an algorithm for theadaptive decision rule suited to the underlying readoutmechanism.Here, we provide an explicit implementation of an

adaptive decision rule for experimental two-state readoutin a regime that lies intermediately between the twoidealized examples of Sec. III. Specifically, we examinefluorescence-based charge-state detection of the NV centerin diamond. As explained below, this system approaches aGaussian latching readout in the limit of high fluorescencerates and long charge-state relaxation times, while it canbegin to resemble state-dependent decay for very lowexcitation powers. In the following, we describe the mainfeatures of the NV charge readout system and use exper-imental data to extract the underlying system parameters.We then both simulate and experimentally implement the

adaptive decision rule with the help of a simple algorithmthat uses our knowledge of the dynamics to update thelikelihood ratio in real time. For our experimental param-eters, we find a speedup of tf=T ≈ 2. The resultingimprovement in detection bandwidth can improve thesensitivity of an NV-center magnetometer using spin-to-charge conversion [12].

A. Charge dynamics of the NV center in diamond

The NV center in diamond is typically observed in twocharge states, namely, the negatively charged NV− and theneutral NV0. In keeping with our notation, we label NV−and NV0 by jþi and j−i, respectively. When the NV−impurity is illuminated with yellow light (≈594 nm),transitions between the NV− ground and excited statesscatter photons, which are detected at a rate γþ. Thefrequencyof theyellow light is below theminimum thresholdrequired to resonantly excite optical transitions of NV0, andthus only a residual detection rate γ− ≪ γþ is observed in theneutral state. Note that the rates γ� include the effect of darkcounts and imperfect detection efficiency. This difference influorescence rates enables direct readout of the NV-centercharge state [12,20]. In addition, the incident light can causeionization (recombination) of the NV− (NV0) state withprobability per unit time Γþ (Γ−). As illustrated in Fig. 3(a),this leads to an alternating process of ionization and

FIG. 3. Schematic representation of (a) the NV-center chargedynamics described in Sec. IVA, (b) the experimental apparatus,and (c) the experimental sequence.

FIG. 2. Adaptive decision rule for the state-dependent decayreadout. The readout is stopped as soon as the likelihood ratio λt(solid magenta) satisfies λt < −λ̄ or λt > λ̄. In this case, this pairof conditions is equivalent to stopping the readout as soon as anevent is detected before a maximum time tM ¼ λ̄τ (dot-dashedblack). We choose the state according to the sign of λt at thestopping time (dashed blue for jþi and dotted red for j−i). In thisexample, we choose j−i. For the magenta trajectory shownabove, this choice results in an error since an event occurs aftertime tM, as indicated by the sudden divergence in λt.

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recombination that limits readout fidelity [40]. Typically, thecondition minðγþ; γ−Þ ≫ maxðΓþ;Γ−Þ is satisfied, so manytransitions occur between ionization and recombinationevents. The trajectory ψ t can therefore be modeled as ahidden two-state Markov process subject to state-dependentPoissonian noise of intensity γ� [12,20]. Because of detectorbandwidth limitations, the maximum readout time tM isusually separated into N bins of duration δt ¼ tM=N, so weobserve δni photons in bins i ¼ 0; 1; 2;…; N − 1. Thus, themeasured trajectory can be represented as an N-componentvector ψN ¼ ðδnN−1;…; δn0Þ.Note that the charge readout of the NV center approaches

the two cases discussed in Sec. III in different limits. Whenmin ðγþ; γ−Þ ≫ δt−1, the noise statistics of the NV-centercharge readout are approximately Gaussian, albeit withasymmetric signal-to-noise ratios per unit time:

r� ¼ ðγþ − γ−Þ24γ�

: ð7Þ

To perform a high-fidelity readout, the rate of informationgain must be faster than state relaxation, min ðrþ; r−Þ ≫max ðΓþ;Γ−Þ. For an adaptive advantage, the rate ofinformation gain must also be slower than the samplingrate, max ðrþ; r−Þ ≪ δt−1. Combined with min ðγþ; γ−Þ ≫δt−1, the last inequality implies the necessary conditionmaxðrþ;r−Þ≪minðγþ;γ−Þ or jγþ − γ−j ≪ 2min ðγþ; γ−Þ.We conclude that when state relaxation is negligible andwhen γþ and γ− are both large and comparable inmagnitude, the NV-center charge readout is approximatelydescribed by the model of Sec. III A with signal-to-noiseratio r≈rþ≈r−. In contrast, when γþ ≫ maxðγ−;Γþ;Γ−Þ,the NV-center charge readout can be approximately mod-eled by the state-dependent decay of Sec. III B withτ ¼ γ−1þ . Indeed, in this limit, the presence or absence ofa single photon on a time scale of a few γ−1þ is sufficient tochoose the state with high confidence. Therefore, thedetection of additional photons adds little information,and the state-dependent decay model is a good approxi-mation. For our parameters, the experiments describedbelow do not fall strictly into either of these limits. It istherefore interesting to characterize the speedup in thisintermediate regime.

B. Experimental setup and data

We characterize the performance of the adaptivedecision rule using experimental data. The data wereacquired using a home-built confocal microscope with594-nm excitation. The 594-nm wavelength lies in betweenthe zero-phonon line of the NV− (637 nm) and NV0

(575 nm), so it only efficiently excites the negativelycharged state. The excitation light is focused through ahigh numerical aperture objective (NA 1.35) onto a h111icut chemical-vapor-deposition-grown diamond in whichsingle defects can be resolved. Photons emitted in the

wavelength range 645–800 nm are collected and detectedwith a single-photon counter. This detection range overlapsstrongly with the NV− fluorescence spectrum and onlyweakly with the NV0 fluorescence, while rejecting Ramanscattering from the diamond. The experimental setup isillustrated schematically in Fig. 3(b).To acquire fluorescence trajectories, 594-nm excitation is

applied continuously and photon counts are recorded for30 s in bins of 100 μs, with a small separation of 8.3 nsnecessary for our field-programmable gate array (FPGA)card to process the counts. These time scales are chosensuch that the total duration (30 s) is much greater than thetime scale for ionization and recombination, and each timebin (100 μs) is much shorter. A subset of such a time traceis illustrated in Fig. 4(a) with data rebinned to 10 ms. The

FIG. 4. (a) Experimental photon-count trajectory illustratingthe state-dependent fluorescence rate and two-level switching ofthe NV-center charge state for an illumination power of 0.55 μW.Here, the photon detection rate fluctuates between γþ ≈ 720 Hzfor NV− and γ− ≈ 50 Hz for NV0, as discussed in Sec. IV B. Thecorresponding average counts per 10-ms bin are indicated by thehorizontal dotted black lines. The NV-center ionization andrecombination rates are found to be Γþ ≈ 3.6 Hz andΓ− ≈ 0.98 Hz, respectively. (b) Log-likelihood ratio λt (solidmagenta) and counts per 100- μs bin (dashed red) for the first60 ms of the data shown in (a). The log-likelihood ratio wascalculated using Eq. (8) and the experimental parameters given inSec. IV B. The discontinuities in λt indicate the detection of aphoton.

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experiment is repeated to obtain statistics on fluorescenceand ionization rates, interleaved with a tracking step (532-nm green laser) that ensures the sample does not driftrelative to the focal point of the microscope. The exper-imental sequence is illustrated in Fig. 3(c). Such data wereacquired for excitation powers ranging from 0.55 μW to5 μW (as measured going into the objective). For com-parison, the saturation intensity at 594 nm is estimated to be2.5 mW. We verified that γþ and γ− scale as the laserintensity and that Γþ and Γ− scale as the laser intensitysquared, as shown in Ref. [20]. Data sets were measured fortwo h111i oriented NV centers, which yielded similarresults. Very weak excitation is used to ensure that γþ andγ− are much larger than Γþ and Γ−.For definiteness, we restrict the following analysis to a

single power setting. Specifically, the data sets for thelowest laser power setting of 0.55 μW were used to extractthe parameters γ� and Γ�, as detailed in Appendix E 1. Weobtain the rates γþ ≈ 720 Hz, γ− ≈ 50 Hz, Γþ ≈ 3.6 Hz,and Γ− ≈ 0.98 Hz, which are required to construct analgorithm for an adaptive decision rule.

C. Readout error analysis

In Refs. [12,20], single-shot readout of the charge statewas performed by counting the total number of photonsdetected in the interval ½0; tf� (see Appendix D). In contrast,implementing the adaptive decision rule requires an effi-cient algorithm to update the likelihood ratio after each timebin [2,29,37,41–44]. A hidden-Markov-model algorithmsuitable to the NV-center charge readout has been devel-oped in, e.g., Refs. [42,44]. Here, we use an equivalentquantum trajectory approach, which is more easily gener-alized to quantum systems with multiple levels andcoherent internal dynamics. Moreover, we find an updaterule that is valid for an arbitrary value of the bin size δt. Inparticular, more than one ionization or recombination eventmay occur during one time bin, an occurrence that becomesmore likely at high illumination power [20]. The updaterule of Ref. [44] is recovered when Γ�δt ≪ 1.As shown in Appendix B, the likelihood ratio ΛN ¼

PðψN jþÞ=PðψN j−Þ after N bins is given by

ΛN ¼ Tr½MðδnN−1Þ � � �Mðδn0ÞlðþÞ0 �

Tr½MðδnN−1Þ � � �Mðδn0Þlð−Þ0 �

: ð8Þ

In Eq. (8), lðþÞ0 ¼ ð1; 0ÞT and lð−Þ

0 ¼ ð0; 1ÞT are initial statevectors representing jþi and j−i, respectively. Here, thetrace Tr of a vector is defined as the sum of its elements.The matrixMðδnÞ is a 2 × 2 update matrix for the detectionof δn photons in a bin. The exact form of the updatematrices is given in Appendix B. They can be stored for allδn ≤ δnmax, where δnmax is the maximum number ofphotons with a non-negligible probability to occur. Thelikelihood ratio, Eq. (8), can then be updated in real time

through simple matrix multiplication. An example of theapplication of Eq. (8) to experimental data is given inFig. 4(b). Note that M is nothing but the measurementsuperoperator for a classical Markov process subject toPoissonian noise. The approach can thus be generalized toany Markovian quantum trajectory by substituting M withthe appropriate measurement superoperator for direct

detection and the vectors lð�Þ0 by any pair of initial states

to be discriminated. In particular, the standard spin detec-tion of the NV center modeled in, e.g., Refs. [45–47] can beprocessed in this way.In the following, we first use Monte Carlo simulations to

assess the ideal theoretical performance of the adaptivedecision rule. We then apply the adaptive decision rule toexperimental data to demonstrate a speedup under realexperimental conditions. In all cases, we choose theexperimental time bin δt ¼ 0.1 ms and a maximum acquis-ition time tM ¼ Nδt ¼ 25 ms. We use this value of δt andthe rates given in Sec. IV B to calculate and store the updatematrices MðδnÞ for δn ¼ 0; 1;…; δnmax. Here, δnmax ¼ 5is chosen to match the maximum number of photonsobserved in any one time bin of the experimental datasets. The residual probability of observing more than δnmax

photons in one time bin is ≈10−10.

1. Monte Carlo simulations

For both initial states j�i, we use the proceduredescribed in Appendix C to generate 1 × 106 randomtrajectories ψN in the interval ½0; Nδt�. We then apply boththe nonadaptive and the adaptive MLE to each trajectory toobtain a Monte Carlo estimate of the average readout timeT and error rate ϵ, as detailed in Appendix C. The resultingerror-rate curves are shown in Fig. 5. For comparison, wealso show the error rate of the photon-counting methodmodeled in Ref. [12] and summarized in Appendix D. Inthis approach, the total number of detected photons iscompared to a threshold to choose the state. As expected,the nonadaptive and adaptive MLE achieve a somewhatlower minimum error rate than photon counting. Moreimportantly, however, we see that the adaptive MLEachieves the minimum error rate of the photon-countingmethod in approximately half the time necessary for thenonadaptive MLE to achieve the same goal. Thus, thebandwidth of the already efficient NV-center charge read-out is doubled through signal processing alone. We haveverified that a similar advantage also exists for theexperimental parameters of Ref. [12]. Since this speedupenables a larger number of NV-center charge measurementsto be performed in a given amount of time, it should lead toa substantial improvement in the sensitivity of the spin-to-charge conversion magnetometer in cases where the meas-urement duty cycle is limited by readout time.We note that the parameters given in Sec. IV B satisfy the

condition for which the NV-center charge readout reducesto the state-dependent decay readout of Sec. III B,

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τ−1 ¼ γþ > maxðγ−;Γþ;Γ−Þ ¼ γ−. It is therefore plau-sible that the speedup ≈2 obtained in Fig. 5 has at leastpartially the same origin as that given in Eq. (6). Indeed, thespeedup ≈2 observed in Ref. [4] was essentially explainedby the state-dependent decay model for a readout whosedynamics are formally the same as the NV-center chargereadout dynamics. However, the model of Sec. III B failswhen the maximum readout time tM becomes comparableto γ−1− ≈ 20 ms, resulting in a much higher saturation errorrate in Fig. 5 than that predicted by Eq. (5) with τ ¼γ−1þ ≈ 1.4 ms and tf ¼ tM ¼ 25 ms. In our experiment, it istherefore necessary to use the update rule of Eq. (8) toobtain an accurate estimate of the error rate. In a contextwhere it may be favorable or necessary to move towards theregime of the high-fidelity Gaussian latching readout ofSec. III A, speedups larger than 2 should be possible.Entering this regime may not be advantageous for the NV-center charge readout since the increase in excitationintensity required to reach the Gaussian regime results ina detrimental increase in the ionization and recombinationrates. However, such an advantage may be possible in othersystems with similar dynamics.

2. Experimental verification

In the simulations of Sec. IV C 1, the statistics of thesimulated fluorescence trajectories are perfectly described

by the two-state model assumed for readout. In realexperimental conditions, however, the charge dynamicsmay deviate from this model. In this section, we show thatthe adaptive-decision speedup is robust to imperfections inour modeling by applying the adaptive decision rule toexperimental data. To independently verify our model ofcharge dynamics, we split our data sets into a calibration setand a testing set. The calibration set is used solely to extractthe experimental values of the rates γ� and Γ� given inSec. IV B. The extracted parameters and Eq. (8) are thenused on the testing set to verify the model. The details aregiven in Appendix E 2.We first perform a preliminary verification by comparing

the experimental distribution of log-likelihood ratios to thedistribution predicted by Monte Carlo simulations [seeAppendix E 2]. We find a close agreement between experi-ment and theory, showing that the model of chargedynamics discussed in Sec. IVA provides a good descrip-tion of the statistics of the experimental trajectories. We cantherefore use our model to perform approximate prepara-tion of the charge state in postselection. We use thispreparation to verify that the adaptive-decision speedupexists when the adaptive decision rule is applied toexperimental data. From the testing set, we prepare 8307(30693) trajectories of 25 ms with initial states jþi (j−i).We then read out the state for each trajectory using thephoton-counting, nonadaptive MLE, and adaptive MLEmethods (i.e., assuming equal prior probabilities).Comparing the result of the readout to the preparationallows us to estimate the experimental error rate~ϵ ¼ ð~ϵþ þ ~ϵ−Þ=2, where ~ϵ� are the error rates conditionedon the preparation in state j�i. To better compare theexperimental results with theory, we then fit the exper-imental error-rate curves to the theoretical prediction,accounting for an additional preparation error η [seeEq. (E7)]. This procedure gives η ¼ 2.22% as a singlefit parameter. We emphasize that this adjustment is auniform transformation of the error rate for all times andhence does not affect the measured speedup. The exper-imental error rate ϵ obtained after compensating forpreparation error is shown along with the error ratedetermined by Monte Carlo simulation in Fig. 5 [theuncompensated error rate ~ϵ is shown in Appendix E 2for comparison]. The experimentally measured error ratethus fits the theoretical prediction very well. Therefore, weconclude that the adaptive-decision speedup ≈2 discussedin Sec. IV C 1 persists when the adaptive decision rule isapplied to real experimental data in spite of possiblesystematic errors and imperfect modeling.

3. Unequal prior probabilities

A balanced probability distribution for the initial state,PðþÞ ¼ Pð−Þ, is often desirable because it enables theextraction of one bit of information per measurement.However, in applications such as the magnetometry

FIG. 5. Error rate as a function of average readout time forthe experimental parameters of Sec. IV B. The lines give theMonte Carlo simulated error rate, and the symbols give theexperimental values corrected for preparation errors usingEq. (E7). We display the error rate for the photon-countingmodel discussed in Appendix D (solid blue and blue circles), forthe nonadaptive MLE (dashed orange and orange squares), andfor the adaptive MLE (dotted green and green triangles). Thehorizontal dotted black lines indicate the minimum error rate ofthe photon-counting (1.9%) and MLE methods (1.5%). Asindicated by the vertical dotted black lines, the speedup of theadaptive MLE compared to the nonadaptive MLE is tf=T ≈ 1.9when the target error rate is the minimum error rate of the photon-counting method.

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protocol of Ref. [12], the probability distribution of theinitial state is, in reality, unbalanced, PðþÞ ≠ Pð−Þ. Ingeneral, the dependence of the adaptive-decision speedupon the prior probabilities Pð�Þ is nontrivial and maydepend on the particularities of the readout dynamics. Ageneral analysis lies beyond the scope of this work. Here,we nevertheless discuss how to account for unequal priorprobabilities. Moreover, we show that the adaptive-decisionspeedup of the experimentally relevant NV-center chargereadout discussed in Sec. IV C 1 persists even for unequalprior probabilities. We distinguish the case where Pð�Þ areunknown from the case where Pð�Þ are known.When the prior probabilities are unknown, the best

strategy is to calibrate the readout thresholds λth and λ�

by assuming that the prior probabilities are equal. Thisleads to the MLE readout discussed in Secs. IV C 1 and IVC 2. Under this assumption, we obtain the error rates ϵð�Þ

and average times Tð�Þ conditioned on the state j�i. Whenthe MLE readout is used on an unbalanced sample ofcharge states, the error rate and average readout timesare then given by ϵ ¼ PðþÞϵðþÞ þ Pð−Þϵð−Þ andT ¼ PðþÞTðþÞ þ Pð−ÞTð−Þ, respectively. The simulatederror rate as a function of average readout time in thisscenario is shown in Fig. 6(a) for the experimentalparameters of Sec. IV B and PðþÞ ¼ 0.25. We also showthe corresponding error-rate curve for the photon-countingmethod (see Appendix D). Using the minimum error rate ofthe photon-counting method as a reference, we find that asubstantial speedup tf=T ≈ 1.6 persists even when the priorprobabilities are unknown.When the prior probabilities are known, they can be used

to improve the MLE readout by adjusting the thresholds todirectly minimize ϵ ¼ PðþÞϵðþÞ þ Pð−Þϵð−Þ for constantT ¼ PðþÞTðþÞ þ Pð−ÞTð−Þ. This leads to the MAP readoutmentioned in Sec. II (see Appendix C for details). We notethat in a given application, the prior probabilities can easilybe determined by using the MLE to estimate the relativeproportion of jþi and j−i. The simulated error rate as afunction of average readout time when using the MAP isshown in Fig. 6(b) for the same experimental parametersand PðþÞ ¼ 0.25. We find an adaptive-decision speeduptf=T ≈ 1.8 using the minimum error rate of the photon-counting method as a reference, similar to the valueobtained in Sec. IV C 1 for PðþÞ ¼ 0.5. Therefore, weconclude that the adaptive-decision speedup persists forsignificant deviations from a balanced initial-statedistribution.

V. PARAMETRIC IMPROVEMENT IN SPEEDUP

From the examples of Secs. III and IV, we might betempted to conclude that the speedup is fundamentallybounded by a constant of order unity. Here, we show thatthere exist readout schemes where the speedup can becomearbitrarily 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 B. We nowsuppose that the states jþi and j−i both decay to a thirdstate j0i through different decay channels R and L,respectively. The two channels may be associated with,e.g., different polarizations of emitted photons or twodifferent leads into which the electron of a double-quantum-dot charge qubit can tunnel (see Fig. 7). If thetwo decay channels can be discriminated, e.g., with the helpof a polarization analyzer, the state j�i can be read out. Forsimplicity, we assume that both states decay with the sameprobability per unit time τ−1.When using a nonadaptive decision rule, we wait a time

tf ≳ τ and base the decision on whether a R or L decay

FIG. 6. Monte Carlo simulated error rate for an unbalancedinitial charge distribution with PðþÞ ¼ 0.25when (a) the value ofPðþÞ is unknown (MLE) and (b) the value of PðþÞ is known(MAP). In both cases, we show the error rate for the photon-counting method of Ref. [12] (solid blue), the nonadaptivereadout (dashed orange), and the adaptive readout (dotted green).The horizontal dotted black lines indicate the minimum errorrates of the photon-counting method (1.6% in both cases) and ofthe optimal methods (1.4% for the MLE and 1.3% for the MAP).The dotted vertical lines show that significant adaptive-decisionspeedups of tf=T ≈ 1.6 and tf=T ≈ 1.8 are obtained when usingthe MLE and MAP, respectively. The discontinuous features in(a) occur because the MLE uses a nonoptimal discriminationthreshold when Pð�Þ ≠ 0.5.

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event has been detected. If no event is detected, we choosethe state at random with equal probability. Assuming, forsimplicity, that the channels R and L can be perfectlydiscriminated, the error rate is given by the probability thatno decay event has occurred up to time tf, e−tf=τ, multipliedby the probability 1=2 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 maximumtime tM is reached. Following the same argument as for thenonadaptive decision rule, the error rate is ϵ ¼ e−tM=τ=2. Toachieve the same error rate as the nonadaptive decision rule,we must therefore choose tM ¼ tf. However, we find thatthe average readout time is now T ¼ ð1 − e−tf=τÞτ. As theerror rate decreases, tf → ∞, the average readout timetends to a constant τ so that

tfT∼ ln

�1

2ϵ

�→ ∞; ðϵ → 0Þ: ð9Þ

Therefore, the speedup increases without bound as theerror rate ϵ is decreased. This parametric improvement inspeedup is achieved by transferring all the informationabout the state to the channel degree of freedom, R or L.This means that the readout no longer relies on discrimi-nating decay from the absence of decay. We can thusarrange the readout so that both j−i and jþi decay on atime scale τ. Therefore, it is no longer necessary to wait forthe entire duration tf when the state is j−i.Such a scheme could be applied in a variety of scenarios.

For example, suppose that two atomic excited states, jþi ¼je; Ri and j−i ¼ je; Li, both decay to the ground statej0i ¼ jgi by emitting photons with right and left circularpolarizations, respectively. This situation is depicted inFig. 7(a). A polarization analyzer could then identify bothstates in a time ≈τ, leading to the parametric improvementof Eq. (9). In another example, suppose that we want todiscriminate between an electron being in the rightmost(jþi ¼ jRi) and leftmost (j−i ¼ jLi) islands of a doublequantum dot. We imagine that, as shown in Fig. 7(b), thepotential barriers between the dots and leads are lowered atthe beginning of the readout phase. An electron in the right(left) dot will then tunnel out into the right (left) lead and bedetected, with the left and right detectors playing the role ofthe polarization analyzers in the previous example. Inperfect analogy with Fig. 7(a), stopping readout as soonas an electron is detected in either lead instead of waitingfor a fixed time tf then leads to a parametric improvementin speedup, Eq. (9). Note that while the assumption ofperfect detection efficiency is difficult to realize for photondetection, it is usually not a limitation for the detection ofelectron charges in quantum dots.

VI. CONCLUSION

In summary, we have established the maximum achiev-able “adaptive-decision speedups” for several physicalreadout models in the high-fidelity limit. To achieve this,we formulated the adaptive-decision problem in terms of afirst-passage time problem. We obtained bounds for thespeedup of two commonly encountered readout models.Specifically, we have shown that the adaptive decision rulecan speed up the Gaussian latching readout (Sec. III A) byup to a factor of 4 in the limit of high readout fidelity, whileit can speed up readouts relying on state-dependent decay(Sec. III B) by up to a factor of 2. To study the achievablespeedup in a real-world scenario, we applied the adaptivedecision rule to a readout of the charge state of a NV centerin diamond using a quantum trajectory formalism thatincorporates the NV-center charge dynamics and accountsfor experimental imperfections such as dark counts andimperfect collection efficiency. Although the NV-centercharge readout reduces to the two aforementioned modelsin distinct limits, its dynamics are in an intermediate regimefor typical experimental parameters. We have shown that asignificant speedup can be achieved under these conditions.

FIG. 7. (a) Readout scheme based on state-dependent polari-zation of an emitted photon. When the state is jþi ¼ je; Ri(j−i ¼ je; Li) in the excited subspace, a right (left) circularlypolarized photon is emitted. Both states decay to the ground statej0i ¼ jgi on a time scale τ. (b) Similar scheme for readout of thecharge state of a double quantum dot. When the electron is inthe right (left) island jRi (jLi), it is detected after tunneling intothe right (left) lead. Both states tunnel into the leads on a timescale τ, leaving the double quantum dot in the empty state j0i.

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Specifically, we found a speedup≈2 both in our experimentand using the experimental parameters of Ref. [12]. Finally,we have proposed a readout scheme that leads to anunbounded speedup as the fidelity is increased, in starkcontrast to the common readout models discussed previ-ously. This scheme relies on discriminating between twodistinct decay channels instead of discriminating betweendecay and absence of decay. We have further providedavenues to realize such a scheme in atomic or quantum-dotsystems.Our results are already applicable to a wide variety of

systems, and they provide a direction for the optimizationof measurement bandwidth in experimentally relevantreadouts of quantum states. In particular, we have shownthat magnetometry based on the NV-center spin-to-charge-conversion readout can be improved with currently achiev-able experimental parameters. Moreover, our results showthat apparent limitations to the adaptive speedup can beovercome by careful redesign of the readout dynamics. Inthis work, our goal was to analyze the fundamentallimitations of the adaptive-decision speedup for extremelimits of several physical readout schemes. We expectdirect extensions of the first-passage time formalismdeveloped here to allow for the derivation of analyticalspeedup bounds in the presence of, e.g., unwanted staterelaxation, dark counts, or imperfect photon collectionefficiency. On a more fundamental level, our analysisprovides the necessary framework to study how theaddition of measurement feedback and coherent readoutdynamics can modify the maximal speedups discussedhere. Moreover, this formalism may allow for the directcharacterization of the achievable speedups for parameterestimation, 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.

APPENDIX A: ANALYTICAL TREATMENTOF THE ADAPTIVE GAUSSIAN

LATCHING READOUT

In this section, we sketch the derivation of the error rateand average readout time for both the nonadaptive and theadaptive decision rule applied to the Gaussian latchingreadout. According to Eq. (1), the expectation and covari-ance of the trajectory ψ tðt0Þ conditioned on the state j�i are

E½ψ tðt0Þj�� ¼ �1;

Cov½ψ tðt0Þ;ψ tðt00Þj�� ¼ r−1δðt0 − t00Þ; ðA1Þ

for t0 < t and t00 < t. Because the noise is Gaussian, allhigher cumulants vanish. Correspondingly, the expectation

and variance of the log-likelihood ratio λt ¼ 2rRt0 dt

0ψ tðt0Þare Eðλtj�Þ ¼ �2rt and Varðλtj�Þ ¼ 4rt. Therefore, λt is asimple drift-diffusion process distributed according to

Pðλtj�Þ≡Gð�Þðλt; tÞ

¼ 1ffiffiffiffiffiffiffiffiffi8πrt

p exp�− ðλt∓2rtÞ2

8rt

�: ðA2Þ

Here, Gð�Þðλ; tÞ is the Green’s function that solves theassociated drift-diffusion equation for λt.For a fixed readout time tf, the state is chosen according

to the sign of λtf . The error rates conditioned on the initial

state being j�i, ϵð�Þ, are thus

ϵðþÞ ¼Z

0

−∞dλtfPðλtf jþÞ;

ϵð−Þ ¼Z

∞

0

dλtfPðλtf j−Þ: ðA3Þ

By symmetry of the problem, ϵðþÞ ¼ ϵð−Þ. Using Eq. (A2),we can then calculate the error rate ϵ ¼ ½ϵðþÞ þ ϵð−Þ�=2 ¼ ϵð�Þ:

ϵ ¼ 1

2erfc

� ffiffiffiffiffiffirtf2

r �: ðA4Þ

To assess the performance of the adaptive decision rule,we must calculate both the error rate ϵ ¼ ½ϵðþÞ þ ϵð−Þ�=2and the average readout time T ¼ ½TðþÞ þ Tð−Þ�=2, whereTð�Þ is the average readout time conditioned on the statebeing j�i. Using the symmetry of the problem again, wehave ϵ ¼ ϵð�Þ and T ¼ Tð�Þ. We may therefore assume thatthe state is jþi without loss of generality. We nowreformulate the calculation of ϵ and T as a first-passagetime problem [26,27] (an alternative formalism is given inRef. [35]). More precisely, let εðλ; tÞ and T ðλ; tÞ be theerror rate and average readout time conditioned on knowingthat the log-likelihood ratio is λ at time t. We wish to obtainϵ ¼ εð0; 0Þ and T ¼ T ð0; 0Þ. Because drift diffusion is aMarkov process, we can use Eq. (A2) to condition thevalues of ε and T on their possible values at time tþ δt:

εðλ; tÞ ¼Z

∞

−∞dxGðþÞðx; δtÞεðλþ x; tþ δtÞ;

T ðλ; tÞ ¼ δtþZ

∞

−∞dxGðþÞðx; δtÞT ðλþ x; tþ δtÞ: ðA5Þ

We then expand ε and T around x ¼ 0 and δt ¼ 0on both sides and keep all terms of order δt. In the limitδt → 0, Eqs. (A5) then reduce to Kolmogorov backwardpartial differential equations [26,27] of the drift-diffusiontype [48]:

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− 1

2r∂εðλ; tÞ

∂t ¼ ∂εðλ; tÞ∂λ þ ∂2εðλ; tÞ

∂λ2 ;

− 1

2r∂T ðλ; tÞ

∂t ¼ ∂T ðλ; tÞ∂λ þ ∂2T ðλ; tÞ

∂λ2 þ 1

2r: ðA6Þ

The adaptive decision rule is implemented by settingappropriate boundary conditions. Since we have assumedthat the state is jþi, an error occurs only when we stopwith λt < 0. We must therefore have εð�λ̄; tÞ ¼ θð∓λ̄Þand εðλ; tMÞ ¼ θð−λÞ. Similarly, the remaining readouttime after stopping must vanish, T ð�λ̄; tÞ ¼ 0 andT ðλ; tMÞ ¼ 0. Here, θðxÞ is the Heaviside step function.To solve Eqs. (A6) analytically, we first take tM → ∞.

The corresponding solutions ε∞ðλÞ and T ∞ðλÞ are inde-pendent of t and satisfy the ordinary differential equations:

∂ε∞ðλÞ∂λ þ ∂2ε∞ðλÞ

∂λ2 ¼ 0;

∂T ∞ðλÞ∂λ þ ∂2T ∞ðλÞ

∂λ2 þ 1

2r¼ 0; ðA7Þ

subject to the boundary conditions ϵ∞ð�λ̄Þ ¼ θð∓λ̄Þ andT ∞ð�λ̄Þ ¼ 0. Solving Eqs. (A7) gives Eq. (3):

ϵ∞ ¼ ε∞ð0Þ ¼1

1þ eλ̄;

T∞ ¼ T ∞ð0Þ ¼λ̄

2rtanh

�λ̄

2

�: ðA8Þ

To obtain the solution for finite tM, we write εðλ; tÞ ¼ε∞ðλÞ þ ηðλ; tÞ and T ðλ; tÞ ¼ T ∞ðλÞ þ ζðλ; tÞ. Sub-stituting these expressions in Eqs. (A6), we find that ηand ζ satisfy the homogeneous equations

− 1

2r∂ηðλ; tÞ

∂t ¼ ∂ηðλ; tÞ∂λ þ ∂2ηðλ; tÞ

∂λ2 ;

− 1

2r∂ζðλ; tÞ

∂t ¼ ∂ζðλ; tÞ∂λ þ ∂2ζðλ; tÞ

∂λ2 ; ðA9Þ

subject to the boundary conditions ηð�λ̄; tÞ ¼ 0, ηðλ; tMÞ ¼θð−λÞ − ε∞ðλÞ and ζð�λ̄; tÞ ¼ 0, ζðλ; tMÞ ¼ −T ∞ðλÞ,respectively. For each of Eqs. (A9), we decompose theboundary condition at time tM in terms of the basis of righteigenfunctions of the drift-diffusion operator ∂λ þ ∂2

λ.The eigenfunctions are chosen to vanish for λ ¼ �λ̄.We then propagate each component backwards in timeto find ηð0; 0Þ and ζð0; 0Þ. This gives the exact analyticalexpressions:

ϵ ¼ ϵ∞ þX∞m¼0

Ame−αmrtM ;

rT ¼ rT∞ þX∞m¼0

Bme−αmrtM : ðA10Þ

Here, we define

Am ¼ 2λ̄

4π2ðmþ 1=2Þ2 þ λ̄2;

Bm ¼ − 16πλ̄2 coshðλ̄2Þð−1Þmðmþ 1=2Þ

½4π2ðmþ 1=2Þ2 þ λ̄2�2 ;

αm ¼ 1

2þ 2π2ðmþ 1=2Þ2

λ̄2: ðA11Þ

Equation (A10) reduces to Eq. (A8) when rtM ≫minð1; λ̄2Þ.

APPENDIX B: DERIVATION OF THEUPDATE MATRICES

In this section, we derive the form of the update matricesMðnÞ using a number-resolved quantum trajectory formal-ism [49,50]. The NV-center charge state at any given time tcan be described by a state vector ρ ¼ ðρþ; ρ−ÞT, where ρ�is the probability of finding the charge state j�i. The statevector obeys the equation of motion of a Markovian two-level fluctuator:

_ρðtÞ ¼ LρðtÞ: ðB1Þ

The state at time t ¼ 0 is ρð0Þ≡ ρ0. Here, L is theLindblad superoperator of the two-level fluctuator (in thebasis fjþihþj; j−ih−jg):

L ¼�−Γþ Γ−

Γþ −Γ−

�: ðB2Þ

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

ρðtÞ ¼X∞n¼0

lðn; tÞ; ðB3Þ

where lðn; tÞ is the unnormalized state vector after meas-urement of n photons in the interval ½0; t�. More precisely,the probability of detecting n photons after time t is

Pðn; tÞ ¼ Trlðn; tÞ; ðB4Þ

where the trace Tr is defined as the sum of all elements inthe vector. Our goal is to find the measurement super-operator Mðn; tÞ so that Pðn; tÞ can be expressed as

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Pðn; tÞ ¼ Tr½Mðn; tÞρ0�: ðB5Þ

We must first obtain the equations of motion for lðn; tÞ.The dynamical evolution of lðn; tÞ is illustrated in Fig. 8.At each time step, there are two possible types of transition.Either an ionization or recombination occurs (n isunchanged) or a photon is detected (n is increased by1). Thus, lðn; tÞ obeys the following set of coupled rateequations:

_lðn; tÞ ¼ Llðn; tÞ −K½lðn; tÞ − lðn − 1; tÞ�: ðB6Þ

The initial condition is lð0; 0Þ≡ l0 ¼ ρ0. Here, Kencodes the state-dependent photon detection:

K ¼�γþ 0

0 γ−

�: ðB7Þ

Note that summing Eq. (B6) over all n recovers Eq. (B1)after applying Eq. (B3), as required by conservation ofprobability. The effect of dark counts and imperfectdetection efficiency is simply to modify the rates γ�.Equation (B6) can be solved by introducing the char-

acteristic function (or Fourier transform) [50]

lðχ; tÞ ¼Xn

lðn; tÞeinχ ;

lðn; tÞ ¼ 1

2π

Z2π

0

dχlðχ; tÞe−inχ ; ðB8Þ

where χ is a counting field. Substituting Eq. (B8) intoEq. (B6), we have

_lðχ; tÞ ¼ RðχÞlðχ; tÞ; ðB9Þ

where

RðχÞ ¼ L −Kþ eiχK: ðB10ÞSolving Eq. (B9) and inverting the characteristic functionthen gives

lðn; tÞ ¼ Mðn; tÞl0; ðB11Þwhere the measurement superoperator Mðn; tÞ is

Mðn; tÞ ¼ 1

2π

Z2π

0

dχeRðχÞte−inχ ;

Mðn; tÞ ¼ 1

n!dn

dzneRðzÞtj

z¼0: ðB12Þ

In the last line, we expressed the result in terms of thez-transform variable, z≡ eiχ [49]. We see that the meas-urement superoperators are generated by the matrix func-tion Gðz; tÞ ¼ eRðzÞt.We can now calculate the probability of the trajectory

ψN ¼ ðδnN−1;…; δn0Þ given the initial state as

PðψN jl0Þ ¼ Tr½MðδnN−1Þ…Mðδn0Þl0�; ðB13Þwhere MðδnÞ ¼ Mðδn; δtÞ is the desired update matrix.Equation (B13) can be used to compare the likelihoodfunction for two arbitrary initial states l0, in particular, the

two charge states lðþÞ0 ¼ ð1; 0ÞT and lð−Þ

0 ¼ ð0; 1ÞT . If wechoose a single time bin, δt ¼ tf, MðnÞ generates thephoton number distributions given in Ref. [12] (seeAppendix D). Moreover, expanding MðδnÞ for small δtyields the continuous-time measurement superoperatorfor direct detection [51]. Note also that Eq. (B13) is anexact solution of the continuous-time filtering equationsdiscussed in Ref. [42].

APPENDIX C: DETAILS OF MONTECARLO SIMULATIONS

In this section, we describe the Monte Carlo simulationsof the error rates in more detail. For a given initial state ρ0,we generate a trajectory iteratively using Eq. (B13) asfollows:(1) Knowing the state ρk in bin k, calculate the corre-

sponding photon probability distribution PkðnÞ ¼Tr½MðδnÞρk� for δn ¼ 0; 1; 2;…; δnmax.

(2) Sample a value δnk at random from the distribu-tions PkðδnÞ.

(3) Find the postmeasurement state ρkþ1 ¼ MðδnkÞρk=PkðδnkÞ.

(4) Go back to step 1 with initial state ρkþ1.We use this procedure to simulate 1 × 106 trajectories usingδnmax ¼ 5, δt ¼ 0.1 ms, tM ¼ 25 ms, and the ratesextracted from experimental data (see Appendix E 1).Each trajectory is then processed using both the non-adaptive and adaptive MLE or MAP, depending on theassumed prior probabilities Pð�Þ.

FIG. 8. Unraveling of the NV-center charge readout dynamics.The state jþi (j−i) represents NV− (NV0). Photons are detectedat state-dependent rates γ�. Each detection increases the numberof observed photons n. The rates Γ� describe ionization orrecombination processes that are not associated with the detectionof a photon.

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We first simulate the nonadaptive decision rule.We choosetimes ranging from tf ¼ 0.1 ms to tf ¼ 25 ms. For eachtrajectory, we use Eq. (8) to calculate Λtf . An error occurs ifΛtf < Λth (Λtf > Λth)when the initial state is jþi (j−i).Here,Λth ¼ Pð−Þ=PðþÞ ¼ eλth is the optimal decision threshold.Averaging the errors over all trajectories for each state j�igives the conditional error rate ϵð�Þ for each time tf. The errorrate is then given by ϵ ¼ PðþÞϵþ þ Pð−Þϵ−.Next, we simulate the adaptive decision rule. We

reformulate the stopping condition in terms of the stoppingprobabilities p� ¼ eλ�−λth=ð1þ eλ�−λthÞ. Here,

pt ¼Λt=Λth

1þ Λt=Λth¼ eλt−λth

1þ eλt−λthðC1Þ

is the probability of the initial state being jþi when thelikelihood ratio is Λt ¼ eλt . To vary the average readouttime, we choose stopping probabilities ranging from pþ ¼0.9 to pþ ¼ 1 and from p− ¼ 0 to p− ¼ 0.1. For eachtrajectory, we then use Eq. (8) to find the first time t forwhich pt ≥ pþ, pt ≤ p− or t ≥ tM. An error occurs if pt <1=2 (pt > 1=2) when the initial state is jþi (j−i).Averaging the stopping times and errors over all trajectoriesfor each state j�i gives the conditional average readout timeTð�Þ and the conditional error rate ϵð�Þ for each pair ofstopping probabilities fpþ; p−g. The average readout timeand the error rate are calculated as T ¼ PðþÞTðþÞ þPð−ÞTð−Þ and ϵ ¼ PðþÞϵðþÞ þ Pð−Þϵð−Þ, respectively.We then numerically choose the pairs of stopping proba-bilities that minimize the error rate under the constraint of aconstant average readout time. An example of the

application of the adaptive decision rule is illustrated inFig. 9 for both initial states.

APPENDIX D: PHOTON-COUNTING METHOD

Here, we summarize the analytical photon-countingreadout model used in Ref. [12]. In this method, thenumber n of photons detected during the interval ½0; tf�is compared to a threshold ν. If n > ν, we choose jþi,while if n ≤ ν, we choose j−i. To calculate the error rate,we need the conditional photon distributions Pðnj�Þ.These can be obtained within our formalism viaEq. (B5). However, from a computational point of view,it is more efficient to use the expressions given in Ref. [12].These are

Pðnj�Þ ¼ Pðn; Ej�Þ þ Pðn;Oj�Þ: ðD1Þ

Here, E (O) corresponds to an even (odd) number ofionizations or recombinations having occurred after time tf.Each term in Eq. (D1) is given by

Pðn; Ej�Þ ¼ Pðn; μð�Þtf Þe−Γ�tf

þZ

tf

0

dtPðn; μð�Þt Þ xtI1ð2xtÞ

tf − te−Γ�te−Γ∓ðtf−tÞ;

Pðn;Oj�Þ ¼Z

tf

0

dtPðn; μð�Þt ÞΓ�I0ð2xtÞe−Γ�te−Γ∓ðtf−tÞ;

ðD2Þ

where IkðzÞ is the kth order modified Bessel function of thefirst kind [39] and where we define

Pðn; μÞ ¼ μn

n!e−μ;

μð�Þt ¼ γ�tþ γ∓ðtf − tÞ;xt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓþΓ−tðtf − tÞ

q: ðD3Þ

The conditional error rates are given by

ϵðþÞ ¼Xn≤ν

PðnjþÞ; ϵð−Þ ¼Xn>ν

Pðnj−Þ: ðD4Þ

Substituting Eq. (D2) into Eq. (D4), we obtain

ϵð�Þ ¼ ϵð�ÞE þ ϵð�Þ

O ; ðD5Þ

where

FIG. 9. Evolution of the NV− probability pt for a trajectorygenerated assuming jþi (solid blue) and j−i (dashed red) for thesame parameters as in Fig. 5. In this particular case, we assumedthat PðþÞ ¼ Pð−Þ. The discontinuities in pt correspond to thedetection of a photon. Data acquisition is stopped when pt firstcrosses the stopping thresholds pþ and p− (dotted black) or whent reaches the maximum acquisition time tM ¼ 25 ms.

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ϵð�ÞE ¼ q≶ðν; μð�Þ

tf Þe−Γ�tf

þZ

tf

0

dtq≶ðν; μð�Þt Þ xtI1ð2xtÞ

tf − te−Γ�te−Γ∓ðtf−tÞ;

ϵð�ÞO ¼

Ztf

0

dtq≶ðν; μð�Þt ÞΓ�I0ð2xtÞe−Γ�te−Γ∓ðtf−tÞ: ðD6Þ

Here, the q≶’s are cumulative distribution functions of thePoisson distribution Pðn; μÞ:

q<ðν; μÞ ¼ Qðνþ 1; μÞ;q>ðν; μÞ ¼ 1 −Qðνþ 1; μÞ; ðD7Þ

where Qða; zÞ is the regularized incomplete Gammafunction [39]. For a given time tf, we compute the integralsin Eq. (D6) numerically to calculate the error rate ϵ ¼PðþÞϵðþÞ þ Pð−Þϵð−Þ when PðþÞ is known, or ϵ ¼½ϵðþÞ þ ϵð−Þ�=2 when PðþÞ is unknown. We then increasethe threshold starting from ν ¼ 0 until we find a localminimum in the error rate. We repeat the procedure atvarious times to plot the error rate as a function of time, asshown in Fig. 5.

APPENDIX E: SYSTEM RATES ANDMODEL VERIFICATION

In this section, we describe how the rates γ� and Γ� wereextracted from experimental data and how our readoutmodel was validated. We first acquire 125 photon-counttrajectories, each of duration 30 s and with time binδt ¼ 0.1 ms. We separate the trajectories into a calibrationset (60 trajectories) and a testing set (65 trajectories). Therates are extracted using only the calibration set. The testingset is then used to independently verify the validity of thecharge dynamics model extracted from the calibration set.

1. Extraction of the system rates

To determine the rate γ�, we split all trajectories of thecalibration set into a total of 1800 subtrajectories of 1 s,each of which is rebinned in bins of δt ¼ 10 ms. We thenmake a histogram of photon counts in all bins of 10 ms, asillustrated in Fig. 10(a). We fit the histogram to a mixture ofPoisson distributions

PðδnÞ ¼ Pðδn; γþδtÞPðþÞ þ Pðδn; γ−δtÞPð−Þ; ðE1Þ

where Pð�Þ are prior probabilities for each charge state andwhere Pðn; μÞ is given in Eq. (D3). The histogram is wellfitted by Eq. (E1), indicating that Γ�δt ≪ 1, i.e.,Γ� ≪ 100 Hz. From the means of the Poisson distribu-tions, we extract γþ ≈ 720 Hz and γ− ≈ 50 Hz. Whenassuming no prior information on the state, the countthreshold ν that best discriminates between NV− and NV0

in one time bin is given by Pðν; γþδtÞ ¼ Pðν; γ−δtÞ, or

ν ¼ γþ − γ−lnðγþγ−Þ

δt: ðE2Þ

We find ν ≈ 2.5. Thus, we choose NV− for δn > 2 and NV0

for δn ≤ 2, with a ≈2% rate of error for the simplifiedmodel of Eq. (E1) (Fig. 5 suggests that the actual error rateis closer to 3% at a measurement time of 10 ms).To determine Γ�, we use this thresholding procedure on

the first 10-ms bin of each 1-s subtrajectory to postselectthe initial state of the subtrajectory. This gives 872subtrajectories with the initial state identified as jþi and2878 trajectories with the initial state identified as j−i. Wethen obtain the average ϱ�ðtÞ of all trajectories for a giveninitial state j�i, as illustrated in Fig. 10(b). We fit both

FIG. 10. (a) Histogram of photon counts per 10-ms time bin(black dots). The solid green curve is a fit to a mixture of Poissondistributions, Eq. (E1). The dashed blue and dotted red curves arethe Poisson distributions conditioned on states jþi and j−i,respectively. We extract γþ ≈ 720 Hz and γ− ≈ 50 Hz from themeans of the conditional distributions. (b) Average photon countsper 10 ms, ϱ�ðtÞ, for trajectories postselected on the initial statesjþi (blue circles) and j−i (red squares). The solid black lines area simultaneous fit of ϱþðtÞ and ϱ−ðtÞ to Eq. (E3). From the fit, weextract Γþ ≈ 3.6 Hz and Γ− ≈ 0.98 Hz. The horizontal dottedblack line is the steady-state average B obtained from the fitto Eq. (E3).

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curves simultaneously [44] to the rate equation model ofthe two-level fluctuator, Eq. (B2), which predicts

ϱ�ðtÞ ¼ A�e−Γt þ B; ðE3Þ

where

Γ ¼ Γþ þ Γ−; B ¼ Γþðγ−δtÞ þ Γ−ðγþδtÞΓþ þ Γ−

: ðE4Þ

From the fitted values of Γ and B, we extract Γþ ≈ 3.6 Hzand Γ− ≈ 0.98 Hz. Note that in the future, the system ratescould be extracted more accurately using machine learning[52]. The rates would then be chosen by training the MLEor MAP readout to minimize the empirical error rate.

2. Verification of the readout model

Finally, we verify the validity of the theoretical readoutmodel in two complementary ways using the remainingtesting set and the rates extracted from the calibration set.We first compare the experimental log-likelihood ratiodistribution to the theoretical prediction. We then directlycalculate the error rate by preparing and subsequentlymeasuring the charge state in postprocessing.To calculate the experimental probability density of the

log-likelihood ratio,PðλÞ, we split the testing set into 78000subtrajectories of duration 25 ms. Using Eq. (8), wedetermine the value of the log-likelihood ratio λ at theend of each subtrajectory. We use these values to calculate ahistogram ofPðλÞ using a bin size of δλ ¼ 0.1. The resultingdistribution is shown in Fig. 11(a). We must compare theexperimental distribution to the theoretical prediction.We first calculate the expected conditional distributionPtheoðλj�Þ using the Monte Carlo simulations describedin Appendix C. We then calculate the full distribution

PtheoðλÞ ¼Xi¼þ;−

PtheoðλjiÞPtheoðiÞ: ðE5Þ

We assume that the prior probabilities are given by theirsteady-state values,

PtheoðþÞ ¼ 1 − Ptheoð−Þ ¼ Γ−Γþ þ Γ−

: ðE6Þ

The theoretical distribution is compared to the experimentaldistribution in Fig. 11(a). The prediction reproduces theexperimental data without any fitting parameters. Ourreadout model can therefore be used to prepare the chargestate with high fidelity in postselection.To directly obtain the experimental time dependence of

the error rate, we split the testing set into 39000 sub-trajectories of duration 50 ms. We prepare the state bycalculating the postmeasurement state ρ after the first 25 msof each subtrajectory, assuming a completely mixed initial

state ρ0 ¼ ð1=2; 1=2ÞT (see the procedure described at thebeginning of Appendix C). If ρþ > 1=2 (ρþ < 1=2), wedecide that the state is most likely jþi (j−i). In this way, weidentify 8307 (30693) subtrajectories as being in the statejþi (j−i) after the first 25 ms. We then use the MLE,adaptive MLE, and photon-counting methods to read outthe state using the last 25 ms of each subtrajectory. Theoutcome of the readout is compared to the preparation togive the measured conditional error rates ~ϵ�. The measurederror rate is then calculated as ~ϵ ¼ ð~ϵþ þ ~ϵ−Þ=2 and isplotted as a function of readout time in Fig. 11(b).Figure 11(b) shows that although the adaptive-decision

speedup is unchanged, the measured error rate ~ϵ is muchlarger than the theoretical prediction ϵ. This could be the

FIG. 11. (a) Experimentally measured probability density PðλÞof the log-likelihood ratio λ (black circles) for subtrajectories ofduration 25 ms. The theoretical prediction PtheoðλÞ (dotted black)and the corresponding weighted conditional distributionsPtheoðλjþÞPtheoðþÞ (solid blue) and Ptheoðλj−ÞPtheoð−Þ (dashedred) are shown for comparison. The theoretical prior probabilitiesPtheoð�Þ are taken to be the stationary values corresponding tothe extracted rates, Eq. (E6). (b) Experimental error rate ~ϵ of thephoton-counting method (blue circles), the nonadaptive MLE(orange squares), and the adaptive MLE (green triangles). Thetheoretical error rate ϵ of Fig. 5 is reproduced for comparison(solid blue for the photon-counting method, dashed orange for thenonadaptive MLE, and dotted green for the adaptive MLE). Thediscrepancy between experiment and theory can be attributed topreparation errors [see Eq. (E7)].

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result of either imperfect preparation or imperfect model-ing. We now show that this discrepancy is completelyconsistent with imperfect preparation. To account forpreparation errors, we introduce an average preparationerror rate η, assuming equal prior probabilities. The trueerror rate is related to the measured error rate by the relation~ϵ ¼ ηð1 − ϵÞ þ ð1 − ηÞϵ or

ϵ ¼ ~ϵ − η

1 − 2η: ðE7Þ

We simultaneously fit the experimental error-rate curves,using η as the only fit parameter. This gives a best-fit valueof η ¼ 2.22%. We find a good agreement between theoryand experiment, as shown in Fig. 5. We conclude that theadditional errors are likely due to preparation errors and notto imperfect modeling. We note that the transformation ofEq. (E7) does not affect the adaptive-decision speedup.

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