Optimal Signal Processing for Continuous Qubit Readout

Optimal post-processing for a generic single-shot qubit readout

W.A. Coish Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada

April 21, 2022

Abstract

We analyze three different post-processing methods applied to a single-shot qubit readout: the average-signal (boxcar filter), peak-signal, and maximum-likelihood methods. In contrast to previous work, we account for a stochastic turn-on time associated with the leading edge of a pulse signaling one of the qubit states. This model is relevant to spin-qubit readouts based on spin-to-charge conversion and would be generically reached in the limit of large signal-to-noise ratio for several other physical systems, including fluorescence-based readouts of ion-trap qubits and nitrogen-vacancy center spins. We derive analytical closed-form expressions for the conditional probability distributions associated with the peak-signal and boxcar filters. For the boxcar filter, we find an asymptotic scaling of the single-shot error rate when is stochastic, in contrast to the result for deterministic . Consequently, the peak-signal method outperforms the boxcar filter significantly when is stochastic, but is only marginally better for deterministic (a result that is consistent with the widespread use of the boxcar filter for fluorescence-based readouts and the peak-signal for spin-to-charge conversion). We generalize the theoretically optimal maximum-likelihood method to stochastic and show numerically that a stochastic turn-on time will always result in a larger single-shot error rate. Based on this observation, we propose a general strategy to improve the quality of single-shot readouts by forcing to be deterministic.

pacs:

03.67.Lx, 42.50.Lc, 03.65.Ta

I Introduction

A prerequisite for many quantum information processing applications is the ability to perform a strong projective single-shot measurement of a quantum bit (qubit) in the computational basis, Sohnet al. (1997). In general, the readout procedure depends on the particular measurement apparatus and physical system used to encode the qubit. However, a wide variety of high-fidelity single-shot readouts rely on the conditional amplification of one of the qubit states, say , by means of a cycling process [see Fig.1(a)]. For example, the state of a nitrogen-vacancy center (NV-center) in diamondNeumannet al. (2010); Robledoet al. (2011); Dréauet al. (2013), of trapped ionsMyersonet al. (2008), and of quantum-dot spinsVamivakaset al. (2010); Delteilet al. (2013); Puriet al. (2013) can be mapped to a fluorescence signal when the system is driven by a laser. Similarly, the spin state of electrons in semiconductor quantum dots or phosphorus donors in silicon can be mapped to a current through a nearby quantum point contact (QPC) or single-electron transistor (SET) via spin-to-charge conversionEngelet al. (2004); Elzermanet al. (2004); Morelloet al. (2010); Nowacket al. (2011); Simmonset al. (2011); Plaet al. (2013). Hybrid optical/electrical approaches to single-spin readout have also been demonstratedYinet al. (2013). Readouts of semiconductor singlet-triplet qubitsHansonet al. (2005); Johnsonet al. (2005); Pettaet al. (2006); Barthelet al. (2009); Pranceet al. (2012) and superconducting qubitsLinet al. (2013) also rely on similar amplification mechanisms. These cycling processes result in a time-dependent analog signal related to the number of cycles per unit time. If cycling is observed, it is inferred that the qubit state must have been ; otherwise, it is inferred that the qubit state must have been . Such readouts are quantum non-demolition (QND) since each cycle preserves the information associated with the initial qubit state. The fact that these readouts are QND is one reason they can reach high fidelities. To convert analog information associated with the noisy cycling signal , a post-processing procedure must be chosen to minimize the frequency of errors in the assignment of the binary state. Although higher-level and hardware-independent protocols can be used to minimize the uncertainty in quantum process tomographyMedfordet al. (2013); Merkelet al. (2013), an understanding of the dependence of the error rate on the underlying physical parameters is useful in further improving state reconstruction.

Interestingly, different communities have used different post-processing protocols to optimize the fidelity of their readouts, even though the various physical readouts are based on very similar cycling processes. In particular, fluorescence-based experiments have typically relied on integrating the signal over time to detect cycling (the so-called boxcar filter)Myersonet al. (2008); Neumannet al. (2010); Robledoet al. (2011); Dréauet al. (2013), although more sophisticated Bayesian inference procedures have also been usedMyersonet al. (2008); Sunet al. (2013). Spin-to-charge conversion experiments for semiconductor spin qubits, on the other hand, have typically been analyzed through a measurement of the peak of the signal (the peak-signal filter)Elzermanet al. (2004); Morelloet al. (2010); Simmonset al. (2011); Plaet al. (2013). In light of the striking similarities between these readouts, it is natural to ask whether the disparity in post-processing originates from a fundamental difference between the experiments. In fact, the only qualitative difference between the two cases is the mechanism triggering the cycling process. For spin-to-charge conversion, there is a random turn-on time after the beginning of the readout phase, where follows a Poisson process. In contrast, for fluorescence-based readouts cycling typically starts on a very short time scale as driving is turned on. Here we demonstrate that the uncertainty resulting from this stochastic turn-on time indeed accounts for the disparity in post-processing procedures, and we propose an avenue for increasing the fidelity of such readouts by making the turn-on time deterministic.

Quantum measurements based on cycling processes have been the subject of various theoretical studies. In particular, considerable attention has been given to the readout of a qubit using a non-QND cycling process (with back-action on the qubit) for several distinct physical systemsGurvitz (1997); Shnirman and Schon (1998); Makhlinet al. (2000); Korotkov (2001); Gurvitz and Berman (2005); Gilad and Gurvitz (2006); Jiaoet al. (2007); Kreisbecket al. (2010). Protocols for the optimal readout of a qubit using a QND cycling process have been studied in great detail in Ref.Gambettaet al. (2007) for the case , although the statistics for the peak signal were not derived in that work. In this paper we analytically obtain the statistics of the peak signal in the case of a stochastic turn-on time and Gaussian white noise. We demonstrate the validity of our approach by fitting our analytical probability distribution for the peak signal to that measured in the readout of single spin qubits in silicon in Ref.Morelloet al. (2010). Most importantly, we show that a significant improvement in fidelity is obtained by using the peak-signal filter over the boxcar filter if the turn-on time is stochastic. More precisely, we prove that for large signal-to-noise ratio , the boxcar-filter error rate scales like when is fixed while the error rate scales like when follows a Poisson process. The key observation is that the loss of information associated with a stochastic can be largely compensated by using a simple peak-signal filter instead of a boxcar filter. This result explains the disparity between post-processing methods used in different experiments and indicates which method should be used in future experiments. Furthermore, we generalize the optimal maximum-likelihood filter developed in Ref.Gambettaet al. (2007) and show numerically that the stochasticity of reduces the fidelity significantly even when this theoretically optimal procedure is followed. This result leads to the conclusion that physical readouts with stochastic turn-on time can be generically improved by engineering them such that becomes deterministic.

The remainder of the text is organized as follows. In Sec.II, we express the readout error rate in terms of the probability distributions of measured observables. In Sec.III, we introduce a model for the noisy cycling signal and formally define the peak-signal and boxcar filters. In Sec.IV, we analytically derive the conditional probability distributions for the observables derived from the peak-signal and boxcar filters. We then fit the peak-signal distributions to experimental data presented in Ref.Morelloet al. (2010). In Sec.V, we numerically obtain the error rate for the peak-signal and boxcar filters and analytically derive the scaling of the boxcar-filter error rate for large signal-to-noise ratio. Finally, in Sec.VI we generalize the maximum-likelihood filter of Ref.Gambettaet al. (2007) to the case of stochastic . We conclude in Sec.VII.

Figure 1: (Color online) (a) Schematic representation of a generic cycling process. This diagram represents, e.g., the fluorescence cycle of an NV-center or trapped ion, or the flow of electrons through an SET or QPC during spin-to-charge conversion. If the system is initially in the excited state (dashed blue line) at , it can trigger cycling between two cycling states (solid green line) at time . The cycling ends at time when the system falls into the ground state (dotted red line). If the system is initially in the state , cycling cannot occur because of either selection rules or energy conservation requirements, as indicated by the crosses. (b) Noisy time-dependent signal resulting from the cycling process of (a) when the initial state is . The readout phase starts at and acquisition starts after an arming time . Initially, cycling does not occur and the signal takes the average value . At a random time , cycling is triggered and the signal rises to . At a subsequent random time , cycling stops and the signal again drops to . Acquisition stops after a measurement time . Throughout, we assume that , that and follow Poisson processes and that the noise is white and Gaussian.

Ii Error rate

For the most general readout, the goal is to infer the initial qubit state from some observable . For example, could be the peak (obtained from the peak-signal filter) or the time average (obtained from the boxcar filter) of some analog signal . These quantities are defined in Eqs. (12) and (13), below. In Sec.VI, we will take to be the full measurement record (appropriate for the maximum-likelihood filter). To infer the state, we define the likelihood ratioTsang (2012):

where is the probability density of measuring the observable given the state and where the last equality is obtained using Bayes' theorem. If is greater than the threshold , the state is most likely ; otherwise, the state is most likely . For simplicity, we assume that the prior probabilities for the initial state are balanced, , in which case the threshold is . The average error rate is then given by:

where and are the error rates conditional on the initial qubit state. These expressions are valid for an arbitrary observable .

In the common case where the observable is a real scalar, as is the case for the peak-signal and boxcar filters, the threshold is equivalent to a threshold for , satisfying . The conditional error rates are then given byGambettaet al. (2007); Barthelet al. (2009); Tsang (2012):

and the fidelity is simply . For the maximum-likelihood filter, such simple thresholding is not possible since is a multi-dimensional object, namely the signal given at all times . In this case, the error rate (2) must be obtained from Monte-Carlo simulations (see Sec.VI)Gambettaet al. (2007).

Iii Model of the signal and noise

iii.1 Noisy cycling signal

We now model the time-dependent signal resulting from the cycling process of Fig.1(a). This could be, for example, a cycling fluorescence transition in NV-centers and ion traps, or the current flowing through an SET or QPC in the case of spin-to-charge conversion in semiconductor spin qubits. If the qubit is initially in the ground state , cycling does not occur. Thus, the average of over realizations of the noise is the same at all times . We choose the convention that (ensemble averages are indicated by angular brackets throughout):

If the qubit is initially in the excited state , cycling begins at a random turn-on time and ends at a random turn-off time . The stochasticity of and typically results from coupling the qubit states to a broadband continuum (the radiation field in the case of a fluorescence readoutMyersonet al. (2008); Neumannet al. (2010); Vamivakaset al. (2010); Robledoet al. (2011), or a Fermi sea of electronic states in the case of spin-to-charge conversionElzermanet al. (2004); Morelloet al. (2010); Simmonset al. (2011)), leading to a Markovian process, hence a Poissonian (exponential) distribution of and . As illustrated in Fig.1(b), the result is a noisy time-dependent signal such that:

Here, the turn-on time and pulse width each follows an independent Poisson process. Therefore, the probability distribution for and has the exponential form:

Here and throughout, time is measured in units of the average pulse width and is the ratio of to the average turn-on time . We recover the case of a deterministic turn-on time when . This is typically the relevant case for fluorescence-based readoutsMyersonet al. (2008); Neumannet al. (2010); Vamivakaset al. (2010); Robledoet al. (2011). As indicated in Fig.1(b), the stochastic turn-on time must be distinguished from a deterministic arming time Elzermanet al. (2004); Gambettaet al. (2007) during which the qubit may relax [see Fig.1(b)]. Indeed, the uncertainty in will affect the readout error rate even if the qubit relaxation time is infinite or . In the following analysis we will neglect qubit relaxation and take the arming time to be negligible, .

For simplicity, we also assume that is subject to Gaussian white noise, i.e. that the signal autocorrelation function is:

where . Here, is the (power) signal-to-noise ratio integrated over an interval :

The assumption of Gaussian noise is only valid when the number of cycling events is much larger than one, so that we can treat as a continuous variable. Furthermore, for simplicity we assume shot noise is negligible compared to other sources of stationary Gaussian white noise (e.g. due to amplifier electronics). In the opposite limit where the readout is limited by the shot-noise power, the error rate is simply given by the probability that no cycling event occursRobledoet al. (2011).

iii.2 Peak-signal and boxcar filters

We take the signal to be measured during a time [see Fig.1(b)]. In practice, each data point on such a trace is necessarily acquired over a finite bin time , corresponding to the inverse bandwidth of either a measurement apparatus or of a low-pass filter applied for post-processing. For simplicity, we assume that is separated in bins of length (see Fig.2). The bin, starting at time , is then assigned its time-averaged value:

With Gaussian white noise, Eq. (7), the probability distribution for in bin is:

where is the normal distribution of zero mean and of variance . Here, is the average of over realizations of the noise. It will also be useful to define the cumulative distribution function corresponding to :

We can now define the peak signal on as the maximum of over all bins:

The time-averaged signal , corresponding to the boxcar filter, is then recovered as a special case of the peak signal with :

Note that in the end, the error rate (2) must be optimized with respect to both the bin time and the measurement time , in addition to the threshold .

The form (5) of the signal suggests an alternative two-time boxcar filter of the form

Iv Statistics of the peak-signal and boxcar filters

To obtain the error rate for the peak-signal and boxcar filters, Eqs. (2) and (3), we must first determine the probability distributions and in the presence of a stochastic turn-on time . In order to extract a maximum of information associated with the qubit state, we need precise knowledge of these distributions. Indeed, the tails of the experimental distributions obtained for similar readoutsMyersonet al. (2008); Morelloet al. (2010); Dréauet al. (2013), which determine the error rates (3), often strongly deviate from simple Gaussian-like behavior. These distributions can be found numerically from a Monte Carlo analysis of this model Morelloet al. (2010). However, an analytical description is helpful in understanding the benefits of one post-processing scheme over another. Moreover, an analytical understanding of the statistics of the filters enables a fast extraction of the fidelity from the data, eliminating the need for time-consuming Monte-Carlo simulations. Therefore, in the following we derive exact analytical expressions for the peak-signal and boxcar distributions. Since the boxcar filter is a special case of the peak-signal filter, we first focus on obtaining .

iv.1 Probability distributions for a stochastic turn-on time

iv.1.1 Peak-signal distribution

As illustrated in Fig.2, the turn-on time and the turn-off time must each fall in a random bin of length  (see Sec.III.2). In the following, we assume that ( ) falls in the ( ) bin. Therefore, using Bayes' rule to account for all possibilities, we write the peak-signal distributions as:

where is the probability that and fall in bins and , respectively:

Figure 2: (Color online) Regions of the plane. The turn-on time (turn-off time ) falls in the ( ) discrete bin of length , where is the number of bins contained in the measurement time . Here, the ( ) bin of the ( ) axis starts at time ( ). The shaded region is forbidden since we must necessarily have . The finite measurement time divides the plane in three regions , Eq. (22). Each region gives a distinct, mutually-exclusive contribution to the peak-signal distribution .

In the last expression, the integrals are taken over the square labeled by in the plane (see Fig.2). Note that we allow for the possibility that and fall outside the measurement window ( and ). Likewise, is the probability distribution for conditional on and falling in a given square :

i.e. it is the average of the distribution over a square of the plane. Here, is the distribution (6) renormalized so that and lie in the cell . We proceed to evaluate expressions (15) and (16), which we then substitute into Eq. (14).

First, we obtain , Eq. (15), by direct integration of Eq. (6). Unsurprisingly, we find the discrete counterpart to the exponential form (6):

where we define the normalization constants:

Next, we derive the probability distributions , Eq. (16). Using the definition (12) of the peak signal and a combinatorial argument, we show in AppendixA that the peak-signal distributions for fixed and are given by:

where is the subset of bins in the measurement window having identical distributions and , Eqs. (10) and (11), with average signal . We note that .

To illustrate Eq. (19), first assume that the qubit state is . In this case, all bins have the same average signal . Thus, Eq. (19) contains a single term:

where , Eq. (10). Since the peak signal manifestly does not depend on and when the state is , the average (16) and the sum (14) are trivial and we obtain:

In the case where the initial qubit state is , the distribution (19) takes a different form in each of the regions of the plane depicted in Fig.2:

As an example, consider the case where and fall in region . If and fall in the same bin , bin has an average signal and the remaining bins have . Thus, in this particular case, Eq. (19) takes the form:

where , Eq. (10). Substituting Eq. (23) into Eq. (16), we obtain:

where

is the average probability distribution in a bin containing both and . Similarly, when and fall in different bins ( ), bin has and bin has . Of the remaining bins, there are with and with . Thus, we find:

where

is the average probability distribution in a bin containing only .

In AppendixB, we give similar expressions for in regions and [Eqs. (58) and (59)] as well as analytical expressions for the distributions , and and their respective cumulative distributions [Eqs. (61), (62) and (63)]. We then perform the sum (14) analytically and find that the probability distribution has a contribution from each region of Fig.2:

The contribution from region arises from events in which the entire pulse occurs outside the measurement window, causing additional errors that could not occur if the turn-on time were deterministic ( ). Explicit expressions for each term in Eq. (28) are given in Eqs. (66) and (67) of AppendixB.

In Fig.3, we fit the analytical expressions (21) and (28) to the experimental data from the spin-to-charge conversion readout of Ref.Morelloet al. (2010). We find the fitted values of the SET current and of the prior probabilities to be in good agreement with the measured values. The theoretical probability distributions provide a good fit to the data, allowing for a fast extraction of the readout error rate. Moreover, the importance of describing the probability distributions with precision is apparent from the non-Gaussian features of the distributions in Fig.3. Indeed, both the protuberance in the left tail of , which is masked by in an experiment, and the asymmetry in the distribution must be accurately described to obtain a genuine estimate of the error rates (3).

Figure 3: (Color online) Fit of the analytical distributions (dashed red line) and (solid blue line), Eqs. (21) and (28), to the experimentally determined distribution extracted from data in Fig. 4(b) of Ref.Morelloet al. (2010) (open circles). The dotted black line is the full peak-signal distribution . The peak SET current for the spin-to-charge conversion is mapped to the reduced peak signal via , where is the average SET current. We set and to their measured values and and we find fitted values of , , and for the current, signal-to-noise ratio, bin time and prior probabilities respectively. The values of and are in good agreement with the experimentally measured values of and . Furthermore, the bin time is of the same order of magnitude as the inverse bandwith of the low-pass filter used in Ref.Morelloet al. (2010), corresponding to . If the data were to be post-processed using the square binning defined in Eq. (9), our model could be used to obtain more accurate estimates of the bin time and signal-to-noise ratio.

iv.1.2 Boxcar-filter distribution

The boxcar-filter distributions are obtained from Eqs. (21), (28), (66) and (67) by setting ( ) and :

Here, and are given by Eqs. (61) and (62) with . The first term of is the contribution from the case where both and fall within the measurement window (region of Fig.2). The second term comes from the case where only falls within the measurement window (region of Fig.2). The last term is the contribution coming from the possibility of the pulse occurring outside the measurement window (region of Fig.2).

iv.2 Limit of a deterministic turn-on time ( )

iv.2.1 Peak-signal distribution

In this section we obtain analytical expressions for the distributions when the turn-on time is deterministic, . This is typically the relevant limit for fluorescence-based readouts Myersonet al. (2008); Neumannet al. (2010); Vamivakaset al. (2010); Robledoet al. (2011), where the cycling process starts almost immediately at the beginning of the readout phase ( ).

If the state is , the distribution (21) is independent of and the peak-signal distribution remains unchanged:

Next, we consider the case where the state is . When and , the normalization constants (18) simplify to:

Moreover, when the condition is satisfied, we asymptotically expand the error functions in Eq. (64) and obtain:

Physically, the condition corresponds to the requirement that the signal fluctuations on the time interval be larger than the signal itself, making it impossible to resolve the jump occurring at . Note that when , the condition is already implied by . From these considerations we see that, for any finite , the effects due to the stochasticity of become relevant at sufficiently large signal-to-noise ratio . Substituting Eqs. (31) and (32) into Eqs. (61), (62) and (63) and using and once again, we obtain:

where and are given by Eq. (64). With these simplifications, the expressions (67) for become:

where is given by Eq. (68). We see that in this limit, has no contribution from region since the pulse cannot fall outside the measurement window. Expressions for in the limit [Eqs. (30) and (34)] are plotted in Fig.4(d).

iv.2.2 Boxcar filter distribution

The boxcar filter for deterministic turn-on time, , is obtained as before by setting ( ) and in Eq. (34). We thus obtain the result derived in Ref.Gambettaet al. (2007):

where , Eq. (63), is given by its limiting expression (33) evaluated at . The first term of comes from the case where falls within the measurement window, , with probability (region of Fig.2). The second term comes from the case where falls outside the measurement window, , with probability (region of Fig.2). Note again that there is no contribution from region of Fig.2 since never falls outside the measurement window.

The conditional probability distributions given by Eq. (35) are shown in Fig.4(d) alongside the corresponding distributions for the peak-signal filter. There is relatively little qualitative difference between the distributions for the boxcar and peak-signal filters when [Fig.4(d)]. In contrast, it is clear that these two post-processing strategies generate very different conditional distributions when is finite [see Fig.4(b)].

V Error rate for the peak-signal and boxcar filters

Figure 4: (Color online) (a),(c) Optimized error rates, Eq. (2), as a function of the signal-to-noise ratio for the boxcar filter (solid blue line), peak-signal filter (dashed red line) and maximum-likelihood filter (dotted green line) for (a) (e.g. spin-to-charge conversion in semiconductor qubits) and (c) (e.g. fluorescence-based readouts in NV-centers or trapped ions). The error rates have been optimized with respect to the threshold , measurement time and bin time (when applicable). In the case , the error rate is significantly decreased for large by using the peak-signal filter instead of the boxcar filter, whereas the advantage is much smaller in the case . The error rate for the maximum-likelihood filter (dotted green line), obtained from the Monte-Carlo solution of Eqs. (75) and (77), is the lowest theoretically achievable error rate. The fluctuations in the maximum-likelihood error rates are due to the finite sample size ( ) of the Monte-Carlo simulation. (b),(d) Optimized probability distributions (dashed red line), (solid blue line), (dotted red line) and (dot-dashed blue line) for in the cases (b) , Eqs. (21), (28) and (29), and (d) , Eqs. (30), (34) and (35). For , the weight of the distribution for is visibly shifted to the right by using the peak-signal filter compared to the boxcar filter, decreasing the error rate significantly. When , the probability distributions for the peak-signal and boxcar filters are qualitatively the same in both cases and no advantage is gained. The dotted black vertical lines indicate the optimal threshold for each case, satisfying .

We are now in a position to compute the average error rate for the peak-signal and boxcar filters. We first numerically integrate the analytical probability distributions derived in Sec.IV to obtain the conditional error rates (3). We then numerically minimize , Eq. (2), with respect to the measurement time , bin time and threshold .

The optimized error rate is plotted as a function of the signal-to-noise ratio in Fig.4(a) for the case of a stochastic turn-on time ( ) ^{1 1} 1In addition to matching the experimental value found in Ref.Morelloet al. (2010), is a natural choice for a spin-to-charge conversion readout – see the discussion in the final paragraph of Sec.VI and in Fig.4(c) for the case of a deterministic turn-on time ( ). The advantage gained by measuring the peak signal instead of the time-averaged signal (employing the boxcar filter) is significant when , but only marginal when . For example, when , using the peak-signal instead of the boxcar filter increases the fidelity from to for , whereas when , the fidelity only increases from to for the same signal-to-noise ratio. The qualitative difference between the two cases is apparent from the corresponding optimized probability distributions plotted in Figs.4(b) and 4(d) for a signal-to-noise ratio of . In the case , the weight of the probability distribution for is shifted to higher values of by using the peak signal filter over the boxcar filter, whereas in the case , the peak-signal and boxcar distributions are qualitatively very similar.

We can better understand why this occurs by studying the asymptotic behavior of the error rate for the boxcar filter at large signal-to-noise ratio. Expression (29) for the boxcar filter probability distributions can be integrated analytically to give the unoptimized error rate:

The optimal threshold is given by the condition . Thus, according to Eq. (29), is the solution of:

When the turn-on time is stochastic, there is a finite lower bound on the optimal measurement time . Indeed, in such a case there is a finite probability that falls outside the measurement window. Therefore, we must necessarily choose an optimal measurement time to minimize the possibility of completely missing the pulse. This implies that as the signal-to-noise ratio increases, , the typical width of the distribution on the right-hand side of Eq. (37) goes as while the distribution on the left-hand side remains delocalized [see Eqs. (61) and (63)]. Thus, the solution of Eq. (37) must be such that the optimal threshold approaches . Therefore, we expand the condition (37) asymptotically in the limit and and find:

where:

Next, we expand Eq. (36) in the same limit and use Eq. (38) to obtain :

Since the first term decreases exponentially with and the second term increases polynomially with , the optimal measurement time must diverge logarithmically when , . Thus, we use and optimize Eq. (40) with respect to when . We find the following leading logarithmic asymptotic form for the error rate:

This result is to be compared to the case of a deterministic turn-on time . In Ref.Gambettaet al. (2007), it was shown that in this case, the average error rate for the boxcar filter scales instead as

This qualitative difference in scaling arises from the fact that when , the optimal measurement time approaches when . This ensures that the turn-off time falls outside the measurement window, , and thus that when the state is [see Fig.4(d)]. As we already argued, this is not possible in the presence of a stochastic turn-on time since the optimal measurement time must always be such that in order to avoid the possibility of missing the pulse. Since the pulse can occur anywhere in the measurement window, we have when the state is , increasing the error rate [see Fig.4(b)]. We have numerically verified that the two-time boxcar filter described in Sec.III.2 suffers from the same limitation. The peak-signal filter partly overcomes this shortcoming by gaining additional information on the location of the pulse within the measurement window, moving the average of the distribution back to [see Fig.4(b)].

To better illustrate this effect, we plot the numerically optimized measurement time and number of bins for the peak-signal filter as a function of signal-to-noise ratio in Fig.5, for both and . When , it becomes advantageous to increase the number of bins as increases since the measurement time remains finite and the location of the pulse is unknown. When , the advantage gained by binning is not significant since the measurement time can become arbitrarily small. These results suggest an explanation for why the peak-signal filter is typically used for spin-to-charge conversion readouts using semiconductor qubits (having a stochastic turn-on time)Elzermanet al. (2004); Morelloet al. (2010); Simmonset al. (2011), whereas fluorescence-based readouts (having a deterministic turn-on time) typically rely on the simple boxcar filterMyersonet al. (2008); Neumannet al. (2010); Vamivakaset al. (2010); Robledoet al. (2011). More importantly, we emphasize that there is a crossover from ( ) to ( ) as increases (see Sec.IV.2). Thus, for any readout with finite it will become necessary to use the peak-signal filter instead of the boxcar filter as the signal-to-noise ratio improves.

Figure 5: (Color online) (a) Optimal measurement time as a function of the signal-to-noise ratio for the peak-signal filter in the cases (purple solid line) and (purple dashed line). When , the measurement time diverges logarithmically with [see the discussion following Eq. (40)], whereas when , approaches as increases. (b) Optimal number number of bins as a function of the signal-to-noise ratio for the peak-signal filter in the cases (purple solid line) and (purple dashed line). Although we have derived our model for , we treat as a continuous variable for numerical optimization: fractional values of must be seen as an interpolation between integer values. Because must remain finite when , it becomes advantageous to increase the number of bins in order to locate the pulse within the measurement window. When ( ), can approach as increases, eliminating the need for binning.

Vi Maximum likelihood filter

In the previous sections, we have shown that for readouts relying on a cycling process (e.g. spin-to-charge conversion in semiconductor qubits), the presence of a stochastic turn-on time for the cycling can significantly decrease the fidelity when simple filters are used. In this section, we generalize the maximum-likelihood filter developed in Ref.Gambettaet al. (2007) for a deterministic turn-on time to the case of a stochastic turn-on time. We show that even for this theoretically optimal Bayesian inference procedure, the fidelity of the readout can be significantly degraded by the uncertainty in the turn-on time.

The maximum-likelihood filter uses all the information contained in a given measurement record to infer the state of the qubit. The likelihood ratio, Eq. (1), now takes the form:

When the qubit state is , the average signal is , Eq. (4), so that the probability distribution for is:

where is a normalization constant. When the qubit state is , the average signal for fixed and , Eq. (5), is , so that the probability distribution for is, using Bayes' rule:

Using these expressions, the likelihood ratio (43) can be rewritten as:

The integral has a contribution from each domain illustrated in Fig.2:

where:

In principle, we can evaluate these integrals numerically to obtain given a particular measurement record . If , we declare the state to be and if , we declare the state to be . If we choose randomly by throwing an unbiased coin. However, we can avoid the triple integrals and the potentially large numerical values of in Eq. (48) by using an equivalent set of stochastic differential equationsRisken (1984); Gambettaet al. (2007) for the estimator (see Appendix C). If , we infer that the state is and if , we infer that the state is .

Although expressions for the case of deterministic turn-on time have already been given in Ref.Gambettaet al. (2007), we reproduce them here in our notation for completeness. Taking the limit in Eq. (48), we find that the likelihood ratio only has contributions from regions and in Fig.2:

where:

A set of stochastic differential equations equivalent to these integrals is also given in Appendix C.

To obtain the error rate of the maximum-likelihood filter, we generate random records by randomly choosing the initial state with equal probability. For each record, we solve the stochastic differential equations (75) and (77) using a standard fourth-order Runge-Kutta methodPresset al. (2002) to obtain the estimator . Typical solutions for as a function of are illustrated in Fig.6. We see that the estimator reaches a constant value as increases. Thus, it is sufficient to choose a sufficiently large measurement time, to obtain the optimal error rateGambettaet al. (2007). The average error rate is then given by the fraction of records that are misidentified by the estimator. We plot as a function of the signal-to-noise ratio in Fig.4(a) for and in Fig.4(b) for . The readout error rate is substantially larger at large when compared to . Quantitatively, we find that to achieve an error rate in the case , it is necessary to have a signal-to-noise ratio of . To achieve the same error rate when , it is sufficient to have a signal-to-noise ratio . Thus, we conclude that even for this optimal post-processing procedure, the additional uncertainty in can significantly degrade the single-shot fidelity of the readout.

Figure 6: (Color online) Estimator as a function of the measurement time for , obtained for a randomly generated record in the cases (solid blue line) and (dashed purple line). When , initially slowly decreases in the interval and suddenly jumps to when the pulse occurs. When , and immediatley jumps to . In both cases the estimator reaches a constant value at large when all the available information on the pulse has been acquired.

An important consequence of this result is that it should always be possible to increase the fidelity of spin-to-charge conversion readouts by making the turn-on time deterministic in the sense of Sec.IV.2. For example, suppose (similar to the experiment of Ref.Morelloet al. (2010)) that an electron spin qubit in a localized orbital is coupled (with tunneling rate ) to each of nearly degenerate orbital states in a neighboring empty SET, initially in the Coulomb-blockade regime. The electron then tunnels from the excited spin state onto the SET at a rate , after which current can flow through the SET in the sequential tunneling regime, and the SET occupation fluctuates between 1 and 0 electrons. An electron will tunnel back to the spin-qubit ground state at a -independent rate assuming the average SET occupation is ^{2 2} 2 is realized when the tunneling rate to the drain lead ( ) is balanced with the tunneling rate from the source lead ( ), i.e. . This choice has the added benefit of minimizing the contribution of shot noise to the signal-to-noise ratio. and that the SET is occupied with spin-up and spin-down electrons with equal probability. The ratio of time scales setting is then . The choice corresponds, in this case, to a single non-degenerate level of the unoccupied SET ( ). Indeed, this happens to be the value found experimentally for the readout of Ref.Morelloet al. (2010). However, the degeneracy could be any value, in principle. The readout fidelity could be improved by increasing such that , entering the regime where the asymptotic form, , of Fig. 4(c) applies. In the typical case where the noise in an individual bin is small compared to the signal, , we find the very simple condition on the degeneracy and signal-to-noise ratio :

Nanostructures with a large density of single-particle states (e.g. a one-dimensional nanowire with a singularity in the density of states), could be used to realize the limit given in Eq. (52), even in the limit of large signal-to-noise ratio .

Vii Conclusions

In conclusion, we have shown that the fidelity of readouts relying on a QND cycling process with a stochastic turn-on time can be significantly increased by measuring the peak of the cycling signal (peak-signal filter) instead of its time average (boxcar filter). The origin of this discrepancy is that the peak-signal filter, by increasing the number of bins in the measurement window, can acquire additional information on the time at which the cycling signal occurs. Our results may explain why spin-to-charge conversion experiments in semiconductor qubits have typically used the peak-signal filter, whereas fluorescence-based readouts have normally relied on the simpler boxcar filter. Moreover, we predict that for any system with a stochastic turn-on time, however small, it will become advantageous to employ the peak-signal filter rather than the boxcar filter when the signal-to-noise ratio becomes larger than the dimensionless inverse average turn-on time ( ). Furthermore, we have generalized the maximum-likelihood filter developed in Ref.Gambettaet al. (2007) to the case of a stochastic turn-on time. We have shown that even when this theoretically optimal procedure is followed, the presence of a stochastic turn-on time can significantly reduce the fidelity of the readout. Thus, we propose that the fidelity of such cycling readouts may be increased by making the turn-on time deterministic. In the case of a semiconductor qubit coupled to a nearby SET, this could be achieved by engineering the density of single-particle states for the SET to enhance tunneling from the qubit to the SET. It should be possible, in principle, to extend our approach to include shot-noise and non-Gaussian types of noise relevant in experiments with a small number of cycling events (e.g. in the presence of dark counts and near-Poissonian noise in ion-trap experimentsMyersonet al. (2008)).

Acknowledgments

The authors are indebted to J.P. Dehollain and A. Morello for sharing their data and experimental insight and to L. Childress for useful discussions. We acknowledge financial support from NSERC, CIFAR, FRQNT, and INTRIQ.

Appendix A Derivation of the general form of the peak-signal distribution

In this appendix, we derive the general form of the peak-signal distribution , Eq. (19). For fixed turn-on time and turn-off time , the probability of a given peak signal is the probability that at least one of the bins has while the remainder have :

where runs over the subsets of bins in the measurement window having identical distributions and , Eqs. (10) and (11). The total number of bins is . In Eq. (53), we introduced the binomial form:

which gives the probability that bins in the subset are such that and that bins are such that . Next, we perform the binomial sum in Eq. (53) and find:

In the continuum limit for , we have . Thus, we can expand the first term of Eq. (55) to linear order to obtain the desired result, Eq. (19):

Appendix B Analytical expressions for

In this appendix, we derive an explicit analytical expression for the distribution , Eq. (28). Following the reasoning of Sec.IV.1.1, we assume that the turn-on time and the turn-off time fall in the and time bin, respectively. We then write out Eq. (56) for and falling in each region of the plane depicted in Fig.2 and perform the average (16) over in bin and in bin .

For in region , we find Eqs. (23) and (24):

where . For in region , we find:

Finally, for in region , we find:

Here, is the average probability distribution in a bin that contains both and and is the average probability distribution in a bin that contains only :

Above, the integrals are taken over the square in the plane, Fig.2. In Eq. (60), , and are the Gaussian distributions (10) with average signal , and , respectively.

Using Eqs. (6) and (10) to perform the integrals (60), we find that, in a bin that contains both and ( ), the average probability distribution and its cumulative function are:

In a bin that contains only the turn-on time , we find the average distributions:

and in a bin that contains only the turn-off time , they are:

Here, we introduce the functions:

where and , Eq. (10).

Next, we substitute Eqs. (17), (57), (58) and (59) into Eq. (14) to obtain . We find that has a contribution from each region :

where the contribution from region has distinct contributions from and :

We perform the sum (14) directly and obtain an analytical form for :

Eq. (65), together with Eqs. (66) and (67), is the central result of this appendix. Here, we have introduced the functions:

where the coefficients are given by:

and:

The functions and are geometric sums arising from performing the sum (14):

Appendix C Stochastic differential equations for the maximum-likelihood filter

In this appendix, we derive stochastic differential equations for the estimator plotted in Fig.6. We first derive equations for the likelihood ratio and then reexpress them in terms of .

We directly differentiate each member of Eq. (48) with respect to to obtain a set of linear differential equations for :

From Eq. (48), we see that these equations must be solved subject to the initial conditions , and .

Using Eq. (1), we may express the estimator in terms of the likelihood ratio (47) as:

Defining , we write the estimator as:

In terms of the 's, the equations (72) transform into a set of non-linear, first-order differential equations:

These equations must be solve with the initial conditions , and . Note that when , for all , and no information can be acquired on the qubit state.

When , we follow a similar procedure and find that the likelihood ratio is the solution of the following pair of equationsGambettaet al. (2007):

with the initial conditions and . Similarly, the estimator is the solution of:

with the initial conditions and .

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