Soft Decoding of a Qubit Readout Apparatus

# Soft Decoding of a Qubit Readout Apparatus

B. D’Anjou Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada    W.A. Coish Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
August 3, 2019
###### Abstract

Qubit readout is commonly performed by thresholding a collection of analog detector signals to obtain a sequence of single-shot bit values. The intrinsic irreversibility of the mapping from analog to digital signals discards soft information associated with an a posteriori confidence that can be assigned to each bit value when a detector is well characterized. Accounting for soft information, we show significant improvements in enhanced state detection with the quantum repetition code as well as quantum state or parameter estimation. These advantages persist in spite of non-Gaussian features of realistic readout models, experimentally relevant small numbers of qubits, and finite encoding errors. These results show useful and achievable advantages for a wide range of current experiments on quantum state tomography, parameter estimation, and qubit readout.

###### pacs:
03.65.Ta,03.67.Ac,03.65.Wj

In most quantum-measurement tasks, the goal is to extract information encoded in a stream of quantum states. For a single-qubit expectation value, the stream is a collection of identically prepared single qubits. To perform a Bell-inequality measurement, the stream is a collection of entangled qubit pairs. For quantum error detection or correction, the stream consists of many qubits that make up the code, on which multiqubit syndrome measurements are performed. In practice, in all of these scenarios, information is commonly extracted by measuring individual qubits or joint observables in a single shot Häffner et al. (2005); Schindler et al. (2011); Bernien et al. (2013); Chow et al. (2011); Reed et al. (2012); Chow et al. (2013); Saira et al. (2014); Barends et al. (2014). While this strategy can be optimal for extracting a single bit of information, e.g., the state of a single qubit, it is generally suboptimal when considering streams of data. A single-shot qubit readout typically involves the irreversible conversion of an analog outcome from a readout apparatus (e.g., a current or voltage pulse, the quadrature of a microwave tone, etc.) into a binary outcome via thresholding Myerson et al. (2008); Barthel et al. (2009); Morello et al. (2010); Neumann et al. (2010); Robledo et al. (2011); Pla et al. (2013); Lin et al. (2013); Liu et al. (2014); Harty et al. (2014) (see Fig. 1). Thresholding erases information about the posterior probability that can be ascribed to each bit value given . In contrast, when a sequence of analog readout outcomes is fed into a decoder that accepts analog values as input, the frequency of decoding errors can be significantly reduced Chase (1972). Such soft-decoding techniques have been central to the development of capacity-achieving classical codes Guizzo (2004) now used in deep-space communications and high-bandwidth 3G/4G cellular networks. Soft-decision decoding has been applied to quantum codes Poulin (2006); Duclos-Cianci and Poulin (2010) and to schemes for fault-tolerant quantum computing Goto and Uchikawa (2013), in which the soft decision is made by correlating multiple single-shot qubit readout outcomes. Soft decoding has also been identified as an important tool for continuous-variable quantum key distribution Mondin et al. (2010).

In this Letter, we explicitly demonstrate the advantages of soft decoding through two experimentally significant examples: enhanced state detection via the quantum repetition code, and state or parameter estimation. In particular, the number of qubits required for efficient enhanced state detection can be reduced, through data processing alone, by up to a factor of , an advantage which persists for a small number of qubits and finite encoding errors. Because additional qubits are an expensive resource, this result is of immediate practical importance. We also extend a result of Ref. Ryan et al. (2013) by making use of soft information to further improve the precision of measurements of Pauli operators required for state tomography. In both cases, we benchmark the improvement by comparing the performance of the widely used and efficient maximum-likelihood estimation Cramér (1946) when applied to analog instead of thresholded qubit readout outcomes. Crucially, we find and characterize significant improvements not only for the idealized Gaussian readout, but also for the realistic non-Gaussian readout investigated in Ref. D’Anjou and Coish (2014) and relevant to many experiments Elzerman et al. (2004); Morello et al. (2010); Pla et al. (2013); Veldhorst et al. (2014).

Enhanced state detection. In the quantum repetition code, a logical qubit (with basis ) in the state is encoded into physical qubits (with basis ):  Deuar and Munro (1999); DiVincenzo (2001); Schaetz et al. (2005). The redundant outcomes of independent measurements of the physical qubits are then correlated to reduce measurement errors. The simplest approach is to measure each qubit in a single shot and assign the binary value to the th qubit. When the encoded states and are equally likely a priori, the optimal approach is to calculate the likelihood ratio Wozencraft and Jacobs (1965) of the data :

 Λc≡P({ci}|1)P({ci}|0)=N∏i=1P(ci|+)P(ci|−). (1)

In Eq. (1), is the probability to obtain the value given that the th qubit is in the state . The projected state of the logical qubit is most likely () if (). We may rewrite the likelihood ratio, Eq. (1), in terms of the conditional single-shot error rates :

 Λc=(1−ϵ+ϵ−)n+⋅(ϵ+1−ϵ−)N−n+, (2)

where is the number of measurements for which the outcome occurred. For a binary symmetric readout, , Eq. (2) results in a simple majority vote since () implies ().

Equation (2) is a maximum-likelihood estimator applied to single-shot readout outcomes. However, a physical readout apparatus typically yields an analog readout outcome that need not be thresholded to a binary value . The observable could be, for example, the time average of a fluorescence signal Schaetz et al. (2005); Myerson et al. (2008); Robledo et al. (2011), the peak of a current pulse through a single-electron transistor or quantum point contact Elzerman et al. (2004); Morello et al. (2010); Veldhorst et al. (2014), the quadrature of a microwave tone Lin et al. (2013); Liu et al. (2014); Jeffrey et al. (2014), or even the likelihood ratio of a single-shot readout Gambetta et al. (2007); Hume et al. (2007); Myerson et al. (2008); Harty et al. (2014); D’Anjou and Coish (2014). Thresholding leads to an irreversible loss of information about the confidence in each bit value. Soft decoding, which makes full use of that information, is achieved by instead applying the maximum-likelihood estimator to the analog readout outcomes:

 ΛO≡P({Oi}|1)P({Oi}|0)=N∏i=1P(Oi|+)P(Oi|−), (3)

where is the probability density for outcome given that the th qubit is in the state .

To take full advantage of soft decoding, it is necessary to have an accurate representation of the conditional probability distributions for the analog qubit readout outcomes. A common idealization for a readout is the Gaussian readout, , where is the power signal-to-noise ratio. Soft decoding of the Gaussian readout with maximum-likelihood estimators such as Eqs. (1) and (3) has been extensively studied in the context of classical communication theory Wozencraft and Jacobs (1965); Chase (1972). Since a projective measurement collapses to either or , the advantage obtained by soft decoding of the readout apparatus translates directly to the quantum case. More precisely, for , the number of qubits and required to achieve a target error rate using and , respectively, are related (for odd) by sup ():

 NO=Nc+12+Nc−12lnrr+O(Ncr). (4)

Thus, soft decoding can reduce the number of required physical qubits by up to a factor of compared to the majority vote (asymptotically, , or alternatively, for fixed , up to a logarithmic prefactor in ). Intuitively, this advantage arises since the majority vote ignores all information contained in strings for which more than half of the bits are corrupted, while soft decoding associates every string with some confidence. A similar asymptotic advantage exists for arbitrary block codes transmitted through a Gaussian communication channel Chase (1972). Importantly, Eq. (4) is valid for the regime of reasonably small relevant to recent experiments Monz et al. (2011); Schindler et al. (2011); Reed et al. (2012); Chow et al. (2013); Saira et al. (2014); Barends et al. (2014); Goodwin et al. (2014). Moreover, the form of the subleading corrections in Eq. (4) suggests that they can be small for realistic experimental values of . We have indeed verified, using the exact analytical expressions for the error rates sup (), that an advantage persists for low signal-to-noise ratio and relatively small . For example, we require instead of to reach an error rate for .

Realistic qubit readouts are typically not well represented by Gaussian probability distributions Gambetta et al. (2007); Myerson et al. (2008); Morello et al. (2010); Robledo et al. (2011); D’Anjou and Coish (2014). To verify that soft decoding of the readout apparatus still provides an advantage in experimentally relevant cases, we apply the estimators in Eqs. (1) and (3) to the realistic non-Gaussian “peak-signal” readout implemented in Refs. Elzerman et al. (2004); Morello et al. (2010) and for which the distributions were analyzed in Ref. D’Anjou and Coish (2014). In this measurement, the analog outcome is the peak value of a finite-duration current pulse signalling the excited state and subject to Gaussian white noise (see Ref. sup () for a summary). A typical pair of distributions for this readout is illustrated in Fig. 1. Even though the distribution has strong non-Gaussian features, soft decoding still gives an appreciable advantage. For example, Monte Carlo simulations with random records show that, similar to the Gaussian readout, we require instead of to reach an error rate for a signal-to-noise ratio sup ().

To account for errors during encoding, we allow for uncorrelated bit flips with probability for both states . The probability distributions for the analog readout outcomes then become , giving a modified version of the likelihood ratio, Eq. (3). We find that when encoding errors are sufficiently large, the soft-decoding procedure reduces to a simple thresholding procedure sup (). However, for the Gaussian readout, soft decoding can still give an advantage over thresholding if

 η≲e−2r (5)

when . Thus, the encoding bit-flip rate must merely be smaller than some power of the single-shot readout error rate ( up to logarithmic corrections). To verify this, we have performed a Monte Carlo simulation of the error rate for the Gaussian readout by generating random measurement records taking into account the bit-flip rate sup (). For example, we find that for and , we require instead of qubits to achieve . Similarly, for the peak-signal readout described in Ref. D’Anjou and Coish (2014), we find from a simulation of random measurement records that for a signal-to-noise ratio of and , we require instead of qubits to achieve sup ().

State and parameter estimation. Many quantum information processing applications, such as state and process tomography Häffner et al. (2005); Chow et al. (2011); Merkel et al. (2013); Medford et al. (2013); Ryan et al. (2013) and parameter estimation Shulman et al. (2014), benefit from accurate and precise estimation of qubit observables (e.g., the Pauli operators). Analog data processing has been used extensively, e.g., for parameter Wiseman and Killip (1997) and state Banaszek et al. (1999) estimation in quantum optical systems, where it is often natural to process quasicontinuous field quadratures or photon counts. For many qubit systems, the common approach is instead to threshold the data. Thresholding the data is generally suboptimal, as we now illustrate.

For definiteness, we consider estimating the quantum expectation value of the single-qubit Pauli operator (in the basis ) from the independent readout of identically prepared copies of a qubit. As in the case of the repetition code, we compare the standard maximum-likelihood estimator (MLE) Cramér (1946) applied to the analog data set instead of the thresholded data set in order to benchmark the improvement. In both cases, the MLE is the value that maximizes the likelihood function under the constraint . In practice, the MLE is obtained by maximizing the equivalent log-likelihood function . The MLE is asymptotically unbiased, normally distributed, and minimizes the variance [i.e. saturates the Cramér-Rao bound, see Eq. (7), below] for large  Cramér (1946).

When the analog data are thresholded, the MLE is the (bias-corrected) thresholded average considered, e.g., in Ref. Ryan et al. (2013). This estimate does not make use of the soft information contained in the distributions for reconstruction of . In contrast, the soft-decoded estimate obtained by applying the MLE to the analog data set makes full use of the distributions . In Ref. Ryan et al. (2013), the alternative soft average was also employed as an estimator for , but this approach is also suboptimal sup ().

We will measure the deviation of an estimate from the true value with the mean squared error (MSE) , given by the sum of the variance and of the squared bias of the estimator, . Here, the statistical average is taken with respect to the distribution of outcomes:

 P(O/c|s0)=1+s02P(O/c|+)+1−s02P(O/c|−). (6)

For this distribution, is a concave function with a unique maximum. A general expression for the asymptotic MSE of the thresholded average can be derived sup (). For the Gaussian readout, it takes the simple form reported in Ref. Ryan et al. (2013), . For the soft-decoded MLE estimate, the asymptotic MSE can be computed directly from the Fisher information of :

 ζSD∼−1N⟨⟨∂2lnP(O|s0)∂s20⟩⟩−1. (7)

Here, we use the symbol “” to indicate a strict asymptotic equality. From Eqs. (6) and (7), an explicit asymptotic form for can be found in terms of the distributions sup ():

 ζSD∼1N⋅1−s201−I,I=∫dOP(O|+)P(O|−)P(O|s0). (8)

In Eq. (8), the integral contains all information about the noise introduced by the readout apparatus. The remaining contribution when is the quantum shot noise (projection noise), which reflects the choice of a particular measurement basis. Since has the form of an overlap integral, it is especially important to understand the tails of the (generally non-Gaussian) readout distributions .

To quantitatively verify that soft decoding can improve state estimation, we set and evaluate Eq. (8) numerically for both the Gaussian readout and the peak-signal readout of Ref. D’Anjou and Coish (2014). We plot the asymptotic MSE as a function of the signal-to-noise ratio in Fig. 2. We also plot the asymptotic MSE of the bias-corrected thresholded average, . Figure 2 confirms that soft decoding always outperforms thresholding (i.e., ). As increases, and exhibit an approximate power-law approach to the projection-noise limit for the (non-Gaussian) peak-signal readout of Ref. D’Anjou and Coish (2014), whereas they decrease exponentially for the Gaussian readout. In the intermediate regime for , there is a clear advantage in soft decoding with the MLE for both readouts, demonstrating substantial benefits in the experimentally relevant regime of signal-to-noise ratios, for the Gaussian readout Ryan et al. (2013), and for the peak-signal readout.

To show that the asymptotic advantage persists when is finite, we calculate from a Monte Carlo simulation with . We first randomly generate measurement records from the distribution for both the Gaussian readout and non-Gaussian peak-signal readout of Ref. D’Anjou and Coish (2014). For each measurement record, we calculate and optimize the log-likelihood function to obtain . We then directly obtain from the variance and bias of the simulated estimates. The results are displayed (open symbols) in Fig. 2 for . The simulated data points coincide with the asymptotic predictions.

In conclusion, we have shown that making use of the soft information contained in the analog outcomes of a qubit readout apparatus, as opposed to irreversibly thresholding each qubit to a binary value, can significantly improve the performance of quantum information processing tasks involving the measurement of many qubits. We have focused on two examples of practical importance. In the case of enhanced state detection with the quantum repetition code, the number of qubits required to achieve a given error rate can be reduced by up to a factor of through improved data processing alone. Importantly, we have shown that an advantage persists for small numbers of qubits and finite encoding errors. In addition, we have shown that optimal processing of analog qubit readout outcomes can appreciably increase the precision on the measurement of qubit observables (e.g., the Pauli operators). Crucially, in both cases we have demonstrated a significant improvement for an experimentally relevant, non-Gaussian qubit readout model Elzerman et al. (2004); Morello et al. (2010); Pla et al. (2013); Veldhorst et al. (2014); D’Anjou and Coish (2014).

Our results offer encouraging prospects for direct improvements of both small and large scale quantum information processing applications through soft decoding of the qubit readout. For example, readout error models for decoders of topological codes Duclos-Cianci and Poulin (2010); Fowler et al. (2012) could be modified using the ideas presented here to accept analog readout outcomes with realistic statistics at the single-physical-qubit level, improving error detection rates. While there are many possible extensions of this work, the direct improvements we have shown to enhanced state detection and quantum state or parameter estimation are both practical and immediately realizable in a wide array of current experiments.

We thank L. Childress, A. Fowler, and D. Poulin for useful discussions. We acknowledge financial support from the National Sciences and Engineering Research Council of Canada (NSERC), the Canadian Institute for Advanced Research (CIFAR), the Fonds the Recherche du Québec Nature et Technologies (FRQNT), and the Institut Transdisciplinaire d’Information Quantique (INTRIQ).

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## Supplemental Material

To show that soft decoding of a readout leads to an advantage for realistic readouts, we have investigated the performance of soft decoding for the experimentally relevant Elzerman et al. (2004); Morello et al. (2010); Veldhorst et al. (2014) ‘peak-signal’ readout analyzed in Ref. D’Anjou and Coish (2014). In this section, we briefly summarize this readout for completeness.

In this readout, the ground and excited states and are mapped to a time-dependent signal subject to Gaussian white noise. When the state is , the signal has a constant value on average. When the state is , however, the signal is a square pulse starting at a random turn-on time and ending at a random turn-off time , as illustrated in Fig. S.1. The times and each follow their own Poisson statistics. The measurement time is divided into bins of length , with the average signal on the bin being . The observable is then chosen to be the maximum of over all bins. Finally, the measurement time , bin time and threshold are chosen to optimize the single-shot readout fidelity. Other choices for the observable have also been discussed in Refs. Gambetta et al. (2007); D’Anjou and Coish (2014).

Typical probability distributions for the peak-signal readout are shown in Fig. S.2 for two different values of the power signal-to-noise ratio (integrated over a time ). We see that the distributions have prominent non-Gaussian features. To perform fast sampling of these distributions, we first cut off the tails of each distribution (the lost probability weight is smaller than about ) and renormalize them. Next, we numerically integrate the analytic expressions for given in Ref. D’Anjou and Coish (2014) to construct a linear interpolation of the inverse cumulative distribution function associated with , as shown in Fig. S.2. An independent sample of is then given by , where is a random number generated from a uniform distribution between and  Rubinstein and Kroese (2008).

## Appendix B Enhanced state detection with the quantum repetition code

### b.1 Advantage of soft decoding

In this section, we give a brief derivation of the advantage obtained for enhanced state detection through soft decoding of the quantum repetition code for a Gaussian readout, Eq. (4) of the main text. Note that to benefit from enhanced state detection, coherent encoding of the logical state into the state is not necessary. For example, encoding into the mixed state (e.g. by allowing the qubits to purely dephase) gives the same advantage.

For the Gaussian readout, each qubit measurement yields an analog value conditioned on the qubit state according to the probability distributions:

 P(O|±)=√r2πe−(O∓1)2r2. (S.1)

Here, is the power signal-to-noise ratio and the signal is normalized to have average values . If the qubit is read out in a single shot, each analog outcome is converted to a binary outcome by setting a threshold . The single-shot error rates conditional on the qubit state are:

 ϵ−=P(c+|−)=∫∞νdOP(O|−),ϵ+=P(c−|+)=∫ν−∞dOP(O|+). (S.2)

Assuming equal a priori probabilities for the ground and excited states, the average single-shot readout error rate is minimized by choosing . An explicit calculation of the integrals gives:

 ϵ=ϵ±=12erfc(√r2). (S.3)

Eq. (S.3) defines the binary symmetric readout associated with the Gaussian readout.

We assume for simplicity that when the qubits of the quantum repetition code are measured, the -qubits state collapses to or with equal probability. The resulting dataset consists of analog readout outcomes . In the main text, we gave two likelihood ratios and for thresholded and analog readout outcomes, respectively. In both cases, if () we infer that the qubit state is (). For the Gaussian readout, the likelihood ratios reduce to:

 Λc=(1−ϵϵ)2n+−N,ΛO=exp(2Nr¯O), (S.4)

where is the number of times that the outcome is converted to if the qubits are read out in a single shot and where is the sample average of the analog outcomes 111In Ref. Schaetz et al. (2005), a two-qubit repetition code was implemented using two trapped ions. The enhanced detection was obtained by collecting the total fluorescence of both ions. We note that according to Eq. (S.4), this effectively implements soft decoding of the repetition code if the fluorescence counts follow a Gaussian readout distribution. For a general readout, however, knowledge of is not sufficient to determine ; in this case the analog observable must be recorded for each qubit..

Since , the likelihood ratio is equivalent to majority vote decoding of the repetition code. The corresponding average error rate is given by the probability that given that is encoded, which is the same as the probability that given that is encoded. If is odd, is given by:

 εc=N∑n+=N+12(Nn+)ϵn+(1−ϵ)N−n+=Iϵ(N+12,N+12). (S.5)

Here, is the regularized incomplete beta function Press et al. (1992). The error rate for is the same since the case provides no information on the qubit state for a binary symmetric readout. For large enough (), Eq. (S.5) takes the approximate form:

 εc≃(NN+12)1(2πr)N+14e−(N+1)r4. (S.6)

Eq. (S.6) must be contrasted to the error rate for the likelihood ratio in Eq. (S.4). The corresponding average error rate is given by the probability that given that is encoded, which is the same as the probability that given that is encoded. Since and are also Gaussians centered at with signal-to-noise ratio , the average error rate for the soft decoding of the readout apparatus is simply:

 εO=∫∞0d¯OP(¯O|0)=12erfc(√Nr2). (S.7)

When , Eq. (S.7) becomes:

 εO≃1√2πNre−Nr2. (S.8)

Inspection of Eqs. (S.6) and (S.8) suggests that decreases at approximately twice the rate of when increases. Indeed, let and be the number of qubits required to achieve a target error rate , i.e. . Using Eqs. (S.5) and (S.7), we solve this equation for to subleading order in and obtain the result discussed in the main text:

 NO=Nc+12+Nc−12lnrr+O(Ncr). (S.9)

This expression is valid for any odd . The asymptotic advantage, , has been discussed for arbitrary block codes in Ref. Chase (1972).

### b.2 Encoding errors

In this section, we expand on the effect of encoding errors on the repetition code. As discussed in the main text, we consider only uncorrelated bit flip errors for simplicity. We consider both the Gaussian readout and the realistic peak-signal readout analyzed in Ref. D’Anjou and Coish (2014) and summarized above.

Let be the probability for any qubit of the code to flip during the encoding sequence. The likelihood ratio for analog readout outcomes takes the modified form:

 ΛO≡N∏i=1ΛO,i, (S.10)

where is the likelihood ratio for a single qubit measurement:

 ΛO,i=P(Oi|1)P(Oi|0)=(1−η)P(Oi|+)+ηP(Oi|−)(1−η)P(Oi|−)+ηP(Oi|+). (S.11)

If the a priori probabilities of the logical states and are equal, the single-shot threshold is obtained as usual from . Therefore, sufficiently localized readout probability distributions such as the Gaussian distributions satisfy:

 ΛO,i≈1−ηη(O≫ν),ΛO,i≈η1−η(O≪ν). (S.12)

Because is approximately constant above and below threshold, it seems that soft decoding of the analog readout outcomes is reduced to a simple thresholding procedure when is finite. As illustrated in Fig. S.3, additional information can nevertheless be extracted from values falling near the threshold, where is non-constant, provided that is small enough.

In the case of the Gaussian readout distributions, is non-constant for values of such that , as illustrated in Fig. S.3. In order for a significant fraction of measured values to lie in that interval, we must have . Therefore:

 η1−η≲e−2r⇒η≲e−2r1+e−2r. (S.13)

In the limit , this reduces to . Since Eq. (S.3) implies that up to logarithmic corrections for , we conclude that must be smaller than some power of . As shown in Fig. S.5, a similar upper bound on exists for the non-Gaussian peak-signal readout of Ref. D’Anjou and Coish (2014).

If each analog outcome is instead thresholded to a binary outcome , the likelihood ratio is:

 Λc=[(1−η)(1−ϵ+)+ηϵ−(1−η)ϵ−+η(1−ϵ+)]n+⋅[(1−η)ϵ++η(1−ϵ−)(1−η)(1−ϵ−)+ηϵ+]N−n+, (S.14)

where is the number of qubits that are assigned the value . To show quantitatively that a significant advantage can be obtained by utilizing the analog readout outcomes in the presence of encoding errors, we must compare the performance of Eq. (S.14) to that of Eq. (S.10).

For both the Gaussian readout and the realisitic peak-signal readout, we performed Monte-Carlo simulations of the error rates for maximum-likelihood decoding of the analog and thresholded readout outcomes. In both cases, we take the signal-to-noise ratio to be and choose parameters that optimize the single-shot readout fidelity. We randomly choose the logical state or with equal probability and generate a random measurement record by sampling independent values from the distributions or , respectively. We then infer the state with both Eq. (S.10) and Eq. (S.14) and record an error if the decision is incorrect. We repeat the procedure times ( times) for the Gaussian readout (peak-signal readout) and obtain the error rate from the ratio of errors to the number of trials. The resulting error rates are shown in Figs. S.4 and S.5 respectively. For the Gaussian readout with , we instead plot the analytic expressions, Eqs. (S.5) and (S.7). Numerical values of the error rates for both readouts without encoding errors are tabulated in Table S.1 for convenience.

## Appendix C State and parameter estimation

In this section, we give a brief derivation of the asymptotic mean squared error of the maximum-likelihood estimator for when applied to analog and thresholded readout outcomes. We also review the soft average discussed in Ref. Ryan et al. (2013). In all cases, we estimate with independent analog or thresholded readout outcomes, or , following a distribution of the form:

 P(Oi/ci|s0)=1+s02P(Oi/ci|+)+1−s02P(Oi/ci|−). (S.15)

In the following, we will denote statistical expectation values with respect to Eq. (S.15) by the double brackets . The maximum-likelihood estimator is the value that maximizes the log-likelihood function:

 ℓ(s)=1NN∑i=1lnP(Oi/ci|s), (S.16)

under the constraint .

First we assume that the values are thresholded to a binary outcome , where the threshold is chosen to satisfy .

To obtain the maximum-likelihood estimator, we must maximize the likelihood function, Eq. (S.16). We first note that Bayes’ rule gives the probability of an outcome given the true expectation :

 P(ci|s0)=1+s02P(ci|+)+1−s02P(ci|−). (S.17)

Here, the transition probabilities of the binary readout are given by the conditional single-shot error rates:

 P(c−|+)≡ϵ+=∫ν−∞dOP(O|+),P(c+|−)≡ϵ−=∫∞νdOP(O|−). (S.18)

Thus, Eq. (S.17) becomes:

 P(c+|s0)=1+s02(1−ϵ+)+1−s02ϵ−,P(c−|s0)=1+s02ϵ++1−s02(1−ϵ−). (S.19)

Next, we use the form of Eq. (S.19) in the log-likelihood function, Eq. (S.16), and optimize with respect to . Maximizing without the constraint (i.e. setting ), the optimum is the thresholded average:

 sTA=1NN∑i=1ci, (S.20)

where the binary outcomes are chosen to be:

 c+=1+(ϵ+−ϵ−)1−(ϵ++ϵ−),c−=−1−(ϵ+−ϵ−)1−(ϵ++ϵ−). (S.21)

In the limit of large , the estimate is unlikely to fall outside the region . In this asymptotic limit, the estimate is unbiased:

 ⟨⟨sTA⟩⟩ =P(c+|s0)c++P(c−|s0)c− (S.22) =1+s02[(1−ϵ+)c++ϵ+c−]+1−s02[ϵ−c++(1−ϵ−)c−]=s0. (S.23)

In this case, the asymptotic mean squared error of the maximum-likelihood estimate, Eq. (S.20), is equal to its asymptotic variance and is given by the central limit theorem:

 ζTA=⟨⟨Δs2TA⟩⟩∼⟨⟨Δc2⟩⟩N=P(c+|s0)c2++P(c−|s0)c2−−s20N. (S.24)

In the special case of a binary symmetric readout with , we have and we recover the expression given in Ref. Ryan et al. (2013):

 ζTA∼(1−2ϵ)−2−s20N. (S.25)

The asymptotic mean squared error of the maximum-likelihood estimator applied to the analog readout outcomes is equal to its asymptotic variance, which saturates the Cramér-Rao bound Cramér (1946):

 ζSD∼1NF(s0), (S.26)

where is the Fisher information of the distribution (S.15):

 F(s0)=⟨⟨(∂lnP(O|s0)∂s0)2⟩⟩=−⟨⟨∂2lnP(O|s0)∂s20⟩⟩. (S.27)

The last equality in Eq. (S.27) is obtained through integration by parts. Differentiating Eq. (S.15) twice gives an explicit form for :

 F(s0)=14∫dO[P(O|+)−P(O|−)]2P(O|s0). (S.28)

Expanding the integrand, we have:

 F(s0)=14[∫dOP(O|+)2P(O|s0)+∫dOP(O|−)2P(O|s0)−2∫dOP(O|+)P(O|−)P(O|s0)]. (S.29)

When the readout distributions are very well-separated, the Fisher information only contains the shot noise contribution . We isolate this contribution in Eq. (S.29) and upon simplification we find:

 F(s0)=11−s20−11−s20I,I=∫dOP(O|+)P(O|−)P(O|s0), (S.30)

where is an overlap integral containing all information about the intrinsic measurement noise described by . Therefore, the asymptotic mean squared error of the maximum-likelihood estimator applied to the analog readout outcomes is:

 ζSD∼1−s201−I. (S.31)

### c.3 Bias-corrected soft average

Another possible estimator for the qubit expectation value is the soft average discussed in Ref. Ryan et al. (2013):

 sSA=1NN∑i=1Oi. (S.32)

We compare the performance of this estimator to the previously discussed estimators, and , for completeness. The expectation value of Eq. (S.32) with respect to has the form:

 ⟨⟨sSA⟩⟩=As0+B, (S.33)

where:

 A=⟨⟨O⟩⟩+−⟨⟨O⟩⟩−2,B=⟨⟨O⟩⟩++⟨⟨O⟩⟩−2. (S.34)

Here, we define the conditional expectations . Thus, the soft average of Eq. (S.32) is biased for general readout probability distributions .

To obtain an unbiased estimate, we replace Eq. (S.32) by the soft average of the rescaled values :

 sSA=1NN∑i=1O′i=1NN∑i=1Oi−BA. (S.35)

The asymptotic mean squared error of the unbiased soft average estimate, Eq. (S.35), is equal to its asymptotic variance and is given by the central limit theorem:

 ζSA=⟨⟨Δs2SA⟩⟩=ΔO′2N=⟨⟨O′2⟩⟩−s20N. (S.36)

In terms of the original observable , this becomes:

 ζSA=ΔO2A2N=⟨⟨O2⟩⟩−(As0+B)2A2N. (S.37)

In the special case of the Gaussian readout, Eq. (S.1), we have and . Direct calculation of then yields the result given in Ref. Ryan et al. (2013):

 ζSA=1+r−1−s20N. (S.38)

Fig. 6 compares the asymptotic performance of the soft average to that of the maximum-likelihood estimates and as a function of the signal-to-noise ratio , for both the Gaussian and the peak-signal readouts. As noted in Ref. Ryan et al. (2013), the soft average outperforms the thresholded average for low . This is because the distribution approaches a Gaussian centered at when for both readouts, , and the maximum-likelihood estimator for the mean of a Gaussian coincides with the soft average. In that case, the soft average is therefore the same as the soft-decoded estimate . However, the soft average estimate offers suboptimal performance for finite and suffers from an significant loss in performance compared to and when becomes large. In contrast, the soft-decoded estimate is optimal for all .

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