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

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

B. D'Anjou and W. A. Coish

Phys. Rev. A 89, 012313 – Published 15 January 2014

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 $t_{i}$ 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 $r$ 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 $ɛ \sim ln r / \sqrt{r}$ when $t_{i}$ is stochastic, in contrast to the result $ɛ \sim ln r / r$ for deterministic $t_{i}$ . Consequently, the peak-signal method outperforms the boxcar filter significantly when $t_{i}$ is stochastic, but is only marginally better for deterministic $t_{i}$ (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 $t_{i}$ and show numerically that a stochastic turn-on time $t_{i}$ 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 $t_{i}$ to be deterministic.

Received 12 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.012313

Authors & Affiliations

B. D'Anjou and W. A. Coish

Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada

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Vol. 89, Iss. 1 — January 2014

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Figure 1
(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 $t = 0$ , it can trigger cycling between two cycling states (solid green line) at time $t_{i}$ . The cycling ends at time $t_{f}$ 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 $ψ (t)$ resulting from the cycling process of (a) when the initial state is $| + 〉$ . The readout phase starts at $t = 0$ and acquisition starts after an arming time $t_{arm}$ . Initially, cycling does not occur and the signal takes the average value $〈 ψ 〉 = - 1$ . At a random time $t_{i}$ , cycling is triggered and the signal rises to $〈 ψ 〉 = 1$ . At a subsequent random time $t_{f}$ , cycling stops and the signal again drops to $〈 ψ 〉 = - 1$ . Acquisition stops after a measurement time $τ_{M}$ . Throughout, we assume that $t_{arm} \to 0$ , that $t_{i}$ and $τ = t_{f} - t_{i}$ follow Poisson processes, and that the noise is white and Gaussian.
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Figure 2
Regions of the $t_{i} − t_{f}$ plane. The turn-on time $t_{i}$ (turn-off time $t_{f}$ ) falls in the $m th$ ( $n th$ ) discrete bin of length $τ_{b} = τ_{M} / N$ , where $N$ is the number of bins contained in the measurement time $τ_{M}$ . Here, the $m th$ ( $n th$ ) bin of the $t_{i}$ ( $t_{f}$ ) axis starts at time $m τ_{b}$ ( $n τ_{b}$ ). The shaded region is forbidden since we must necessarily have $t_{f} > t_{i}$ . The finite measurement time $τ_{M}$ divides the plane in three regions $R_{i}$ , Eq. (22). Each region gives a distinct, mutually exclusive contribution to the peak-signal distribution $P (ψ_{p} | +)$ .
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Figure 3
Fit of the analytical distributions $P (ψ_{p} | -) P (-)$ (dashed red line) and $P (ψ_{p} | +) P (+)$ (solid blue line), Eqs. (21) and (28), to the experimentally determined distribution $P (ψ_{p})$ extracted from data in Fig. 4(b) of Ref. [11] (open circles). The dotted black line is the full peak-signal distribution $P (ψ_{p}) = P (ψ_{p} | -) P (-) + P (ψ_{p} | +) P (+)$ . The peak SET current $I_{p}$ for the spin-to-charge conversion is mapped to the reduced peak signal $ψ_{p}$ via $ψ_{p} = (2 I_{p} - I) / I$ , where $I$ is the average SET current. We set $Γ$ and $τ_{M}$ to their measured values $Γ = 4$ and $τ_{M} = 2.5$ and we find fitted values of $I \approx 2.0 nA$ , $r \approx 110$ , $τ_{b} \approx 0.075$ , and $P (+) = 1 - P (-) = 0.47$ for the current, signal-to-noise ratio, bin time, and prior probabilities, respectively. The values of $I$ and $P (\pm)$ are in good agreement with the experimentally measured values of $I^{\exp} \approx 1.9 nA$ and $P {(+)}^{\exp} \approx 0.47$ . Furthermore, the bin time is of the same order of magnitude as the inverse bandwidth of the low-pass filter used in Ref. [11], corresponding to $τ_{b}^{\exp} \approx 0.2$ . 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.
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Figure 4
(a), (c) Optimized error rates, Eq. (2), as a function of the signal-to-noise ratio $r$ for the boxcar filter (solid blue line), peak-signal filter (dashed red line), and maximum-likelihood filter (dotted green line) for (a) $Γ = 4$ (e.g., spin-to-charge conversion in semiconductor qubits) and (c) $Γ \to \infty$ (e.g., fluorescence-based readouts in NV centers or trapped ions). The error rates have been optimized with respect to the threshold $ν$ , measurement time $τ_{M}$ , and bin time $τ_{b}$ (when applicable). In the case $Γ = 4$ , the error rate is significantly decreased for large $r$ by using the peak-signal filter instead of the boxcar filter, whereas the advantage is much smaller in the case $Γ \to \infty$ . The error rate for the maximum-likelihood filter (dotted green line), obtained from the Monte Carlo solution of Eqs. (C4) and (C6), is the lowest theoretically achievable error rate. The fluctuations in the maximum-likelihood error rates are due to the finite sample size ( $5 \times 10^{4}$ ) of the Monte Carlo simulation. (b), (d) Optimized probability distributions $P (ψ_{p} | -)$ (dashed red line), $P (ψ_{p} | +)$ (solid blue line), $P (\bar{ψ} | -)$ (dotted red line), and $P (\bar{ψ} | +)$ (dot-dashed blue line) for $r = 30$ in the cases (b) $Γ = 4$ , Eqs. (21), (28), and (29), and (d) $Γ \to \infty$ , Eqs. (30), (34), and (35). For $Γ = 4$ , 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 $Γ \to \infty$ , 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 $ν$ for each case, satisfying $P (ν | -) = P (ν | +)$ .
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Figure 5
(a) Optimal measurement time $τ_{M}$ as a function of the signal-to-noise ratio $r$ for the peak-signal filter in the cases $Γ = 4$ (purple solid line) and $Γ \to \infty$ (purple dashed line). When $Γ = 4$ , the measurement time diverges logarithmically with $r$ [see the discussion following Eq. (40)], whereas when $Γ \to \infty$ , $τ_{M}$ approaches 0 as $r$ increases. (b) Optimal number number of bins $N = τ_{M} / τ_{b}$ as a function of the signal-to-noise ratio $r$ for the peak-signal filter in the cases $Γ = 4$ (purple solid line) and $Γ \to \infty$ (purple dashed line). Although we have derived our model for $N \in N$ , we treat $N$ as a continuous variable for numerical optimization: fractional values of $N$ must be seen as an interpolation between integer values. Because $τ_{M}$ must remain finite when $Γ = 4$ , it becomes advantageous to increase the number of bins in order to locate the pulse within the measurement window. When $Γ \to \infty$ ( $t_{i} = 0$ ), $τ_{M}$ can approach 0 as $r$ increases, eliminating the need for binning.
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Figure 6
Estimator $P [+ | ψ (t)]$ as a function of the measurement time $τ_{M}$ for $r = 30$ , obtained for a randomly generated record $ψ (t)$ in the cases $Γ = 4$ (solid blue line) and $Γ \to \infty$ (dashed purple line). When $Γ = 4$ , $P [+ | ψ (t)]$ initially slowly decreases in the interval $[0, t_{i}]$ and suddenly jumps to 1 when the pulse occurs. When $Γ \to \infty$ , $t_{i} \to 0$ and $P [+ | ψ (t)]$ immediately jumps to 1. In both cases the estimator reaches a constant value at large $τ_{M}$ when all the available information on the pulse has been acquired.
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