A circuit for transferring electrons with high fidelity

Experimentally, the high-fidelity circuit for electron transfer is realized by a chain of quantum dots in which the first and the last dot are operated as single-electron pumps²⁷ and the central dot provides the error signal as shown in Fig. 1. A clock of frequency (f = 30–300 MHz) drives the pumps to transfer one electron per cycle through the chain (from top to bottom in Fig. 1).

Within one clock cycle, the entrance barrier to the dynamic quantum dot is lowered and raised by the pump stimulus, isolating one electron from the source reservoir and then ejecting it over the high exit barrier; barrier height asymmetry between entrance and exit defines the transfer direction²⁸. The operating points of the pumps are chosen to minimize and approximately balance the error probabilities of transferring either zero or two electrons instead of one (with a slight bias towards zero-electron transfers, as this error rate only increases exponentially and not double-exponentially with deviations from the optimal operating point²⁹). The working points of the pumps are not retuned when operating the full circuit. Reproducible formation of quantum dots³⁰ allows demonstrating the high-fidelity operation of the circuit event at zero magnetic field, at which readout precision is enhanced by cryogenic reflectometry.

The excess charge x from accumulating errors is inferred from a differential measurement by a charge detector capacitively coupled to the central dot, reading out the detector state before and after each sequence transferring t electrons. As tunneling events are only enabled by the clocked stimulus applied to the pumps, a long detector integration-time up to 1 ms can be chosen for unambiguous identification of x with a signal to noise ratio of 17 (Fig. 1b). A full histogram of detector states before and after the transfer sequence allows to reconstruct the shape of the Coulomb blockade peak resonance utilized by the charge detector and provides rigorous classification thresholds for the identification of x. The sequence of electron transfer and charge detection is repeated with the repetition rate limited by the detector integration-time (up to 4 kHz), until a set number of counts (N = 1 × 10⁵ to 2 × 10⁶) is accumulated. Any deviations not aligned with the measurement timing, such as instabilities in the charge detector, are readily recognized and discarded, while unintended charge transitions during the operation of the pump are counted and correctly identified as errors.

Although the individual accuracy of the active components can exceed metrological precision³¹, their simultaneous operation in a mesoscopic circuit³² precludes the prediction of transfer fidelity from component-wise characterization due to interactions and crosstalk between the elements in the chain, exemplifying the need for circuit-level benchmarking. Experimental evidence for strong discord between component-wise and circuit-level characterization is given in Supplementary Note 3.

Implementation of a random-walk benchmark

Here we report the measurement results on two devices: device A introduces the methodology to resolve effects beyond statistical noise of independent-error accumulation in a high-fidelity circuit, while device B demonstrates the effects of memory with increased repetition frequency. Both devices share very similar device geometries and parameters.

Figure 2a shows the counting statistics measured for device A at f = 30 MHz for t up to 10⁴ compared to predictions of the baseline model. General trends expected from the random walk are evident: for short sequences, $$ t<1000\ll {\left(P_{+}{P}_{-}\right)}^{-1/2}$$, the power-law rise of the probabilities $$ p_{\mid x\mid >0}^t$$ corresponds to the exponential decay of error-free transfer fidelity $$ p_0^t$$, which remains close to 1. For longer sequences the distribution spreads and the weight of self-intersecting paths (e.g., orange line in Fig. 1) increases, in accordance with Eq. (1).

Fig. 2

Consistency of random-walk models with respect to the experimental statistics of accumulated errors.

a Measured $$ p_x^t$$ for device A; error bars are given by the standard deviation of the binomial distribution, solid lines show a least-squares fit of Eq. (1). b Likelihood-maximizing P_± (white dots) and p > 0.05 consistency regions estimated separately for each sequence length (coded by color). The inset shows the probability density function of the Dirichlet distribution with parameter α = (2.43 × 10³, 3.50 × 10⁷, 7.47 × 10²). The corresponding global best-fit values for P_± are marked by red lines on the axes of the consistency-region plot. The color scale indicates the level of confidence at different coverage factors k for a symmetric normal distribution; the red circle in both plots and the marker in the color scale indicates the region corresponding to k = 4. c Empirical cumulative distribution of p-values for different models in comparison to the uniform distribution (black line).

The key question for random-walk benchmarking is whether the uncorrelated residual randomness defined by two probabilities P₊ and P₋ predicts the entire probability distribution. This question is answered in three steps: (i) significance testing of deviations from the baseline model as a statistical null-hypothesis to delineate the inevitable sampling error from model error; (ii) extending the model to accommodate correlated excess noise³³ detected in the first step; (iii) perform parameter estimation of the noise model that yields average values of P_± with an estimate of the variability.

For consistency testing, we have increased the number N of samples per sequence by a factor of ~10, and limited t to 100. Fisherian significance tests³⁴ are used to define consistency regions of p-value > 0.05 in the parameter space (P₊, P₋) where the baseline model cannot be rejected at this significance level (see “Methods” section). Figure 2b shows quasielliptic consistency regions computed for each sequence length t separately, randomly clustering in a tight area with the sizes shrinking roughly as $$ ~1/\sqrt{t}$$, as expected. Their overlap is only partial: best-fit global (P₊, P₋) estimated from maximal likelihood (marked on the axes of Fig. 2b) lies outside of 7 regions out of 42. A more rigorous test on whether this inconsistency can be explained by sampling error alone is provided by Fisher’s meta-analysis method (Fig. 2c): under the null-hypothesis, the cumulative distribution of p-values obtained separately for each sequence length t should be uniform (a straight line)^35,36 (Supplementary Note 6), which is not the case for the best-fit baseline model (triangles in Fig. 2c). Quantitatively, the baseline model yields global Fisher’s combined p < 3 × 10⁻⁶, and hence is statistically rejected. We attribute this incompatibility to excess noise due to imperfections in the physical realization of the baseline model. Nevertheless, the partial overlap and the tight clustering observed in Fig. 2b suggests that the excess noise is rather small. We model the excess noise as stochastic variability of P_±, and check whether it can be plausibly explained by the presence of two-level fluctuators³⁷.

To quantify the excess noise, the model is now extended (part (ii) of the outline above) by drawing the step probabilities P_± randomly from a Dirichlet distribution^38,39 (Supplementary Note 8) over the standard 2-simplex; the corresponding parameters $$ \mathit{\alpha }=\left\{\alpha ⟨P_{-}⟩,\alpha \left(1-⟨P_{+}⟩-⟨P_{-}⟩\right),\alpha ⟨P_{+}⟩\right\}$$ are specified by two means, 〈P_±〉, and one additional concentration parameter α which controls the variance, $$ \mathrm{\Delta }{P}_{\pm }^2=⟨P_{\pm }⟩\left(1-⟨P_{\pm }⟩\right)/\left(\alpha +1\right)$$. The Dirichlet distribution is strongly peaked near the mean point for $$ \alpha \gg {⟨P_{\pm }⟩}^{-1}$$, and always guarantees 0 ≤ P_± ≤ 1. This extra randomness can be introduced at different timescales²⁴. Uncorrelated noise (new P_± after each step of a walk) is equivalent to the baseline model with $$ P_{\pm }\to ⟨P_{\pm }⟩$$, and is already ruled out by the significance tests above. We compare a “fast fluctuator” model in which a new pair of P_± is drawn independently after completion of each individual random walk versus a “slow drift” model in which the values of P_± are randomly reset only after all N realizations for a fixed number of steps have been collected (precise excess noise model definitions are given in Supplementary Notes 9 and 11, and the data acquisition timeline is illustrated in Supplementary Fig. S1). Although short of proper time-resolved noise metrology²⁶, contrasting these two correlated-noise models gives an indication of the relevant timescales (nanoseconds versus half-hour in the experiments). The sensitivity of Fisher’s significance testing makes it possible to distinguish between the two models, which cannot be resolved by the second moment of $$ ⟨p_x^t⟩$$ as utilized, e.g., for noise-averaged fidelities in randomized benchmarking of quantum gates³³. The results of Fisher’s combined test (Fig. 2c) favor the “slow drift” (p = 0.71) over the “fast fluctuator” (p < 3 × 10⁻⁶) model. The corresponding best-fitting Dirichlet distribution (parameters indicated by red lines on the axes of Fig. 2b and plotted in the inset) gives 1 σ uncertainty estimates P₋ = (6.92 ± 0.14) × 10⁻⁵ and P₊ = (2.13 ± 0.08) × 10⁻⁵. Parametric instability at only a few-percent level validates a suitably extended random-walk model as a robust representation of error accumulation in this high-fidelity single-electron circuit.

In order to gain insight into a possible physics mechanism for excess noise and illustrate the robustness of statistical methods, we have simulated the experimental timeline using a random-walk model with P_± parameters subjected to 1/f noise from an ensemble of independent two-level fluctuators (Supplementary Note 13). The results follow the general pattern outlined above: (i) for a fixed size of the statistical sample, there is a threshold in the excess noise amplitude above which the data contradict both the baseline and the fast-fluctuator models but remain consistent with the slow-drift model. This threshold corresponds to excess noise sufficiently affecting probabilities of multiple errors per burst to reveal inconsistency with Eq. (1) in the tails (∣x∣ > 1) of the error syndrome distribution $$ p_x^t$$. (ii) The estimated best-fit ΔP_± parameters correlate well with the standard deviation of the P_± in the underlying simulation. (iii) Even a single fluctuator with a fixed switching rate (bimodal distribution of P_± and a Poisson distribution of switching times⁴⁰) can generate detectable excess noise still consistent with our Dirichlet-based statistical models. As for the physics of the real device in a noisy environment, the simulations favor an explanation of the detected excess noise by the presence of multiple charge fluctuators over a single two-level system due to the absence of a bimodal signature in Fig. 2b. In conclusion, accurate statistics of error counts can give enough sensitivity to reliably estimate the baseline error rates P_± and even capture a fingerprint of long-time correlations in the environment.

Benchmarking for memory effects

The methodology to quantify independent-error accumulation described above makes it possible to probe the effect of increased clock frequency on the circuit and thereby investigate response times of the electron shuttle and interactions between subsequent steps. In device B, the error rates are P₋ = (6.31 ± 0.23) × 10⁻³ and P₊ = (2.71 ± 0.043) × 10⁻² at the same frequency of 30 MHz as device A investigated above. A ten-fold increase of the clock frequency to 300 MHz is introduced by uniform time compression of signals controlling the transfer operations; the resulting counting statistics is presented in Fig. 3a (circles). The random-walk model with constant P_±, described by Eq. (1), no longer applies even qualitatively, which raises the question whether the fidelity of the circuit has decreased to a point where errors can no longer be considered rare as outlined in the beginning. This question is answered in the negative with the help of the following theorem defining a spread condition, which sets a precise bound on the applicability of the random-walk approach with possibly non-stationary error rates: If distributions $$ \left(p_x^t\right)$$ and $$ \left(p_x^{t+1}\right)$$ satisfy

Fig. 3

Memory effect probed at the increased clock frequency.

a Measured $$ p_x^t$$ for device B at a clock frequency of 300 MHz and τ_Delay = 0 s (left, t-axis inverted) and $$ {\tau }_{\mathrm{Delay}}=3.\overline{3}\mathrm{ns}$$ (right). Dashed lines represent $$ p_x^t$$ predicted by deconvolved single-step error rates and $$ p_x^{t-1}$$. b Single-step error rates $$ {P_{\pm }}^t$$ for τ_Delay = 0 s (left, t-axis inverted, dashed lines show guide to the eye) and $$ {\tau }_{\mathrm{Delay}}=3.\overline{3}\mathrm{ns}$$ (right, translucent area corresponds to the 1 σ uncertainty estimates). Inset depicts the timing diagram of the sequence—a stimulus of duration τ_op drives the transfer operation followed by delay time τ_Delay before the next step.

We find that the distributions measured on device B do satisfy the spread condition (2) as long as all x are fully resolved in counting (t ≤ 6). We estimate the non-stationary but x-homogeneous single-step error probabilities of the corresponding Markov chains, $$ P_{\pm 1}^{\left(x,t\right)}=P_{\pm }^t$$, by a numerical deconvolution of the Markov process equation (Supplementary Note 2). The resulting error rates $$ P_{\pm }^t$$ in Fig. 3b provide reasonable prediction (dashed lines) of the measured $$ p_x^t$$ in Fig. 3a (circles). The t-dependence of $$ P_{\pm }^t$$ is strong and reproduced well above the noise. This implies memory: probabilities for the next step depend on how many steps have taken place before. $$ P_{\pm }^t$$ do not saturate within t≤6 indicating a long memory time of more than 6 τ_op = 20 ns.

To probe this memory effect, we introduce a delay time τ_Delay between otherwise unaltered signals driving the transfer operations thus extending the physical time f⁻¹ corresponding to a single step of the random walk from τ_op to τ_op + τ_Delay as sketched in Fig. 3b. With increasing delay, a gradual reduction of the t-dependence in $$ P_{\pm }^t$$ is observed until, for τ_Delay > 3 ns (see right part of Fig. 3a, b), the stationary behavior consistent with the baseline model is recovered. Surprisingly, τ_Delay sufficient to recover stationary behavior is on the order of a single-step duration τ_op, significantly shorter than the number of steps with pronounced memory effect at τ_Delay = 0 ns (Fig. 3b). Both times are significantly longer than the expected timescales in GaAs systems for relaxation via electron–electron or phonon interaction^41–43, and raise the need for a dedicated investigation. In Fig. 3b, $$ P_{\pm }^t$$, estimated at each t by deconvolution (squares), are compared with the confidence intervals of the “slow-drift” model with stationary P_± (color bands). The comparison shows good agreement and is consistent with our framework for random-walk benchmarking of high-fidelity single-electron circuits. For the showcased device, circuit-level interactions and memory effects significantly lower the attainable clock speed compared to record frequencies for individual pumps reported in the literature⁴⁴. However, benchmarking by error accumulation introduces a tool to investigate these limitations and identify possible mitigation-techniques since τ_op and τ_Delay can be freely adjusted with error rates still accurately estimated on the circuit level, as long as these remain within the high-fidelity bound monitored by the spread condition.

In conclusion, the view of single-electron components as elements of a digital circuit has enabled an abstract and universal description of fidelity in terms of the random walk of an error syndrome. Accumulation of errors over long sequences allows probing fast and accurate operations beyond the bandwidth of a slow single-charge detector. The accompanying statistical methodology quantifies the stability of the error process and uncovers short memory times, both of which are elusive to direct observation. In quantum metrology, an accurate estimate of the circuit error has an immediate application: the variance of the current I = (I_s + I_d)/2 flowing into (I_s) and out of (I_d) the circuit is given by the variance of the differential charge x, which corresponds to the displacement current I_s − I_d = efx/t. Hence, the variance of x, $$ \mathrm{\Delta }{x}^2\approx \left(⟨P_{+}⟩+⟨P_{-}⟩\right)t+\left(\mathrm{\Delta }{P}_{+}^2+\mathrm{\Delta }{P}_{-}^2\right)t^2$$, provides a bound for the deviation of the current I from the error-free value ef, enabling counting-verification of a primary standard for the ampere. In the broader context, sensitive tests of single-electron circuits create new ground for developing benchmarking techniques of engineered quantum systems.

Devices

Devices A and B were fabricated from GaAs/AlGaAs heterostructures with two dimensional electron gas (2DEG) nominally 90 nm below the surface. Quantum dots are formed by CrAu top gates depleting a shallow-etched mesa³⁰. The charge detector is formed against the edge of a separate mesa and capacitively coupled to the central quantum dot via a floating gate⁴⁵.

Measurement setup

All measurements were performed in a dilution refrigerator at a base temperature of 20 mK and 0 T external field. The charge detector signal is readout by rf reflectometry⁴⁶. Sinusoidal pulses generated by arbitrary waveform generators modulate the entrance barriers of the single-electron pumps and drive the clock-controlled electron transfer²⁷. The drift-stability due to control voltages is estimated to be better than 10⁻⁸. Charge transfer and detector readout are triggered in a sequence: (i) readout of the initial detector state, (ii) application of t sinusoidal pulses to both pumps simultaneously, (iii) readout of the final detector state, (iv) reset by connecting the intermediate dot to source. The difference between initial and final detector state yields the charge x deposited on the central quantum dot by the burst transfer, providing raw data for subsequent statistical analysis.

Consistency testing

Fisher’s p-value for each experimentally measured x-resolved set of N counts is defined as the probability of an equally or more extreme outcome under the null-hypothesis being tested (either the baseline random walk or one of the two excess noise models with Dirichlet-distributed P_±); it is evaluated by Monte Carlo sampling as described in the Supplementary Notes 4 and 8.