Spin-qubit noise spectroscopy from randomized benchmarking by supervised learning

We demonstrate a method to obtain the spectra of $1/f$ noises in spin-qubit devices from randomized benchmarking, assisted by supervised learning. The noise exponent, which indicates the correlation within the noise, is determined by training a double-layer neural network with the ratio between the randomized benchmarking results of pulse sequences that correct noise or not. After the training is completed, the neural network is able to predict the exponent within an absolute error of about 0.05, comparable with existing methods. The noise amplitude is then evaluated by training another neural network with the decaying fidelity of randomized benchmarking results from the uncorrected sequences. The relative error for the prediction of the noise amplitude is as low as 5% provided that the noise exponent is known. Our results suggest that the neural network is capable of predicting noise spectra from randomized benchmarking, which can be an alternative method to measure noise spectra in spin-qubit devices.

Figure 1

An example of the $1/f$ noise used in the simulation. (a) Noise as a function of time. (b) The power spectral density corresponding to the noise shown in (a). $S\left(\omega \right)=A/{\left(\omega t_0\right)}^{1.5}$. The noise amplitude $A$ has been scaled so that $S(\omega =1/t_0)\approx 1/t_0$, and $t_0$ is an arbitrary time unit.

Figure 2

Randomized benchmarking results of single-qubit Clifford gates subject to $1/f$ noise with different exponents. (a), (c), (e) The fidelities for uncorrected and corrected gates are shown as black and red/gray lines, respectively. (b), (d), (f) The ratios between the corrected and uncorrected sequences correspond to the results shown on the left. The noise exponents are (a), (b) $\alpha =0$, (c), (d) $\alpha =1$, and (e), (f) $\alpha =1.5$. The noise amplitude is fixed by $A{t}_0={10}^{-3}$.

Figure 3

Schematics of the neural network used in this work. The inputs are given by RB results: $\kappa$ (corresponding to output of $\alpha$) and $F$ (corresponding to $A$). $\kappa$ and $F$ are discretized into ${\kappa }^i$ and $F^i$ where $i=1,...,N$ and $N=200$. The neural network contains two hidden layers with $N_n$ neurons each.

Figure 4

The averaged error $\mathrm{\Delta }$ for the noise exponent $\alpha$ as functions of the number of epochs. The left-hand column (a, c, e) corresponds to a training set with 5000 entries, while the right-hand column (b, d, f) corresponds to a training set with 10 000 entries. For (a) and (b), the blue (upper) line, red (middle) line, and black (bottom) line show results with learning rates $\eta =0.001$, 0.005, and 0.01 respectively, while other parameters are fixed at $b=10$ and $N_n=50$. For (c) and (d), the blue (bottom) line, red (middle) line, and black (upper) line show results with bin sizes $b=10$, 20, and 30, respectively, while other parameters are fixed at $\eta =0.005$ and $N_n=50$. Lines in (e) and (f) show results with $N_n=10$, 30, and 50, respectively, while other parameters are fixed at $\eta =0.005$ and $b=10$.

Figure 5

The (dimensionless) averaged relative error $\delta$ for the noise amplitude $A{t}_0$ as functions of the number of epochs. The left-hand column (a), (c), (e) corresponds to a training set with 5000 entries, while the right-hand column (b), (d), (f) corresponds to a training set with 10 000 entries. For (a) and (b), the blue (upper) line, red (middle) line, and black (bottom) line show results with learning rates $\eta =0.001$, 0.005, and 0.01, respectively. For (c) and (d), the blue (bottom) line, red (middle) line, and black (upper) line show results with bin sizes $b=10$, 20, and 30 respectively. Lines in (e) and (f) show results with $N_n=10$, 30, and 50. Except where explicitly noted, the parameters are set at their default values as in Table 1.

Figure 6

Schematic of a neural network used to describe the backpropagation algorithm. ${\omega }_{jk}^l$ denotes the weight of the $j\mathrm{th}$ neuron in the $l\mathrm{th}$ layer to the $k\mathrm{th}$ neuron in the ($l-1$)th layer.

Figure 7

The averaged absolute error $\mathrm{\Delta }$ for the exponent $\alpha$ as functions of the number of epochs. There are four hidden layers with each layer 25 neurons, namely, $N_n=25$. The black line corresponds to a training set with 5000 entries while the red (gray) one corresponds to 10 000 entries.

Figure 8

The averaged relative error $\delta$ for the noise amplitude $A{t}_0$ as a function of the number of epochs. There are four hidden layers with each layer 25 neurons, namely, $N_n=25$. The black line corresponds to a training set with 5000 entries while the red (gray) one corresponds to 10 000 entries.