Using Deep Learning to Understand and Mitigate the Qubit Noise Environment

Understanding the spectrum of noise acting on a qubit can yield valuable information about its environment, and crucially underpins the optimization of dynamical decoupling protocols that can mitigate such noise. However, extracting accurate noise spectra from typical time-dynamics measurements on qubits is intractable using standard methods. Here, we propose to address this challenge using deep-learning algorithms, leveraging the remarkable progress made in the field of image recognition, natural language processing, and more recently, structured data. We demonstrate a neural-network-based methodology that allows for extraction of the noise spectrum associated with any qubit surrounded by an arbitrary bath, with significantly greater accuracy than the current methods of choice. The technique requires only a two-pulse echo decay curve as input data and can further be extended either for constructing customized optimal dynamical decoupling protocols or for obtaining critical qubit attributes such as its proximity to the sample surface. Our results can be applied to a wide range of qubit platforms, and provide a framework for improving qubit performance with applications not only in quantum computing and nanoscale sensing but also in material characterization techniques such as magnetic resonance.

Popular Summary

Understanding the environment surrounding quantum systems is key to being able to use them as quantum sensors or qubits, but doing so accurately is challenging using traditional mathematical and experimental techniques. Our results show how machine-learning techniques and specifically neural networks can be used to infer the qubit environment accurately using only simple and readily performed experimental measurements. This is important because it allows the design of bespoke control sequences to protect fragile quantum states from their particular environments for much longer than would otherwise be the case.

Once the specific nature of the qubit environment is known, then it is straightforward to predict how a qubit will evolve over its lifetime. However, it is challenging to do the reverse: to deduce the environment from a measurement of the qubit. By simulating a series of real-world noise environments and the qubit evolution we are able to teach a neural network how the evolution measurement corresponds to its noise environment. This enables the network to successfully predict the noise environment of a new qubit using only a simple measurement of its evolution, which is readily performed in the lab. We use similar techniques to erase unsystematic experimental noise without sacrificing underlying information of the qubit environment, providing an effective route to implement our toolbox using experimental data.

This work represents an early effort to utilize deep learning to understand qubit decoherence and thus has the potential to bring together two growing communities dealing with quantum technologies and artificial intelligence.

Figure 1

The challenge of extracting noise spectra from coherence-decay measurements. (a) Decoherence curves under dynamical decoupling sequences (see inset) as a function of total sequence duration, $n\tau$, simulated using Eq. (2) from some underlying noise spectrum [dashed curve in (b)]. Solid curves in (b) show noise spectra extracted from the coherence decays using the $\delta$-function approximation to the filter function in Eq. (3). (c),(d) Actual filter functions corresponding to sequences comprising one and four $\pi$ pulses at two $\tau$ values. Simulations assume a $\pi$-pulse duration of 100 ns.

Figure 2

Our neural-network approach. (a) Flow chart of the methodology. (b) A detailed view of a long short-term memory (LSTM) cell. $c_t$ is the cell state vector (providing the memory of the network), $x_t$ the cell input (one point in a coherence decay), and $h_t$ the cell output, which is fed both to the output dense layer and the next cell in the series. Red boxes represent dense layers with either sigmoid ($\sigma$) or tanh activations. Blue circles represent pointwise operations. (c) A sketch of the whole network: points on the coherence decay are fed into the LSTM layer, the output of the LSTM layer is then input into a dense layer and the ouptut of this dense layer is compared with a target noise spectrum. (d) An example network training graph showing training loss and validation loss (mean absolute percentage error) over training.

Figure 3

Comparison of two approaches for producing a noise spectrum from a measured coherence-decay curve. Stacked histograms of errors in the predicted noise spectra from the holdout test data set are shown using (a) a $\delta$-function approximation, and (b) our neural network. Three different forms of noise spectrum are studied, each with approximately $8500$ individual noise spectra and associated decay curves. In (b), the tail of the histogram is clipped for clarity and the maximum error is 2.3%. Also shown are the true (solid curves) and predicted (dashed curves) noise spectra obtained under (c) the $\delta$-function approximation and (d) the neural-network approach. The noise spectra used are at the 50th percentile error for each approach.

Figure 4

Comparison between estimated error in noise-spectra predictions for three different neural networks and those obtained using a $\delta$-function approximation. Histograms of error rates for predicting noise spectra generated using Eq. (3) from coherence decays with a coherence-decay time constant $T_2$ between $10-140\mu \mathrm{s}$ obtained by means of (a) the $\delta$-function approach and (b) a trained neural network. Stacked histograms of error rates for predicting noise spectra having three different functional forms: $1/f$, Lorentzian, and double Lorentzian obtained by means of (c) the $\delta$-function approach and (d) a third trained neural network, the tail of the neural-network distribution is clipped for clarity with maximum error in this case 200.1% with high errors caused by noise spectra with large portions very close to 0 where small deviations in predicted noise spectrum magnify mean percentage error. Histograms of error rates for predicting stretched-exponential noise spectra generated from coherence decays constructed by implementing dynamical decoupling protocol comprising 32 $\pi$ pulses extracted by means of (e) the $\delta$-function approach and (f) a second trained neural network; inset in (e) shows mean percent errors along with the maximum deviation as a function of the number of $\pi$ pulses for the conventional technique.

Figure 5

Comparison of the neural-network approach with an established approach. (a) Histograms of errors in noise extraction when employing Alvarez et al. methodology [9], (b) histograms of errors in predicting noise via the trained neural network, (c) a plot comparing predictions of different methodologies for a representative noise spectrum. The inset shows the decoherence curve corresponding to the underlying noise spectrum. Note: all the artificial noise spectra are generated using a $1/f$ model along with a Lorentzian-shaped feature whose parameters are chosen randomly.

Figure 6

Denoising the experimental data using curve fitting or neural networks. Stacked histograms of errors in smoothing synthetic experimental decoherence curves from four classes of noise model are shown using two denoising approaches: (a) curve fitting to Eq. (4), or (b) the neural-network approach. In both cases, insets show mean errors and the maximum deviation. The percent errors are calculated using logarithmic values of decoherence curve. (c) A synthetic noisy decoherence curve (from a $1/f$ noise model) along with a denoised curve obtained from the neural network.

Figure 7

Potential applications of the methodology include defect depth prediction [23] and the generation of customized dynamical decoupling sequences. (a) Simulated double-Lorentzian-type noise-spectra characteristic of near-surface spin defects, for different increasing depths (i–iii). (b) Defect-bath coupling strengths for low- and high-frequency noise corresponding to noise spectra shown in (a); the variations in the coupling strengths are indicative of changes in the defect position with respect to the surface. Generation of optimized dynamical decoupling sequences: (c) specimen decoherence curves derived by applying 32 $\pi$-pulse CPMG to the noise spectra shown in (d). (e) Histograms demonstrating coherence improvement after implementing the SLSQP algorithm to optimize $\pi$-pulse positions; enhancements obtained by employing UDD protocol [6] are also included for comparison. Extracted optimal protocols corresponding to two histograms denoted by $◯1$ and $◯2$ are shown alongside the bar plot, illustrating an increasing divergence from the equivalent CPMG sequence as coherence decreases. Initial coherence refers to the value of coherence for a 32-pulse CPMG sequence, absolute enhancement refers to the increased coherence at the same point in time achieved with a UDD and optimized sequence. If the qubit had an initial coherence of 0.5 at $100\mu \mathrm{s}$ with a CMPG-32 sequence and had a coherence of 0.8 at $100\mu \mathrm{s}$ with an optimized sequence, then the absolute enhancement is 0.3.

Figure 8

Neural network versus $\delta$-function approximation performance for inferring the Lorentzian noise spectrum shown in (b) from the coherence-decay curve shown in (a). The true (target) noise spectrum is shown in solid green in (b), the $\delta$-function approximation result is shown by a dotted pale-green line, the neural-network result is shown by a dashed pale-green line.

Figure 9

Neural-network performance for inferring noise spectra from coherence-decay curves derived from $1/f+$ Lorentzian noise, having been trained only on $1/f$, Lorentzian, and double Lorentzian data. (a) example target noise spectrum in blue, resulting in the coherence-decay curve in the inset, with the approximate noise spectra derived from the $\delta$-function approximation and the untrained neural network. (b) The distribution of errors across a set of data of this functional form. The mean error for the untrained neural network is 11.9% whilst the mean error for the $\delta$-function approximation is 24.4%. The distribution of the untrained neural network has a long tail due to noise spectra with large portions close to 0, where mean percentage error is magnified. (c),(d) The same data, but with the neural network now fine tuned on a small $1/f$ + Lorentzian dataset (entirely separate from the data upon which it is evaluated to maintain train-test split). In this case the mean error drops to 3.5%.

Figure 10

Application of Alvarez-Suter methodology to the simulated data to predict the underlying noise spectrum. (a) Plot demonstrating how $A_k\mathrm{s}$ are associated with the sensitivities of the filter function corresponding to a given pulse delay. (b) Relaxation times extracted via fitting the signal at various pulse delays. The inset shows the actual signal for select pulse delays. Each data point here corresponds to the value of coherence recorded for the given pulse delay and the total duration of the sequence; the total duration can be varied by changing the number of $\pi$ pulses present in the protocol. (c) Comparison between the actual noise spectrum and the predicted noise spectrum.