Method for noise-induced regularization in quantum neural networks

TL;DR

It is demonstrated that noise levels in quantum hardware can be effectively tuned to enhance the ability of quantum neural networks to generalize data, acting akin to regularisation in classical neural networks.

Abstract

In the current quantum computing paradigm, significant focus is placed on the reduction or mitigation of quantum decoherence. When designing new quantum processing units, the general objective is to reduce the amount of noise qubits are subject to, and in algorithm design, a large effort is underway to provide scalable error correction or mitigation techniques. Yet some previous work has indicated that certain classes of quantum algorithms, such as quantum machine learning, may, in fact, be intrinsically robust to or even benefit from the presence of a small amount of noise. Here, we demonstrate that noise levels in quantum hardware can be effectively tuned to enhance the ability of quantum neural networks to generalize data, acting akin to regularisation in classical neural networks. As an example, we consider two regression tasks, where, by tuning the noise level in the circuit, we demonstrated improvement of the validation mean squared error loss. Moreover, we demonstrate the method's effectiveness by numerically simulating quantum neural network training on a realistic model of a noisy superconducting quantum computer.

Figures (5)

• Figure 1: Scheme of the method for regularising quantum neural network models proposed in the work. Blue rectangles indicate single and two-qubit gates. Orange circles after each gate indicate the action of induced controllable noise with the strength optimized as a regularisation hyperparameter.
• Figure 2: The ansatz circuit used to construct the QNN model, with alternating encoding and trainable layers, followed by a Pauli-Z measurement of the first qubit. Features are encoded with $R_x$ gates. The trainable layers are composed of a series of general rotation gates $R(\theta_1,\theta_2,\theta_3)$ followed by a layer of two-qubit $R_{xx}$ gates. The encoding and trainable layers are repeated $L$ times such that the input data is re-uploaded in each encoding layer. The measured qubit undergoes a parameterized general rotation before the measurement.
• Figure 3: Mean MSE of the training (left) and validation (right) averaged over 20 training trajectories for the diabetes dataset. The plot shows results for QNN with various representative levels of $\gamma$ for each individual injected noise: (a) amplitude damping, (b) phase damping, and (c) depolarizing noise channels. The color of the lines indicates the noise level as given by the color bar. Trajectories with non-vanishing noise levels consistently achieve lower validation losses than those with vanishing noise.
• Figure 4: Average value of the minimum validation loss obtained during training, averaged over 20 trajectories, together with the training loss at the same epochs, plotted versus the noise rate $\gamma$ for (a) the diabetes dataset and (b) the concrete compressive strength dataset. Three types of noise are considered, as indicated by the subplot titles: Amplitude-damping (AD), Phase-damping (PD), and Depolarizing (DP). The black line with triangular markers shows the trace of the Fisher Information (FI) matrix averaged over 100 random weights. The error bars indicate the standard error of the mean across trajectories. The minimum values of the validation loss are found at a non-vanishing value of the noise level.
• Figure 5: Training (a) and validation (b) losses from noisy simulations of a QNN on the IBM kingston processor using the diabetes dataset. Idle waiting times $t_\mathrm{wait}$ are varied as indicated by line color. Panels (c) and (d) show the average minimum validation loss and the corresponding training loss versus $t_\mathrm{wait}$ and the strength of stochastic $x$–rotations $\sigma$, respectively. Results are averaged over 20 trajectories trained for $10^5$ epochs; error bars indicate the standard error of the mean. Both approaches yield nonzero optimal values of $t_\mathrm{wait}$ and $\sigma$, demonstrating the potential of noise-induced regularization on realistic hardware.