Training single-electron and single-photon stochastic physical neural networks
Authors: Tong Dou, Shiro Kumara, Josh Burns, Ethan Sigler, Parth Girdhar, David Petty, Gerard Milburn, Jo Plested, Matt Woolley
Organizations: School of Engineering and Technology, UNSW Canberra, Campbell, ACT 2600, Australia · Sussex Centre for Quantum Technologies, University of Sussex, Brighton BN1 9RH, United Kingdom · Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom · National Quantum Computing Centre, Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom · School of Systems and Computing, UNSW Canberra, Campbell, ACT 2600, Australia
The computational demands of deep learning motivate the investigation of alternative approaches to computation. One alternative is physical neural networks~(PNNs), in which learning and inference are performed directly via physical processes. Stochastic PNNs arise when the underlying neurons are realized by the dynamics of a stochastic activation switch. Here we propose novel electronic and photonic stochastic neurons. The electronic realization is implemented by single-electron tunneling through a quantum dot. The photonic realization is implemented via a single-photon source driving one of two modes coupled via a controllable beam-splitter-like interaction. In the electronic case, the charge state of the quantum dot forms the basis for the stochastic neuron, whereas in the photonic case the occupation of the undriven mode serves as the basis for the stochastic neuron. Training of stochastic PNNs is performed with models of stochastic neurons, as well as with coherently-driven, single-photon detector stochastic neurons previously introduced. Several training strategies for MNIST handwritten digit classification have been investigated using single-hidden-layer stochastic PNNs, including varying the number of trials in each layer to control forward pass stochasticity and employing either true probability or empirical outputs in the backward pass to evaluate their influence on gradient estimation. We show that when empirical outputs are used in the backward pass, the network achieves more than 97% test accuracy with few trials per layer. Despite the simplicity of the model architecture, high test accuracy is maintained in the presence of a high degree of noise and model uncertainty. The results demonstrate the potential of embracing stochastic PNNs for deep learning.
Figures & tables
Figure 1: Schematic of an SPD stochastic neuron adapted from [ 17 ] . The mean photon number λ(z) incident on the detector in a given measurement interval is a function of the pre-activation z . The output of the SPD neuron is modeled as the stochastic output from the single-photon detector, given by h=fSPD(z) , where h=0 corresponds to no-clock and h=1 corresponds to a click.
Figure 2: Schematic of a SET stochastic neuron realized in a semiconductor quantum dot system. The pre-activation z modulates the dot energy level ε(z) via the gate electrode. Electron tunneling between the dot and leads (source and drain) is governed by the tunneling rates ΓS/D± , where the total electron tunneling rates in and out of the dot are defined as Γin=ΓS++ΓD− and Γout=ΓD++ΓS− , respectively. The resulting dot occupation can be modeled as the output of the SET stochastic neuron, which is monitored by the current IQPC of a nearby quantum point contact (QPC), provided by h=fSET(z) , where h=0 denotes the unoccupied state ( N−1 electrons) and h=1 denotes the occupied state ( N electrons) of the quantum dot.
Figure 3: Schematic of a TSP stochastic neuron realized in an optomechanical system. The neuron is driven by an incident single-photon pulse ξ(t) with a temporal decay rate ζ . The pre-activation z modulates the interaction between the cavity mode a and the mechanical mode b via the time-dependent coherent control pulse α(t) . The system evolution is subject to the cavity decay rate κ and mechanical dissipation γ . Driven by the control pulse, the single-photon state is stochastically transferred to the mechanical mode, whose occupation at any time t is at most unity, providing a stochastic binary neuron output. The neuron activation can be obtained by measuring the mechanical occupation number, provided by h=fTSP(z) , where h=0 denotes the unoccupied state and h=1 denotes the occupied state of mode b .
Figure 4: Benchmarking stochastic physics-aware training on SPD, SET, and TSP stochastic neurons. The TP approach is applied to both the hidden and output layers. Inset: Activation probabilities pPSN(z) as functions of the pre-activation z for corresponding PSNs. The curves are plotted over different z -ranges: [−3,3] for SPD, [−5,5] for SET, [−50,50] for TSP, and for TSP neuron, parameters are set to t=0.21 , γ=0.02 , κ=30.0 , ζ=10.7 . All three PSNs support effective training even with only 1 trial in the hidden layer, and test accuracy improves as the number of trials increases.
Figure 5: Performance of the EG estimator under finite sampling constraints compared to the TP baseline. The EG estimator applied to the hidden layer only and to both layers enables effective training under finite-sampling constraints ( K≤10 ). Test accuracy quickly approaches TP upper bound as the trial count increases. K=1 is omitted in all plots for the EG estimators in the hidden layer dues to vanishing empirical gradients.
Figure 6: Performance of ST estimators under finite sampling. Applying the identity ST estimator to the hidden layer limits the learning capacity, with test accuracy saturating below roughly 93% across all evaluated output configurations. In contrast, applying the EG estimator in the hidden layer while retaining a ST estimator in the output layer yields highly competitive performance ( >98% ). The configuration applying the TP approach in both hidden and output layers is a baseline. Note that for K=1 , the EG estimator at the output layer is enabled by introducing a smoothing parameter ϵ (detailed in Sec. III.4 ).
Figure 7: Output-layer performance under infinite and few trial regimes. Finite output-layer sampling remains effective under strict trial-budget constraints ( K≤10 ). Test accuracy approaches the infinite-trial limit as the trial count increases.
Figure 8: Comparison of test accuracy using softmax and unnormalized linear activations in the output layer for models with (a) a single hidden layer and (b) two hidden layers. While the linear and MSE formulation underperforms in shallow networks, increasing the network depth to 2 hidden layers substantially improves its accuracy, making it comparable to the 1-layer softmax and CE configuration.
Physical noise in near-term quantum hardware is usually treated as a nuisance to suppress. We ask whether it can instead act as a hardware-native regularizer for photonic hybrid quantum-classical neural networks (PHQCNNs), analogous to noise-injection regularization in classical deep learning. Using Quandela's Perceval simulator and the MerLin framework, we build PHQCNNs for Iris, Digits, and MNIST and inject Perceval's seven-parameter physical noise model directly into training. A genetic algorithm searches the six continuous noise dimensions and 1 boolean parameter to find, per dataset, the configuration maximizing validation accuracy, compared against a noiseless baseline across five seeds. GA-tuned noise yields modest accuracy gains on Iris (+0.82pp) and Digits (+1.45pp), but a clear degradation on MNIST (-1.21pp). Per-parameter sweeps show that no individual noise parameter is consistently beneficial, motivating the joint search, while a second-order loss expansion shows that physical noise induces a Tikhonov-like regularization term whose effect is dataset-dependent. Physical photonic noise can thus act as a free regularizer, but not universally.
Farah Elnakhal, Alberto Marchisio, Nouhaila Innan +2
Science Division, New York University Abu Dhabi, UAE · eBrain Lab, Division of Engineering, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE · Center for Quantum and Topological Systems, NYUAD Research Institute, New York University Abu Dhabi, UAE +1
While backpropagation and automatic differentiation have driven deep learning's success, the physical limits of chip manufacturing and rising environmental costs of deep learning motivate alternative learning paradigms such as physical neural networks. However, most existing physical neural networks still rely on digital computing for training, largely because backpropagation and automatic differentiation are difficult to realize in physical systems. We introduce FFzero, a forward-only learning framework enabling stable neural network training without backpropagation or automatic differentiation. FFzero combines layer-wise local learning, prototype-based representations, and directional-derivative-based optimization through forward evaluations only. We show that local learning is effective under forward-only optimization, where backpropagation fails. FFzero generalizes to multilayer perceptron and convolutional neural networks across classification and regression. Using a simulated photonic neural network as an example, we demonstrate that FFzero provides a viable path toward backpropagation-free in-situ physical learning.
Yaqi Guo, Fabian Braun, Bastiaan Ketelaar +3
Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, Delft, 2628CD, The Netherlands · Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, Delft, 2628CD, The Netherlands · Department of Intelligent Systems, Delft University of Technology, Van Mourik Broekmanweg 6, Delft 2628XE, The Netherlands
Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is of fundamental importance. In this work, we establish the linear convergence of stochastic gradient descent / flow in training over-parameterized two layer PINNs with a general class of activation functions for solving one model second-order elliptic problem, i.e., the Poisson equation, in the sense of high probability. These results extend the existing result [20] in which gradient descent was analyzed. The challenge of the analysis lies in handling the dynamic randomness introduced by stochastic optimization methods. The key of the analysis lies in ensuring the positive definiteness of suitable Gram matrices during the training. The analysis sheds insight into the dynamics of the optimization process, and provides guarantees on physics informed neural networks trained by stochastic algorithms.
Bangti Jin, Longjun Wu
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China