Efficient quantum machine learning with inverse-probability algebraic corrections

TL;DR

Quantum neural networks provide expressive probabilistic models but training is hindered by oscillatory loss landscapes and hardware noise. The authors propose inverse-probability algebraic learning, updating parameters via a pseudo-inverse of the Jacobian in probability space, avoiding gradient-based optimization and learning-rate tuning. In a teacher–student QNN benchmark with finite-shot sampling and dephasing noise, the algebraic method achieves faster convergence and lower final error than gradient descent or Adam, with error scaling following the shot-noise limit of $1/S$ as the number of shots $S$ increases. This principled training paradigm is particularly well suited to near-term quantum devices, offering robust, efficient learning under realistic hardware constraints.

Abstract

Quantum neural networks (QNNs) provide expressive probabilistic models by leveraging quantum superposition and entanglement, yet their practical training remains challenging due to highly oscillatory loss landscapes and noise inherent to near-term quantum devices. Existing training approaches largely rely on gradient-based procedural optimization, which often suffers from slow convergence, sensitivity to hyperparameters, and instability near sharp minima. In this work, we propose an alternative inverse-probability algebraic learning framework for QNNs. Instead of updating parameters through incremental gradient descent, our method treats learning as a local inverse problem in probability space, directly mapping discrepancies between predicted and target Born-rule probabilities to parameter corrections via a pseudo-inverse of the Jacobian. This algebraic update is covariant, does not require learning-rate tuning, and enables rapid movement toward the vicinity of a loss minimum in a single step. We systematically compare the proposed method with gradient descent and Adam optimization in both regression and classification tasks using a teacher-student QNN benchmark. Our results show that algebraic learning converges significantly faster, escapes loss plateaus, and achieves lower final errors. Under finite-shot sampling, the method exhibits near-optimal error scaling, while remaining robust against intrinsic hardware noise such as dephasing. These findings suggest that inverse-probability algebraic learning offers a principled and practical alternative to procedural optimization for QNN training, particularly in resource-constrained near-term quantum devices.

Figures (4)

• Figure 1: Illustration of the quantum neural network and the proposed learning method. a. The architecture of the quantum neural network composed of a variational quantum circuit. b. The description of the gradient descent learning in a classical neural network. c. The description of the gradient descent and the proposed learning methods in a quantum neural network.
• Figure 2: Comparison of the gradient descent methods and the proposed learning method. a. Loss history for a classification task with binary cross-entropy (BCE) loss. c. Loss history for a regression task with mean-squared error (MSE) loss. The shaded area indicates the standard deviation of ten ensemble results.
• Figure 3: Effect of the shot noise in the gradient descent methods and the proposed learning method. a. The final loss values with respect to the number of sampling shots. The vertical errorbars indicate the standard deviations of ten ensemble results. b. A selected loss history for Adam. c. A selected loss history for the proposed learning method.
• Figure 4: Effect of the intrinsic errors in the gradient descent methods and the proposed learning method. a. The final loss values with respect to the dephasing error rates. The vertical errorbars indicate the standard deviations of ten ensemble results. b. Selected loss histories for Adam. c. Selected loss histories for the proposed learning method.