Quantum neural network equipped with backpropagation on a qudit processor

AI summary

Overview

Research area: Quantum machine learning, specifically quantum neural networks (QNNs) implemented on multi-level quantum systems (qudits) using trapped ions. Technical level: Advanced. Scope: This paper reports an experimental demonstration of a backpropagation-trained QNN encoded on a single four-level trapped-ion qudit and applies it to a binary image-classification task.

What This Paper Is About

QNNs are limited by the size of the Hilbert space available on the quantum processor. Qudits, which have more than two levels, provide a larger state space for a given number of physical units and can therefore support more expressive networks. The paper designs and experimentally realizes such a qudit-based QNN on a single trapped ^40Ca^+ ion, training it with a hybrid quantum-classical backpropagation procedure and testing it on Fashion-MNIST images.

Key Contributions

1. A qudit-based QNN architecture that combines amplitude encoding, parameterized transitions between adjacent qudit levels, and a trainable circuit organized into transition stairs for multi-class classification.
2. An experimental implementation on a single trapped ^40Ca^+ ion using a four-dimensional qudit (d = 4), with an ancillary level used to read out amplitude signs.
3. A hybrid quantum-classical backpropagation protocol in which intermediate data vectors are measured on the qudit processor while error vectors and gradients are computed on a classical device; the paper also outlines a possible fully quantum extension.
4. A comparison against baseline models, including a linear support vector machine and a two-qubit QNN, plus a numerical study of robustness to parameter initialization.

Main Findings

• Experimental test accuracy: The qudit QNN reached 95.7% classification accuracy on a 395-sample test set containing 197 trousers and 198 pullovers from Fashion-MNIST.
• Numerical comparison: A noise-free numerical simulation using the experimentally trained parameters reached 96.7% test accuracy after the 13th epoch.
• Training improvement: Experimental classification accuracy rose from 45.6% before training to 95.7% after 13 epochs.
• Training setup: The training set contained 30 samples (15 trousers and 15 pullovers); 15 samples were randomly selected per epoch, with learning rate λ = 0.2 and regularization weight γ = 0.5.
• Data compression: Principal Component Analysis compressed the original 28 × 28 images to 4 × 1 vectors; the first four principal components accounted for 73.69% of the explainable variance in the training set.
• Hardware performance: Single-qudit T(·) operation fidelities exceeded 99%, and the state preparation and measurement error for dark states was approximately 0.2%.
• Readout statistics: Fluorescence detection collected an average of 36.4 (2.7) counts over 300 μs; 1500 shots were used for population measurements and 600 shots for each sign measurement.
• Measurement precision: Finite sampling gave a standard deviation of about 0.013 in the absolute values of the output vector and intermediate data vectors with 1500 shots.
• Baseline comparison: Linear SVM achieved 100.0% training and 97.2% test accuracy; the two-qubit QNN achieved 100.0% training and 95.4% test accuracy; the single-qudit QNN simulation achieved 96.7% training and 96.7% test accuracy; the single-qudit QNN experiment achieved 100.0% training and 95.7% test accuracy.
• Initialization robustness: Across 1000 distinct random parameter initializations, the QNN consistently converged to test accuracy exceeding 95%, with average loss stabilizing around 0.5.
• Output separation: The amplitudes on levels |3⟩ and |4⟩ were used to compute label probabilities, and training produced a distinct classification boundary between the two image classes.

Methodology in Plain English

The researchers stored information in several internal energy levels of a single trapped ^40Ca^+ ion rather than in a two-level qubit. Four levels encoded the data vector, and a fifth ancillary level was used to determine the signs of the amplitudes. Input images were first compressed to four numbers using Principal Component Analysis and then mapped onto the qudit through a sequence of laser-driven transitions.

The trainable part of the network consisted of layers of rotations between neighboring energy levels. After the circuit, the amplitudes on a chosen subset of levels were converted into class probabilities using a softmax function, and the network was trained by minimizing a cross-entropy loss with an added regularization term.

For backpropagation, the circuit was split into sublayers. The researchers measured the intermediate data states on the ion, then used the chain rule on a classical computer to propagate the output error backward through the sublayers and calculate parameter gradients. The parameters were updated iteratively by gradient descent. The experiment used a narrow-linewidth 729 nm laser, controlled with an acousto-optic modulator, to drive transitions, and a sequential measurement scheme to read out all five level populations in one experimental sequence.

Why This Matters

This work shows that multi-level quantum hardware can be used to build and train neural networks with a small number of physical components. It connects qudit control techniques with gradient-based quantum machine learning and provides an experimental template that could be adapted to other qudit-capable platforms.

Real-world applications:

• Image classification and pattern recognition on near-term quantum processors.
• Quantum data learning, where inputs are themselves quantum states.
• Classification tasks in quantum error correction and quantum state identification.
• Resource-constrained quantum machine learning, where qudits replace multiple qubits.

Industry relevance: The results are relevant to quantum hardware developers working with trapped ions, photonics, and superconducting circuits, as well as to teams exploring quantum machine learning algorithms for near-term devices. The demonstrated hybrid training loop also points to practical ways of combining quantum hardware with classical optimization infrastructure.

Future Directions

• Adding nonlinearity to build deeper QNNs, for example through block encoding based on quantum singular value transformation or through classical computation of activation functions.
• Using qudit shadow estimation to reduce the measurement cost of backpropagation during training.
• Extending the architecture from a single qudit to multi-qudit systems for scalable implementations.
• Realizing fully quantum backpropagation by encoding error vectors as quantum states and propagating them backward through the inverse circuit.
• Investigating and mitigating the magnetic-field noise identified as the main source of discrepancy between experimental and numerical results.

Target Audience

This paper is most useful to researchers in quantum information and quantum machine learning, trapped-ion experimentalists, graduate students working on qudit control or variational quantum algorithms, and hardware developers interested in using multi-level systems for near-term quantum applications.

Authors’ abstract

Quantum neural networks (QNNs), one of the fundamental algorithms in quantum machine learning, have been widely used in classification and identification tasks. However, the capabilities of QNNs are constrained by their size, which is determined by the dimension of the Hilbert space of the underlying quantum processor. Multi-level quantum digits (qudits) offer access to a higher-dimensional Hilbert space compared to two-level qubits, enabling the construction of more expressive QNNs. In this work, we report an experimental demonstration of qudit-based QNN using a trapped $\rm ^{40}Ca^+$ ion. We train the QNN using a hybrid quantum-classical implementation of backpropagation and achieve an experimental classification accuracy of $95.7\%$ on a test image set. This demonstration highlights the potential of qudit-based processors to QNN architectures and provides a framework for implementing qudit-based QNNs across various quantum devices.