Can Geometric Quantum Machine Learning Lead to Advantage in Barcode Classification?

TL;DR

This work develops a geometric quantum machine learning (GQML) approach with embedded symmetries that allows for the classification of similar and dissimilar pairs based on global correlations, and enables generalization from just a few samples.

Abstract

We consider the problem of distinguishing two vectors (visualized as images or barcodes) and learning if they are related to one another. For this, we develop a geometric quantum machine learning (GQML) approach with embedded symmetries that allows for the classification of similar and dissimilar pairs based on global correlations, and enables generalization from just a few samples. Unlike GQML algorithms developed to date, we propose to focus on symmetry-aware measurement adaptation that outperforms unitary parametrizations. We compare GQML for similarity testing against classical deep neural networks and convolutional neural networks with Siamese architectures. We show that quantum networks largely outperform their classical counterparts. We explain this difference in performance by analyzing correlated distributions used for composing our dataset. We relate the similarity testing with problems that showcase a proven maximal separation between the BQP complexity class and the polynomial hierarchy. While the ability to achieve advantage largely depends on how data are loaded, we discuss how similar problems can benefit from quantum machine learning.

Figures (5)

• Figure 1: Sketch of the machine learning workflows for distinguishing images (barcodes) based on their correlations. Pairs of barcodes are processed by a classical learner via Siamese deep neural networks. A quantum learner processes data embedded into quantum states via parametrized equivariant measurements, classifying barcodes as correlated or uncorrelated.
• Figure 2: Architectures for geometric quantum machine learning used in the study. (a) Quantum neural network approach based on equivariant unitary circuit (QNN$_{\mathcal{U}}$) parametrized by a vector $\bm{\theta}$ that processes a pair of barcodes encoded into quantum states. (b) QNN$_{\mathcal{M}}$ based on equivariant measurements that are weighted adaptively to form a model for correlated sample processing.
• Figure 3: Comparison of the training performance (a) and generalization (b) between the two GQML architectures, $\text{QNN}_\mathcal{U}$ and $\text{QNN}_\mathcal{M}$. Both models were trained and tested on pairs of barcodes with 16 pixels each, corresponding to an 8-qubit quantum state for each input datum. Solid/dashed lines indicate the mean values across the 10 trials, while the shaded region indicates the standard deviation.
• Figure 4: Generalization of quantum and classical learners for the barcode similarity problem shown as a function of the number of samples from each class used in training. We show final training loss (a) and test accuracy (b) of quantum and classical networks, trained and tested on pairs of 1024-pixel barcodes (20-qubit system). Classical benchmarks correspond to deep (DNN) and convolutional (CNN) Siamese neural networks, which learn the similarity via a metric distance evaluated in feature space (see details in the main text).
• Figure 5: Generalization of learners for the barcode similarity problem shown as a function of system size (number of qubits $n$ for each image, $2n$ in total). Both, training loss (a) and test accuracy (b) remain steady across all system sizes, indicating that the quantum--classical separation is ubiquitous for smaller and larger pairs of barcodes. The plots show simulations run with the same parameters as described in Fig. \ref{['fig:generalization']}, with 5 training samples taken from each class ($M=10$).