Beyond Optimization: Harnessing Quantum Annealer Dynamics for Machine Learning

TL;DR

It is demonstrated that short annealing times yield higher classification accuracy, while longer times reduce accuracy but lower sampling costs, and the participation ratio is introduced as a measure of the effective model size and shows its strong correlation with generalization.

Abstract

Quantum annealing is typically regarded as a tool for combinatorial optimization, but its coherent dynamics also offer potential for machine learning. We present a model that encodes classical data into an Ising Hamiltonian, evolves it on a quantum annealer, and uses the resulting probability distributions as feature maps for classification. Experiments on the quantum annealer machine with the Digits dataset, together with simulations on MNIST, demonstrate that short annealing times yield higher classification accuracy, while longer times reduce accuracy but lower sampling costs. We introduce the participation ratio as a measure of the effective model size and show its strong correlation with generalization.

Figures (3)

• Figure 1: Overview of the QA-based machine learning model and experimental results. (a) Schematic of our quantum machine learning (QML) model implemented on a QA, e.g., the D-Wave Advantage System 7.1. Here, the PCA-based dimensionality compression is expressed as a linear transformation using the matrix $W_\mathrm{PCA}$, and assign each element of this 20-dimensional vector $\boldsymbol{x}'$ to the coupling constants of the final Hamiltonian $\hat{H}_\mathrm{AS}$ of the QA. (b)Structure of the Pegasus graph and a representative subgraph used. (c) Classification accuracy as a function of annealing time for different values of the hyperparameter $\gamma$. We collected 1,000 shots for each image and performed the learning procedure ten times. The solid, dashed, and dash-dotted curves correspond to $\hat{H}_\mathrm{AS}$ and $\gamma\hat{H}_\mathrm{AS}$ cases with $\gamma = 0.25$, and $0.5$, respectively. Square, diamond, and star markers denote experimentally obtained accuracy rates for $\gamma = 0.25, 0.5$. (d) Accuracy rates for $\gamma = 0.25$ in the presence of Hamiltonian randomness. For each image, the output probability distribution was averaged over 100 random realizations of the Hamiltonian. Using these averaged probabilities, we simulated measurements with 1,000 shots. The blue and orange dash-dotted lines show the averaged training and testing accuracy, respectively, each averaged over 10 independent measurement simulations. The colored areas indicate the corresponding standard deviations. (e) Testing accuracy versus number of shots for simulation and experiment. Testing accuracy rates obtained from both numerical simulations and experiments are plotted as a function of the number of measurement shots $M$. Star-shaped points denote the experimentally measured testing accuracy. Dash-dotted curves correspond to the accuracy obtained from numerical simulations including Hamiltonian randomness. The curves represent averages while the shaded areas indicate the associated standard deviations, shown here explicitly only for $M = 250$ and $M = 2000$. In (e)-(d), the red solid line shows the peforamnce of the linear model (without QA).
• Figure 2: Classification performance under finite-shot conditions and associated error. Classification results are shown for two annealing times: $T = 2.0$ (a) and $T = 8.0$ (b). For each case, panel (1) presents the test accuracy using the full theoretical distribution (solid lines), while panel (2) shows the absolute difference between performance based on theoretical versus estimated distributions for various numbers of qubits $N$. In panel (2), the vertical axis corresponds to the performance error, and the horizontal axis is $1/\sqrt{N_s}$.
• Figure 3: Relationship between APR and annealing time for the MNIST dataset. (a) Direct relationship between APR and annealing time. The positions corresponding to $T=2.0$ and $T=8.0$ are highlighted in gray. (b) Relationship between APR and annealing time rescaled by the number of qubits. The result for $T=1.0$, highlighted in gray, clearly deviates from the rest of the data, indicating that insufficient quantum dynamics occur at such short annealing times.