Quantum adiabatic machine learning by zooming into a region of the energy surface

Recent work has shown that quantum annealing for machine learning, referred to as QAML, can perform comparably to state-of-the-art machine learning methods with a specific application to Higgs boson classification. We propose QAML-Z, an algorithm that iteratively zooms in on a region of the energy surface by mapping the problem to a continuous space and sequentially applying quantum annealing to an augmented set of weak classifiers. Results on a programmable quantum annealer show that QAML-Z matches classical deep neural network performance at small training set sizes and reduces the performance margin between QAML and classical deep neural networks by almost 50% at large training set sizes, as measured by area under the receiver operating characteristic curve. The significant improvement of quantum annealing algorithms for machine learning and the use of a discrete quantum algorithm on a continuous optimization problem both opens a class of problems that can be solved by quantum annealers and suggests the approach in performance of near-term quantum machine learning towards classical benchmarks.

Figure 1

Zooming extension. While QAML only performs one anneal, QAML-Z iteratively updates the weight ${\mu }_i$ (indicated by the dot) of the $i\mathrm{th}$ weak classifier in the strong classifier ensemble by performing a binary search over the energy surface using spin-up and spin-down outcomes.

Figure 2

Area under the ROC curve for the QAML-Z extension, simulated annealing (SA-Z), a logistic regression (LR-Z), the original QAML, a deep neural network (DNN), and XGBoost (XGB) [69] as a function of training set size. While QAML-Z matches DNN performance at small training set sizes, it decreases the margin between QAML and DNN by 47% for the largest training sets. Error bars indicate $1\sigma$ error, including both variation over training sets and statistical error estimated by reweighting samples from a Poisson distribution.

Figure 3

QAML-Z performance on the test set vs zooming iteration number (training set size of 1000). Top: Significant improvements by QAML-Z can be separately seen for classifier augmentation and zooming over the original QAML algorithm. Bottom: Ising model energy on the test set improves monotonically, indicating negligible overfitting. Error bars indicate $1\sigma$ error. Here and in all subsequent figures, the energy is given in units corresponding to the Hamiltonian in Eq. (9), which for our purposes may be considered to be dimensionless.

Figure 4

Comparison of quantum and simulated annealing for the new and original algorithms, measured by area under ROC (AUROC) curve. Although QAML-Z outperforms QAML and SA-Z, the inclusion of excited states in the SAE-Z variant reproduces QAML-Z performance to one standard deviation. Error bars indicate $1\sigma$ error.

Figure 5

Difference between the lowest energy of quantum annealing (QA) and simulated annealing (SA). SA finds a lower minimum energy than QA given identical initial Hamiltonians. Error bars show $1\sigma$ error.