Classical Surrogates for Quantum Learning Models

The advent of noisy intermediate-scale quantum computers has put the search for possible applications to the forefront of quantum information science. One area where hopes for an advantage through near-term quantum computers are high is quantum machine learning, where variational quantum learning models based on parametrized quantum circuits are discussed. In this work, we introduce the concept of a classical surrogate, a classical model which can be efficiently obtained from a trained quantum learning model and reproduces its input-output relations. As inference can be performed classically, the existence of a classical surrogate greatly enhances the applicability of a quantum learning strategy. However, the classical surrogate also challenges possible advantages of quantum schemes. As it is possible to directly optimize the Ansatz of the classical surrogate, they create a natural benchmark the quantum model has to outperform. We show that large classes of well-analyzed reuploading models have a classical surrogate. We conducted numerical experiments and found that these quantum models show no advantage in performance or trainability in the problems we analyze. This leaves only generalization capability as a possible point of quantum advantage and emphasizes the dire need for a better understanding of inductive biases of quantum learning models.

Figure 1

A quantum learning model has a classical surrogate if there exists a process that produces an equivalent classical model that is both efficient in the size of the quantum learning model and the desired approximation parameters.

Figure 2

Training and test loss over epochs for three different learning models trained on a standard synthetic dataset over random splits. The dashed lines indicate the mean loss associated with the empirical risk minimizer. We observe the effects of higher expressivity when the number of parameters of the quantum model is increased but the surrogate model outperforms both quantum models in loss and trainability.