Subtleties in the trainability of quantum machine learning models

Abstract

A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called Quantum Machine Learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called Variational Quantum Algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.

Group-Invariant Quantum Machine Learning

“…Similar to their classical counterparts, the ability of QML models to solve a given task hinges on several factors, with one of the most important being the choice of the model itself. If the inductive biases [14] of a model are uninformed, its expressibility is large, leading to issues such as barren plateaus in the training landscape [15][16][17][18][19][20]. Adding sharp priors to the model narrows the effective search space and increases its performance [21][22][23].…”

Exponential data encoding for quantum supervised learning

“…) for some α > 1 [61], this means that ε = O(α −MN/2 ) so that estimations are not mere random guesses [41]. Finally, when exponential encoding is used, we get N gt < O 3 MN(1−log 3 α)/2 , telling us that a feasible quantum advantage entails α < 3.…”

Fock state-enhanced expressivity of quantum machine learning models

“…Similar to the standard noisy and large scale variational circuits, the variational machine learning approach becomes more difficult to train as the dimension of the Fock space increases, likely due to the issue of vanishing cost function gradients [60][61][62][63][64], requiring exponentially-growing precision to optimize the circuit parameters in-situ [65]. In addition, it is expensive to train the quantum gates (in this case the tunable beam splitter meshes) in the noisy-intermediate scale quantum (NISQ) era as it is time-consuming to The classification boundaries for all datasets become more complicated as the number of input photons increases, illustrating the increasing expressive power.…”