Encoding-dependent generalization bounds for parametrized quantum circuits

Abstract

A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

Grover search revisited; application to image pattern matching

“…In contrast, with PQC, the depth and the number of multi qubit gate seems not to increase so fast, i.e., O(poly(n)) in depth and O(n) of multi qubit gates in nqubit system, although PQC does not guarantee a very precise data loading. In fact the power of expressibility of various type of PQC is under actively investigated [35][36][37][38]. For instance we find ansatz with higher expressibility containing a circuit-block or an all-to-all entangler, depending on the hardware architecture.…”

Recent advances for quantum classifiers

“…Second, since the noise in the quantum devices seems inevitable, increasing the fidelity of the quantum operations and developing better error correction techniques are of crucial importance. Third, for the variational quantum classifiers, there are already some works trying to measure their representation power [185][186][187][188][189][190]. To find whether there is a separation between these models and their classical counterparts for potential future applications, further studies are highly desirable.…”

Sample complexity of learning parametric quantum circuits

“…A natural problem is then to generalize the PAC learning model to quantum learning scenarios. Indeed, notable progress has been made along this direction [27][28][29][30][31][32][33][34][35][36][37]. For example, in reference [28] Chung and Lin have studied the sample complexity of learning quantum channels and demonstrated that we can PAC-learn a polynomial-size quantum circuit with a polynomial number of samples.…”