The power of quantum neural networks

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.

Section: Introductionmentioning

confidence: 83%

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias

Gil-Fuster

Meyer

et al. 2021

“…Gradient scaling is one of the few directions of significant progress. The most famous gradient scaling result is the barren plateau phenomenon [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], whereby the gradient of the cost function shrinks exponentially with the number of qubits. Various issues lead to barren plateaus, such as deep ansatzes that lack structure [19,21,31], global cost functions [20,21], high levels of noise [22,32], scrambling target unitaries [24], and large entanglement [28,29].…”

Section: Introductionmentioning

confidence: 99%

Effect of barren plateaus on gradient-free optimization

Arrasmith

Cerezo

Czarnik

et al. 2021

160

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

“…The authors of Ref. [73] have looked at the classical Fisher information of a parametrized quantum circuit to quantify its capacity [73]. They define a new capacity measure they call the effective dimension which can be used to bound how well a variational quantum learning model can generalize on unseen data.…”

Section: Analyzing Quantum Learning Modelsmentioning

confidence: 99%

“…It does not take into account the number of available datapoints and should therefore not be confused with the quantum generalization of the effective dimension of Ref. [73]. It is still an intuitive measure as the rank directly captures in how many directions a varying of the parameters will also result in a varying of the underlying quantum state.…”

Section: Analyzing Quantum Learning Modelsmentioning

confidence: 99%

Fisher Information in Noisy Intermediate-Scale Quantum Applications

Meyer

2021

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.