Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias, Christoph; Gil-Fuster, Elies; Meyer, Johannes Jakob; Eisert, Jens; Sweke, Ryan

doi:10.22331/q-2021-11-17-582

Cited by 65 publications

(37 citation statements)

References 41 publications

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“…In this perspective, the achieved results in this study provide a negative response to their question. Combining the analysis in [70] and our results, a promising research direction is analyzing the non-uniform generalization bounds of QNNs to understand their power.…”

Section: Note Addedmentioning

confidence: 68%

“…During the preparation of the manuscript, we notice that a very recent theoretical study [70] indicated that to deeply understand the power of QNNs, it is necessary to demonstrate whether QNNs possess the ability to achieve zero risk for a randomly-relabeled real-world classification task. Their motivation highly echoes with our purpose such that statistical learning theory can be harnessed as a powerful tool to study the capability and limitations of QNNs.…”

Section: Note Addedmentioning

confidence: 99%

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The dilemma of quantum neural networks

Yang¹,

Wang²,

Du³

et al. 2021

Preprint

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The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bound than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that quantum neural networks (QNNs) have the ability to achieve this goal on specific datasets. With this regard, it is of great importance to understand whether these advantages are still preserved on real-world tasks. Through systematic numerical experiments, we empirically observe that current QNNs fail to provide any benefit over classical learning models. Concretely, our results deliver two key messages. First, QNNs suffer from the severely limited effective model capacity, which incurs poor generalization on real-world datasets. Second, the trainability of QNNs is insensitive to regularization techniques, which sharply contrasts with the classical scenario. These empirical results force us to rethink the role of current QNNs and to design novel protocols for solving real-world problems with quantum advantages.

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Section: Note Addedmentioning

confidence: 68%

Section: Note Addedmentioning

confidence: 99%

The dilemma of quantum neural networks

Yang¹,

Wang²,

Du³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We note that by construction the latent space probability distribution p θ (x) corresponds to a parametrized quantum circuit with feature map encoding [68][69][70][71], and can be analyzed by studying associated Fourier series (for brevity, we omit t dependence in this subsection). We proceed to analyse the model capacity of the phase feature map Ûϕ (x).…”

Section: Phase Feature Map Analysismentioning

confidence: 99%

“…The exponential capacity can then be seen as a problem for certain distributions (see discussion in Ref. [71]), as highly-oscillatoric terms will lead to overfitting and corrupt derivatives when solving differential equations. 2.…”

Section: Phase Feature Map Analysismentioning

confidence: 99%

Protocols for Trainable and Differentiable Quantum Generative Modelling

Kyriienko¹,

Paine²,

Elfving³

2022

Preprint

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We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQCbased model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

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“…This layered form terminates with a circuit encoded by a variable as a final entangling circuit cancels out for kernels based on U † (x)U(y). As with many variational algorithms, when choosing feature maps it is important to have a map expressible enough to represent the solution to the problem whilst also being trainable [65].…”

Section: A Encodingmentioning

confidence: 99%

Quantum Kernel Methods for Solving Differential Equations

Paine¹,

Elfving²,

Kyriienko³

2022

Preprint

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We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits (DQC) approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.

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Encoding-dependent generalization bounds for parametrized quantum circuits

Cited by 65 publications

References 41 publications

The dilemma of quantum neural networks

The dilemma of quantum neural networks

Protocols for Trainable and Differentiable Quantum Generative Modelling

Quantum Kernel Methods for Solving Differential Equations

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