Effects of quantum resources on the statistical complexity of quantum circuits

Bu, Kaifeng; Koh, Dax Enshan; Lü, Li; Luo, Qingxian; Zhang, Yaobo

doi:10.48550/arxiv.2102.03282

Cited by 9 publications

(15 citation statements)

References 63 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Given the fundamental role of generalization bounds, there has recently been a strong and steady stream of works contributing to the derivation of generalization bounds for PQC-based models [24][25][26][27][28][29][30][31][32]. However, as discussed in detail in Section 4, these prior works all differ from our results in a variety of ways.…”

Section: Introductionmentioning

confidence: 70%

“…However, unlike in Ref. [27], the Rademacher complexity bounds of Refs. [26,28] are given in terms of quantities that exhibit an implicit dependence on the data-encoding strategy.…”

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 86%

“…Finally, Ref. [27] has recently initiated a resource-theoretic approach by providing encoding-independent bounds on both the Rademacher and Gaussian complexity of encoding-first PQC-based models, in terms of the number of repetitions of resource channels allowed in the PQC. These Rademacher and Gaussian complexity bounds have then been used to derive generalization bounds, which depend on the same quantities, and therefore provide an encoding-independent resource-theoretic perspective on generalization in encoding-first PQC-based models.…”

Section: Encoding-independent Complexity and Generalization Boundsmentioning

confidence: 99%

“…From a resource theoretic perspective, and complimenting Ref. [27], the series of works [26,28] have further studied the Rademacher complexity of encoding-first PQC-based models. However, unlike in Ref.…”

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

See 3 more Smart Citations

Encoding-dependent generalization bounds for parametrized quantum circuits

Caro,

Gil-Fuster,

Meyer

et al. 2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 70%

“…However, unlike in Ref. [27], the Rademacher complexity bounds of Refs. [26,28] are given in terms of quantities that exhibit an implicit dependence on the data-encoding strategy.…”

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 86%

Section: Encoding-independent Complexity and Generalization Boundsmentioning

confidence: 99%

Section: Encoding-dependent Complexity and Generalization Boundsmentioning

confidence: 99%

See 2 more Smart Citations

Encoding-dependent generalization bounds for parametrized quantum circuits

Caro,

Gil-Fuster,

Meyer

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…1, have attracted great attention from industry and academia. The popularity of VQAs origins from the versatility of PQCs, which guarantees their efficient implementations on the noisy intermediate-scale quantum (NISQ) machines [16], as well as theoretical evidence of quantum superiority [17][18][19][20][21][22][23][24][25]. Moreover, prior studies have exhibited the progress of VQAs of accomplishing diverse learning tasks, e.g., machine learning issues such as data classification [26][27][28][29][30] and image generation [31][32][33], combinatorial optimization [34][35][36][37], finance [38][39][40], quantum information processing [41][42][43], quantum chemistry and material sciences [44][45][46][47], and particle physics [48,49].…”

Section: Introductionmentioning

confidence: 99%

Accelerating variational quantum algorithms with multiple quantum processors

Du¹,

Yang²,

Tao³

2021

Preprint

View full text Add to dashboard Cite

Variational quantum algorithms (VQAs) have the potential of utilizing near-term quantum machines to gain certain computational advantages over classical methods. Nevertheless, modern VQAs suffer from cumbersome computational overhead, hampered by the tradition of employing a solitary quantum processor to handle large-volume data. As such, to better exert the superiority of VQAs, it is of great significance to improve their runtime efficiency. Here we devise an efficient distributed optimization scheme, called QUDIO, to address this issue. Specifically, in QUDIO, a classical central server partitions the learning problem into multiple subproblems and allocate them to multiple local nodes where each of them consists of a quantum processor and a classical optimizer. During the training procedure, all local nodes proceed parallel optimization and the classical server synchronizes optimization information among local nodes timely. In doing so, we prove a sublinear convergence rate of QUDIO in terms of the number of global iteration under the ideal scenario, while the system imperfection may incur divergent optimization. Numerical results on standard benchmarks demonstrate that QUDIO can surprisingly achieve a superlinear runtime speedup with respect to the number of local nodes. Our proposal can be readily mixed with other advanced VQAs-based techniques to narrow the gap between the state of the art and applications with quantum advantage.

show abstract

Generalization in quantum machine learning from few training data

et al. 2022

View full text Add to dashboard Cite

Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as $$\sqrt{T/N}$$ T / N . When only K ≪ T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to $$\sqrt{K/N}$$ K / N . Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.

show abstract

Effects of quantum resources on the statistical complexity of quantum circuits

Cited by 9 publications

References 63 publications

Encoding-dependent generalization bounds for parametrized quantum circuits

Encoding-dependent generalization bounds for parametrized quantum circuits

Accelerating variational quantum algorithms with multiple quantum processors

Generalization in quantum machine learning from few training data

Contact Info

Product

Resources

About