Training Quantum Embedding Kernels on Near-Term Quantum Computers

Hubregtsen, Thomas; Wierichs, David; Gil-Fuster, Elies; Derks, Peter-Jan H. S.; Faehrmann, Paul K.; Meyer, Johannes Jakob

doi:10.48550/arxiv.2105.02276

Cited by 18 publications

(25 citation statements)

References 56 publications

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“…A potential candidate is quantum machine learning [62][63][64][65][66]. Recall that recent studies have exhibited the potential of quantum kernels to earn quantum advantages [67][68][69][70][71]. Note that the core of quantum kernels is calculating the similarity of two quantum inputs, which amounts to accomplishing the fidelity estimation.…”

Section: Discussionmentioning

confidence: 99%

On exploring practical potentials of quantum auto-encoder with advantages

Du,

Tao

2021

Preprint

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Quantum auto-encoder (QAE) is a powerful tool to relieve the curse of dimensionality encountered in quantum physics, celebrated by the ability to extract low-dimensional patterns from quantum states living in the high-dimensional space. Despite its attractive properties, little is known about the practical applications of QAE with provable advantages. To address these issues, here we prove that QAE can be used to efficiently calculate the eigenvalues and prepare the corresponding eigenvectors of a high-dimensional quantum state with the low-rank property. With this regard, we devise three effective QAE-based learning protocols to solve the low-rank state fidelity estimation, the quantum Gibbs state preparation, and the quantum metrology tasks, respectively. Notably, all of these protocols are scalable and can be readily executed on near-term quantum machines. Moreover, we prove that the error bounds of the proposed QAE-based methods outperform those in previous literature. Numerical simulations collaborate with our theoretical analysis. Our work opens a new avenue of utilizing QAE to tackle various quantum physics and quantum information processing problems in a scalable way.

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Section: Discussionmentioning

confidence: 99%

On exploring practical potentials of quantum auto-encoder with advantages

Du,

Tao

2021

Preprint

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“…tation angles depends on the values in x i . While in general the embedding can be in itself trainable [14,57,58], here the embedding is fixed.…”

Section: An Embedding Channel E Ementioning

confidence: 99%

Subtleties in the trainability of quantum machine learning models

Thanasilp¹,

Wang²,

Nghiem³

et al. 2021

Preprint

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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.

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“…We note that the kernel target alignment also weighs the contributions of f depending on the corresponding eigenvalue, i.e., the alignment is better if large |a i | correspond to large γ i . The kernel target alignment was used extensively to optimize kernel functions [31] and recently also used to optimize quantum kernels [35].…”

Section: Supervised Learningmentioning

confidence: 99%

The Inductive Bias of Quantum Kernels

Kübler¹,

Buchholz²,

Schölkopf³

2021

Preprint

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It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high-or infinite-dimensional reproducing kernel Hilbert spaces (RKHS).We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements.In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets. * JMK and SB contributed equally and are ordered randomly.

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Training Quantum Embedding Kernels on Near-Term Quantum Computers

Cited by 18 publications

References 56 publications

On exploring practical potentials of quantum auto-encoder with advantages

On exploring practical potentials of quantum auto-encoder with advantages

Subtleties in the trainability of quantum machine learning models

The Inductive Bias of Quantum Kernels

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