Reduction of finite sampling noise in quantum neural networks

David A. Kreplin; Marco Roth

doi:10.22331/q-2024-06-25-1385

Reduction of finite sampling noise in quantum neural networks

Fraunhofer Institute for Manufacturing Engineering and Automation (IPA), Nobelstraße 12, D-70569 Stuttgart, Germany

Published:	2024-06-25, volume 8, page 1385
Eprint:	arXiv:2306.01639v3
Doi:	https://doi.org/10.22331/q-2024-06-25-1385
Citation:	Quantum 8, 1385 (2024).

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Abstract

Quantum neural networks (QNNs) use parameterized quantum circuits with data-dependent inputs and generate outputs through the evaluation of expectation values. Calculating these expectation values necessitates repeated circuit evaluations, thus introducing fundamental finite-sampling noise even on error-free quantum computers. We reduce this noise by introducing the variance regularization, a technique for reducing the variance of the expectation value during the quantum model training. This technique requires no additional circuit evaluations if the QNN is properly constructed. Our empirical findings demonstrate the reduced variance speeds up the training and lowers the output noise as well as decreases the number of necessary evaluations of gradient circuits. This regularization method is benchmarked on the regression of multiple functions and the potential energy surface of water. We show that in our examples, it lowers the variance by an order of magnitude on average and leads to a significantly reduced noise level of the QNN. We finally demonstrate QNN training on a real quantum device and evaluate the impact of error mitigation. Here, the optimization is feasible only due to the reduced number of necessary shots in the gradient evaluation resulting from the reduced variance.

Featured image: Reduction of variance through a modified Loss function

Popular summary

Quantum neural networks (QNNs) are quantum computing methods that use parameterized quantum circuits with data-dependent inputs, producing outputs through the evaluation of expectation values. In a supervised learning framework, QNNs are trained to minimize a loss function based on labeled data.

Determining the model output by evaluating expectation values necessitates multiple circuit evaluations, known as shots, which introduces intrinsic finite-sampling noise, even in ideal quantum computers. While increasing the number of measurements can reduce finite-sampling noise, practical constraints limit this approach. Therefore, we propose reducing the variance of expectation values as an alternative method to lower the noise. This can be accomplished by adding variance as a regularization term in the loss function. If the QNN follows specific design principles, the variance can be calculated without additional circuit evaluations. This approach also applies to minimizing the number of shots required for evaluating the QNN gradient during training.

The efficacy of this technique is empirically validated through various benchmarks, including function regression and modeling the potential energy surface of water. Our results demonstrate that variance regularization can reduce the variance by an order of magnitude on average, significantly decreasing noise in QNN outputs. Furthermore, the practical feasibility of this method is confirmed on real quantum hardware, highlighting its importance in reducing the number of shots needed for gradient evaluation and enabling more efficient optimization on quantum computers.

► BibTeX data

@article{Kreplin2024reductionoffinite,
  doi = {10.22331/q-2024-06-25-1385},
  url = {https://doi.org/10.22331/q-2024-06-25-1385},
  title = {Reduction of finite sampling noise in quantum neural networks},
  author = {Kreplin, David A. and Roth, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1385},
  month = jun,
  year = {2024}
}

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