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Neural Error Mitigation of Near-Term Quantum Simulations

Nature Machine Intelligence volume 4, pages 618–624 (2022)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Near-term quantum computers provide a promising platform for finding the ground states of quantum systems, which is an essential task in physics, chemistry and materials science. However, near-term approaches are constrained by the effects of noise, as well as the limited resources of near-term quantum hardware. We introduce neural error mitigation, which uses neural networks to improve estimates of ground states and ground-state observables obtained using near-term quantum simulations. To demonstrate our method’s broad applicability, we employ neural error mitigation to find the ground states of the H2 and LiH molecular Hamiltonians, as well as the lattice Schwinger model, prepared via the variational quantum eigensolver. Our results show that neural error mitigation improves numerical and experimental variational quantum eigensolver computations to yield low energy errors, high fidelities and accurate estimations of more complex observables such as order parameters and entanglement entropy without requiring additional quantum resources. Furthermore, neural error mitigation is agnostic with respect to the quantum state preparation algorithm used, the quantum hardware it is implemented on and the particular noise channel affecting the experiment, contributing to its versatility as a tool for quantum simulation.

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Fig. 1: NEM procedure.
Fig. 2: Experimental and numerical NEM results for molecular Hamiltonians.
Fig. 3: Performance of NEM applied to ground states of the lattice Schwinger model.

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Data availability

The experimental and numerical quantum simulation measurement data shown in Fig. 2 for H2 and LiH, as well as the measurement data used in Fig. 3a–d for the eight-site lattice Schwinger model, are available at https://github.com/1QB-Information-Technologies/NEM (see Zenodo repository50).

Code availability

The numerical implementation of the NEM and the code used to numerically implement the quantum simulations studied here are available at https://github.com/1QB-Information-Technologies/NEM (see Zenodo repository50).

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Acknowledgements

E.R.B., F.H., B.K. and P.R. thank 1QBit for financial support. During part of this work, E.R.B. and F.H. were students at the Perimeter Institute and the University of Waterloo and received funding through Mitacs, and F.H. was supported through a Vanier Canada Graduate Scholarship. Research at the Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. E.R.B. was also supported by a scholarship through the Perimeter Scholars International programme. P.R. thanks M. Lazaridis and O. Lazaridis and Innovation, Science and Economic Development Canada for financial support. J.C. acknowledges support from the Natural Sciences and Engineering Research Council, a Canada Research Chair, the Shared Hierarchical Academic Research Computing Network, Compute Canada, a Google Quantum Research Award and the Canadian Institute for Advanced Research (CIFAR) AI chair programme. Resources used by J.C. in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR and companies sponsoring the Vector Institute (www.vectorinstitute.ai/#partners). P.R. thanks C. Muschik for useful conversations. We thank M. Bucyk for carefully reviewing and editing the manuscript.

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Author notes

  1. These authors contributed equally: Elizabeth R. Bennewitz and Florian Hopfmueller.

Authors and Affiliations

  1. 1QB Information Technologies (1QBit), Waterloo, ON, Canada

    Elizabeth R. Bennewitz, Florian Hopfmueller & Bohdan Kulchytskyy

  2. Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, Canada

    Elizabeth R. Bennewitz, Florian Hopfmueller, Juan Carrasquilla & Pooya Ronagh

  3. Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada

    Elizabeth R. Bennewitz, Florian Hopfmueller & Pooya Ronagh

  4. Vector Institute, MaRS Centre, Toronto, ON, Canada

    Juan Carrasquilla

  5. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

    Pooya Ronagh

  6. 1QBit, Vancouver, BC, Canada

    Pooya Ronagh

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Contributions

E.R.B. and F.H. developed the codebase for all studies, performed numerical experiments and analysed the results. E.R.B. performed experiments using the IBM quantum processor. E.R.B. and F.H. focused on the quantum chemistry and the lattice Schwinger model case studies, respectively. All authors contributed to ideation and dissemination. J.C. and P.R. contributed to the theoretical foundations and design of the method.

Corresponding author

Correspondence to Pooya Ronagh.

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Nature Machine Intelligence thanks Alba Cervera-Lierta and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 VQE ansatz circuits for the lattice Schwinger model.

a, Variational quantum circuit used to prepare the approximate ground state of the lattice Schwinger model, using VQE simulated classically. The input state \(|{\Psi}_{0}\rangle\) is \(|01\cdots 01\rangle\) (\(|10\cdots 10\rangle\)) for m ≥ −0.7 (m < −0.7). b, For the results shown in Fig. 3e,f, the entangling layers of (a) are replaced with \({{{{\mathcal{O}}}}}_{N}\) for N sites. The layers of single-qubit gates are left unchanged.

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Supplementary Information

Supplementary Sections 1–8, Figs. 1–7 and Table 1.

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Bennewitz, E.R., Hopfmueller, F., Kulchytskyy, B. et al. Neural Error Mitigation of Near-Term Quantum Simulations. Nat Mach Intell 4, 618–624 (2022). https://doi.org/10.1038/s42256-022-00509-0

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