Quantum Error Mitigation using Symmetry Expansion

Zhenyu Cai

Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom
Quantum Motion Technologies Ltd, Nexus, Discovery Way, Leeds, LS2 3AA, United Kingdom

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Abstract

Even with the recent rapid developments in quantum hardware, noise remains the biggest challenge for the practical applications of any near-term quantum devices. Full quantum error correction cannot be implemented in these devices due to their limited scale. Therefore instead of relying on engineered code symmetry, symmetry verification was developed which uses the inherent symmetry within the physical problem we try to solve. In this article, we develop a general framework named symmetry expansion which provides a wide spectrum of symmetry-based error mitigation schemes beyond symmetry verification, enabling us to achieve different balances between the estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 6 to 9 times below what is achievable by symmetry verification when the average number of circuit errors is between 1 to 2. The corresponding sampling cost for random shot noise reduction is just 2 to 6 times higher than symmetry verification. Beyond symmetries inherent to the physical problem, our formalism is also applicable to engineered symmetries. For example, the recent scheme for exponential error suppression using multiple noisy copies of the quantum device is just a special case of symmetry expansion using the permutation symmetry among the copies.

Mitigating errors using the symmetries of the problem is expected to play key roles in practical near-term quantum applications. A good example is symmetry verification, in which we perform measurements to verify whether the output state is in the right symmetry subspace, so that we can detect certain errors on the state. In this work, we introduce a general framework named symmetry expansion for symmetry-based error-mitigation techniques. It encompasses previous schemes like symmetry verification, symmetry subspace expansion and virtual distillation while also generates a wide range of schemes, offering different balances between the estimation biases and the sampling cost. Furthermore, we devise a way to identify the symmetry expansion schemes that have much smaller estimation biases than symmetry verification. These symmetry expansion schemes allow for cancellation between the biases due to the detectable and undetectable errors, while symmetry verification can only remove the biases due to the detectable errors, leaving behind the biases due to the undetectable errors.

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Cited by

[1] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum (NISQ) algorithms", arXiv:2101.08448.

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[4] Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "Qubit-efficient exponential suppression of errors", arXiv:2102.06056.

[5] Ryan LaRose, Andrea Mari, Sarah Kaiser, Peter J. Karalekas, Andre A. Alves, Piotr Czarnik, Mohamed El Mandouh, Max H. Gordon, Yousef Hindy, Aaron Robertson, Purva Thakre, Nathan Shammah, and William J. Zeng, "Mitiq: A software package for error mitigation on noisy quantum computers", arXiv:2009.04417.

[6] Zhenyu Cai, "Resource-efficient Purification-based Quantum Error Mitigation", arXiv:2107.07279.

[7] Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Unifying and benchmarking state-of-the-art quantum error mitigation techniques", arXiv:2107.13470.

[8] Nobuyuki Yoshioka, Hideaki Hakoshima, Yuichiro Matsuzaki, Yuuki Tokunaga, Yasunari Suzuki, and Suguru Endo, "Generalized quantum subspace expansion", arXiv:2107.02611.

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