Department of Applied Physics and Physico-Informatics & Quantum Computing Center, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama, 223-8522, Japan
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Abstract
The hybrid quantum-classical algorithm is actively examined as a technique applicable even to intermediate-scale quantum computers. To execute this algorithm, the hardware efficient ansatz is often used, thanks to its implementability and expressibility; however, this ansatz has a critical issue in its trainability in the sense that it generically suffers from the so-called gradient vanishing problem. This issue can be resolved by limiting the circuit to the class of shallow alternating layered ansatz. However, even though the high trainability of this ansatz is proved, it is still unclear whether it has rich expressibility in state generation. In this paper, with a proper definition of the expressibility found in the literature, we show that the shallow alternating layered ansatz has almost the same level of expressibility as that of hardware efficient ansatz. Hence the expressibility and the trainability can coexist, giving a new designing method for quantum circuits in the intermediate-scale quantum computing era.
Featured image: Top: Energy versus the iteration step in the VQE problem for the Hamiltonian ${mathcal H}= sum_{i=1}^{4}
(sigma_x^i sigma_x^{i+1} + sigma_y^{i} sigma_y^{i+1} + sigma_z^i sigma_z^{i+1}), $ with the ansatz TEN (left), ALT (center), and HEA (right); TEN is the tensor product ansatz, ALT is the alternating layered ansatz, and HEA is the hardware efficient ansatz. The blue lines and the associated error bars represent the average and the standard deviation of the mean energies in total 100 trials, respectively; in each trial, the initial parameters of the ansatz are randomly chosen. Optimization to decrease $langle {cal H} rangle$ in each iteration is performed by using Adam Optimizer with learning rate $0.001$.
Bottom: Three of 100 trajectories for each ansatz TEN (left), ALT (center), and HEA (right), indicated by red lines. The trajectories are chosen such that the energies at the final iteration step are the three smallest values.
► BibTeX data
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Cited by
[1] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Variational Quantum Algorithms”, arXiv:2012.09265.
[2] 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.
[3] Tyler Volkoff and Patrick J. Coles, “Large gradients via correlation in random parameterized quantum circuits”, Quantum Science and Technology 6 2, 025008 (2021).
[4] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, “Connecting ansatz expressibility to gradient magnitudes and barren plateaus”, arXiv:2101.02138.
[5] Alexey Uvarov and Jacob Biamonte, “On barren plateaus and cost function locality in variational quantum algorithms”, arXiv:2011.10530.
[6] Alba Cervera-Lierta, Jakob S. Kottmann, and Alán Aspuru-Guzik, “The Meta-Variational Quantum Eigensolver (Meta-VQE): Learning energy profiles of parameterized Hamiltonians for quantum simulation”, arXiv:2009.13545.
[7] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, “Solving nonlinear differential equations with differentiable quantum circuits”, arXiv:2011.10395.
[8] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, “Long-time simulations with high fidelity on quantum hardware”, arXiv:2102.04313.
[9] Tobias Haug, Kishor Bharti, and M. S. Kim, “Capacity and quantum geometry of parametrized quantum circuits”, arXiv:2102.01659.
[10] Joonho Kim, Jaedeok Kim, and Dario Rosa, “Universal Effectiveness of High-Depth Circuits in Variational Eigenproblems”, arXiv:2010.00157.
[11] Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, “NISQ Algorithm for Hamiltonian Simulation via Truncated Taylor Series”, arXiv:2103.05500.
[12] Andrew Arrasmith, Zoë Holmes, M. Cerezo, and Patrick J. Coles, “Equivalence of quantum barren plateaus to cost concentration and narrow gorges”, arXiv:2104.05868.
The above citations are from SAO/NASA ADS (last updated successfully 2021-04-19 14:52:34). The list may be incomplete as not all publishers provide suitable and complete citation data.
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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
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Source: https://quantum-journal.org/papers/q-2021-04-19-434/
