Expressibility of the alternating layered ansatz for quantum computation

Kouhei Nakaji and Naoki Yamamoto

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.

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[1] J. Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. https:/​/​doi.org/​10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[2] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5 (1): 1–7, 2014. https:/​/​doi.org/​10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[3] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, 2017. https:/​/​doi.org/​10.1038/​nature23879.
https:/​/​doi.org/​10.1038/​nature23879

[4] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, 9 (1): 1–6, 2018. https:/​/​doi.org/​10.1038/​s41467-018-07090-4.
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[5] E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum, 3: 214, 2019. https:/​/​doi.org/​10.22331/​q-2019-12-09-214.
https:/​/​doi.org/​10.22331/​q-2019-12-09-214

[6] J. Stokes, J. Izaac, N. Killoran, and G. Carleo. Quantum natural gradient. Quantum, 4: 269, 2020. https:/​/​doi.org/​10.22331/​q-2020-05-25-269.
https:/​/​doi.org/​10.22331/​q-2020-05-25-269

[7] N. Yamamoto. On the natural gradient for variational quantum eigensolver. arXiv preprint arXiv:1909.05074, 2019.
arXiv:1909.05074

[8] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature communications, 12 1: 1791, 2021. https:/​/​doi.org/​10.1038/​s41467-021-21728-w.
https:/​/​doi.org/​10.1038/​s41467-021-21728-w

[9] S. Sim, P. D. Johnson, and A. Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quantum Technologies, 2 (12): 1900070, 2019. https:/​/​doi.org/​10.1002/​qute.201900070.
https:/​/​doi.org/​10.1002/​qute.201900070

[10] T. Ali, A. Bhattacharyya, S. S. Haque, E. H. Kim, N. Moynihan, and J. Murugan. Chaos and complexity in quantum mechanics. Physical Review D, 101 (2): 026021, 2020. https:/​/​doi.org/​10.1007/​JHEP04(2017)121.
https:/​/​doi.org/​10.1007/​JHEP04(2017)121

[11] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves. Symmetric informationally complete quantum measurements. Journal of Mathematical Physics, 45 (6): 2171–2180, 2004. https:/​/​doi.org/​10.1063/​1.1737053.
https:/​/​doi.org/​10.1063/​1.1737053

[12] A. Klappenecker and M. Rotteler. Mutually unbiased bases are complex projective 2-designs. In Proceedings. International Symposium on Information Theory, 2005. ISIT 2005., pages 1740–1744. IEEE, 2005. https:/​/​doi.org/​10.1109/​ISIT.2005.1523643.
https:/​/​doi.org/​10.1109/​ISIT.2005.1523643

[13] I. Bengtsson and K. Życzkowski. Geometry of quantum states: an introduction to quantum entanglement. Cambridge university press, 2017. https:/​/​doi.org/​10.1017/​CBO9780511535048.
https:/​/​doi.org/​10.1017/​CBO9780511535048

[14] K. Życzkowski and H.-J. Sommers. Average fidelity between random quantum states. Physical Review A, 71 (3): 032313, 2005. https:/​/​doi.org/​10.1103/​PhysRevA.71.032313.
https:/​/​doi.org/​10.1103/​PhysRevA.71.032313

[15] Z. Puchała and J. Miszczak. Symbolic integration with respect to the haar measure on the unitary groups. Bulletin of the Polish Academy of Sciences: Technical Sciences, 65 (No 1): 21–27, 2017. https:/​/​doi.org/​10.1515/​bpasts-2017-0003.
https:/​/​doi.org/​10.1515/​bpasts-2017-0003

[16] F. G. Brandao, A. W. Harrow, and M. Horodecki. Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346 (2): 397–434, 2016. https:/​/​doi.org/​10.1007/​s00220-016-2706-8.
https:/​/​doi.org/​10.1007/​s00220-016-2706-8

[17] A. Harrow and S. Mehraban. Approximate unitary $ t $-designs by short random quantum circuits using nearest-neighbor and long-range gates. arXiv preprint arXiv:1809.06957, 2018.
arXiv:1809.06957

[18] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
arXiv:1412.6980

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[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] Tyler Volkoff and Patrick J. Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

[3] 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.

[4] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, "Connecting ansatz expressibility to gradient magnitudes and barren plateaus", arXiv:2101.02138.

[5] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", arXiv:2011.10395.

[6] Alexey Uvarov and Jacob Biamonte, "On barren plateaus and cost function locality in variational quantum algorithms", arXiv:2011.10530.

[7] 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.

[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] Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, "NISQ Algorithm for Hamiltonian Simulation via Truncated Taylor Series", arXiv:2103.05500.

[11] Joonho Kim, Jaedeok Kim, and Dario Rosa, "Universal Effectiveness of High-Depth Circuits in Variational Eigenproblems", arXiv:2010.00157.

[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.

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