Expressibility of the alternating layered ansatz for quantum computation

Kouhei Nakaji; Naoki Yamamoto

doi:10.22331/q-2021-04-19-434

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

Published:	2021-04-19, volume 5, page 434
Eprint:	arXiv:2005.12537v2
Doi:	https://doi.org/10.22331/q-2021-04-19-434
Citation:	Quantum 5, 434 (2021).

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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

@article{Nakaji2021expressibilityof,
  doi = {10.22331/q-2021-04-19-434},
  url = {https://doi.org/10.22331/q-2021-04-19-434},
  title = {Expressibility of the alternating layered ansatz for quantum computation},
  author = {Nakaji, Kouhei and Yamamoto, Naoki},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {434},
  month = apr,
  year = {2021}
}

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