End-to-end complexity for simulating the Schwinger model on quantum computers

Kazuki Sakamoto; Hayata Morisaki; Junichi Haruna; Etsuko Itou; Keisuke Fujii; Kosuke Mitarai

doi:10.22331/q-2024-09-17-1474

End-to-end complexity for simulating the Schwinger model on quantum computers

Kazuki Sakamoto¹, Hayata Morisaki¹, Junichi Haruna², Etsuko Itou^3,4, Keisuke Fujii^1,2,5, and Kosuke Mitarai^1,2

¹Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
²Center for Quantum Information and Quantum Biology, Osaka University 560-0043, Japan.
³Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
⁴Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako 351-0198, Japan
⁵Center for Quantum Computing, RIKEN, Hirosawa 2-1, Wako Saitama 351-0198, Japan

Published:	2024-09-17, volume 8, page 1474
Eprint:	arXiv:2311.17388v3
Doi:	https://doi.org/10.22331/q-2024-09-17-1474
Citation:	Quantum 8, 1474 (2024).

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Abstract

The Schwinger model is one of the simplest gauge theories. It is known that a topological term of the model leads to the infamous sign problem in the classical Monte Carlo method. In contrast to this, recently, quantum computing in Hamiltonian formalism has gained attention. In this work, we estimate the resources needed for quantum computers to compute physical quantities that are challenging to compute on classical computers. Specifically, we propose an efficient implementation of block-encoding of the Schwinger model Hamiltonian. Considering the structure of the Hamiltonian, this block-encoding with a normalization factor of $\mathcal{O}(N^3)$ can be implemented using $\mathcal{O}(N+\log^2(N/\varepsilon))$ T gates. As an end-to-end application, we compute the vacuum persistence amplitude. As a result, we found that for a system size $N=128$ and an additive error $\varepsilon=0.01$, with an evolution time $t$ and a lattice spacing a satisfying $t/2a=10$, the vacuum persistence amplitude can be calculated using about $10^{13}$ T gates. Our results provide insights into predictions about the performance of quantum computers in the FTQC and early FTQC era, clarifying the challenges in solving meaningful problems within a realistic timeframe.

Featured image: Runtime for calculating the vacuum persistence amplitude.

► BibTeX data

@article{Sakamoto2024endtoendcomplexity,
  doi = {10.22331/q-2024-09-17-1474},
  url = {https://doi.org/10.22331/q-2024-09-17-1474},
  title = {End-to-end complexity for simulating the {S}chwinger model on quantum computers},
  author = {Sakamoto, Kazuki and Morisaki, Hayata and Haruna, Junichi and Itou, Etsuko and Fujii, Keisuke and Mitarai, Kosuke},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1474},
  month = sep,
  year = {2024}
}

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