Efficient Verification Of Ground States Of Frustration-Free Hamiltonians

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Huangjun Zhu, Yunting Li, and Tianyi Chen

State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China
Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China
Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China

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Abstract

Ground states of local Hamiltonians are of key interest in many-body physics and also in quantum information processing. Efficient verification of these states are crucial to many applications, but very challenging. Here we propose a simple, but powerful recipe for verifying the ground states of general frustration-free Hamiltonians based on local measurements. Moreover, we derive rigorous bounds on the sample complexity by virtue of the quantum detectability lemma (with improvement) and quantum union bound. Notably, the number of samples required does not increase with the system size when the underlying Hamiltonian is local and gapped, which is the case of most interest. As an application, we propose a general approach for verifying Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs based on local spin measurements, which requires only a constant number of samples for AKLT states defined on various lattices. Our work is of interest not only to many tasks in quantum information processing, but also to the study of many-body physics.

Featured image: Optimal edge colorings of the square lattice and honeycomb lattice. These optimal colorings can be used to construct efficient protocols for verifying ground states of frustration-free Hamiltonians, including AKLT states.

Popular summary

We propose a general recipe for verifying the ground states of frustration-free Hamiltonians based on local measurements and determine the sample complexity. When the Hamiltonian is local and gapped, we can verify the ground state with a constant sample cost that is independent of the system size, which is tens of thousands of times more efficient than previous protocols for large and intermediate quantum systems. Notably, we can verify Affleck-Kennedy-Lieb-Tasaki (AKLT) states on arbitrary graphs, and the resource cost is independent of the system size for most AKLT states of practical interest, including those defined on various 1D and 2D lattices. Our work reveals an intimate connection between the quantum verification problem and many-body physics. The protocols we constructed are useful not only to addressing various tasks in quantum information processing, but also to studying many-body physics.

► BibTeX data

@article{Zhu2024efficient,
  doi = {10.22331/q-2024-01-10-1221},
  url = {https://doi.org/10.22331/q-2024-01-10-1221},
  title = {Efficient {V}erification of {G}round {S}tates of {F}rustration-{F}ree {H}amiltonians},
  author = {Zhu, Huangjun and Li, Yunting and Chen, Tianyi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1221},
  month = jan,
  year = {2024}
}

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Cited by

[1] Tianyi Chen, Yunting Li, and Huangjun Zhu, "Efficient verification of Affleck-Kennedy-Lieb-Tasaki states", Physical Review A 107 2, 022616 (2023).

[2] Zihao Li, Huangjun Zhu, and Masahito Hayashi, "Robust and efficient verification of graph states in blind measurement-based quantum computation", npj Quantum Information 9, 115 (2023).

[3] Ye-Chao Liu, Yinfei Li, Jiangwei Shang, and Xiangdong Zhang, "Efficient verification of arbitrary entangled states with homogeneous local measurements", arXiv:2208.01083, (2022).

[4] Siyuan Chen, Wei Xie, and Kun Wang, "Memory Effects in Quantum State Verification", arXiv:2312.11066, (2023).

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