Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi; Daniel Ranard

doi:10.22331/q-2019-07-08-159

Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi^1,2 and Daniel Ranard¹

¹Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305 USA
²Institute for Advanced Study, Princeton NJ 08540 USA

Published:	2019-07-08, volume 3, page 159
Eprint:	arXiv:1712.01850v2
Doi:	https://doi.org/10.22331/q-2019-07-08-159
Citation:	Quantum 3, 159 (2019).

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Abstract

We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``$\textit{k}$-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

► BibTeX data

@article{Qi2019determininglocal,
  doi = {10.22331/q-2019-07-08-159},
  url = {https://doi.org/10.22331/q-2019-07-08-159},
  title = {Determining a local {H}amiltonian from a single eigenstate},
  author = {Qi, Xiao-Liang and Ranard, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {159},
  month = jul,
  year = {2019}
}

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