Matrix product state approximations to quantum states of low energy variance

Kshiti Sneh Rai; J. Ignacio Cirac; Álvaro M. Alhambra

doi:10.22331/q-2024-07-10-1401

Matrix product state approximations to quantum states of low energy variance

Kshiti Sneh Rai^1,2, J. Ignacio Cirac², and Álvaro M. Alhambra^3,2

¹Instituut-Lorentz, Niels Bohrweg 2, Leiden, NL-2333 CA, The Netherlands
²Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
³Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain

Published:	2024-07-10, volume 8, page 1401
Eprint:	arXiv:2307.05200v3
Doi:	https://doi.org/10.22331/q-2024-07-10-1401
Citation:	Quantum 8, 1401 (2024).

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Abstract

We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as $\propto 1/\log N$ can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.

Featured image: Illustration of the energy distribution of an initial product state, with standard deviation $\sigma$, and of the filtered distribution, with a smaller standard deviation $\delta$ and potentially a slightly shifted average (dashed line).

Multimedia: QIP 2024 Matrix product state approximations to quantum states of low energy variance

Popular summary

The study of the entanglement structure of quantum many body systems is of central interest in quantum information and condensed matter physics. The presence and character of the entanglement in these systems often underlies some of its deepest physical features. For instance, a lack of entanglement is usually thought to imply large energetic fluctuations, which underline abrupt non-equilibrium dynamics. Here, we defy this intuition by proving that in all local Hamiltonian models there exist states with both low entanglement and vanishingly small energetic fluctuations.

In our results, we use algorithms based on matrix product states to approximate those states of low energy variance and bounded entanglement. Even for decreasing variances, the entanglement of states in the finite energy regime does not abruptly increase. To study this algorithm, we put forward a type of central limit theorem – the Berry Esseen theorem, to bound the energetic fluctuations of states with a matrix product state description. We do this through a novel method that shows that the fluctuations of our quantum states of interest strongly resemble random coin tosses.

Our work provides fundamental insights into the energy distributions of product and low-entangled states in local Hamiltonian systems. The results also justify previous numerical findings of tensor network-based algorithms which show thermalization of local observables. Furthermore, through the rigorous study of complexity of our algorithm, we are able to set a benchmark for future quantum algorithms for studying entangled states at finite temperature.

► BibTeX data

@article{Rai2024matrixproductstate,
  doi = {10.22331/q-2024-07-10-1401},
  url = {https://doi.org/10.22331/q-2024-07-10-1401},
  title = {Matrix product state approximations to quantum states of low energy variance},
  author = {Rai, Kshiti Sneh and Cirac, J. Ignacio and Alhambra, {\'{A}}lvaro M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1401},
  month = jul,
  year = {2024}
}

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