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Certified algorithms for equilibrium states of local quantum Hamiltonians

Author

Listed:
  • Hamza Fawzi

    (University of Cambridge)

  • Omar Fawzi

    (LIP)

  • Samuel O. Scalet

    (University of Cambridge)

Abstract

Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.

Suggested Citation

  • Hamza Fawzi & Omar Fawzi & Samuel O. Scalet, 2024. "Certified algorithms for equilibrium states of local quantum Hamiltonians," Nature Communications, Nature, vol. 15(1), pages 1-6, December.
    • Handle: RePEc:nat:natcom:v:15:y:2024:i:1:d:10.1038_s41467-024-51592-3
    DOI: 10.1038/s41467-024-51592-3
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    References listed on IDEAS

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    1. K. Temme & T. J. Osborne & K. G. Vollbrecht & D. Poulin & F. Verstraete, 2011. "Quantum Metropolis sampling," Nature, Nature, vol. 471(7336), pages 87-90, March.
    2. Johannes Bausch & Toby S. Cubitt & James D. Watson, 2021. "Uncomputability of phase diagrams," Nature Communications, Nature, vol. 12(1), pages 1-8, December.
    3. Toby S. Cubitt & David Perez-Garcia & Michael M. Wolf, 2015. "Undecidability of the spectral gap," Nature, Nature, vol. 528(7581), pages 207-211, December.
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