Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity

• URL: http://arxiv.org/abs/2402.05552v1
• Date: Thu, 8 Feb 2024 10:42:47 GMT
• Title: Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity
• Authors: Ales Wodecki and Jakub Marecek
• Abstract summary: We consider the problem of learning local quantum Hamiltonians given copies of their state at a known inverse temperature. Our main technical contribution is a new flat approximation of the exponential function based on the Chebyshev expansion.
• Score: 3.2634122554914
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a new flat polynomial approximation of the exponential function based on the Chebyshev expansion, which enables the formulation of learning quantum Hamiltonians as a polynomial optimization problem. This, in turn, can benefit from the use of moment/SOS relaxations, whose polynomial bit complexity requires careful analysis [O'Donnell, ITCS 2017]. Finally, we show that learning a $k$-local Hamiltonian, whose dual interaction graph is of bounded degree, runs in polynomial time under mild assumptions.

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