Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

We study the problem of learning a Hamiltonian H to precision \varepsilon, supposing we are given copies of its Gibbs state \rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H)) at a known inverse temperature \beta. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample compl...

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Veröffentlicht in:Proceedings / annual Symposium on Foundations of Computer Science S. 135 - 146
Hauptverfasser: Haah, Jeongwan, Kothari, Robin, Tang, Ewin
Format: Tagungsbericht
Sprache:Englisch
Veröffentlicht: IEEE 01.10.2022
Schlagworte:
Complexity theory
Computer science
Hamiltonian learning
Markov random field
partition function
quantum algorithm
Quantum computing
real-time evolution
Real-time systems
Time complexity
ISSN:2575-8454
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Zusammenfassung:We study the problem of learning a Hamiltonian H to precision \varepsilon, supposing we are given copies of its Gibbs state \rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H)) at a known inverse temperature \beta. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of \rho needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low \beta) regime, their algorithm has sample complexity poly (N, 1/\beta, 1/\varepsilon) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error \varepsilon with sample complexity S=O(\log N/(\beta\varepsilon)^{2}) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary e^{-i t H} in a small t regime with similar sample and time complexity.
ISSN:2575-8454
DOI:10.1109/FOCS54457.2022.00020