Learning many-body Hamiltonians with Heisenberg-limited scaling
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Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In this work, we propose the first algorithm to achieve the Heisenberg limit for learning an interacting N-qubit local Hamiltonian. After a total evolution time of 𝒪(ϵ^-1), the proposed algorithm can efficiently estimate any parameter in the N-qubit Hamiltonian to ϵ-error with high probability. The proposed algorithm is robust against state preparation and measurement error, does not require eigenstates or thermal states, and only uses polylog(ϵ^-1) experiments. In contrast, the best previous algorithms, such as recent works using gradient-based optimization or polynomial interpolation, require a total evolution time of 𝒪(ϵ^-2) and 𝒪(ϵ^-2) experiments. Our algorithm uses ideas from quantum simulation to decouple the unknown N-qubit Hamiltonian H into noninteracting patches, and learns H using a quantum-enhanced divide-and-conquer approach. We prove a matching lower bound to establish the asymptotic optimality of our algorithm.
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