Learning $k$-body Hamiltonians via compressed sensing

• URL: http://arxiv.org/abs/2410.18928v1
• Date: Thu, 24 Oct 2024 17:16:19 GMT
• Title: Learning $k$-body Hamiltonians via compressed sensing
• Authors: Muzhou Ma, Steven T. Flammia, John Preskill, Yu Tong,
• Abstract summary: We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $epsilon$ with total evolution time.
• Score: 0.5867838258848337
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• Abstract: We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $\epsilon$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/\epsilon)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.

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