Learning quantum Gibbs states locally and efficiently
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912307608551424 |
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| author | Chen, Chi-Fang Anshu, Anurag Nguyen, Quynh T. |
| author_facet | Chen, Chi-Fang Anshu, Anurag Nguyen, Quynh T. |
| contents | Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature $β$, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a $n$-qubit $D$-dimensional Hamiltonian to an additive error $ε$ with sample complexity $\tilde{O}\left(\frac{e^{\mathrm{poly}(β)}}{β^2ε^2}\right)\log(n)$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to $n,ε$ and is polynomially tight with respect to $β$. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on $n$ but worse on $β, ε$. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_02706 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning quantum Gibbs states locally and efficiently Chen, Chi-Fang Anshu, Anurag Nguyen, Quynh T. Quantum Physics Statistical Mechanics Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature $β$, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a $n$-qubit $D$-dimensional Hamiltonian to an additive error $ε$ with sample complexity $\tilde{O}\left(\frac{e^{\mathrm{poly}(β)}}{β^2ε^2}\right)\log(n)$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to $n,ε$ and is polynomially tight with respect to $β$. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on $n$ but worse on $β, ε$. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures. |
| title | Learning quantum Gibbs states locally and efficiently |
| topic | Quantum Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2504.02706 |