Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.18928 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915060204437504 |
|---|---|
| author | Ma, Muzhou Flammia, Steven T. Preskill, John Tong, Yu |
| author_facet | Ma, Muzhou Flammia, Steven T. Preskill, John Tong, Yu |
| contents | 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 $ε$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/ε)$ 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 most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning $k$-body Hamiltonians via compressed sensing Ma, Muzhou Flammia, Steven T. Preskill, John Tong, Yu Quantum Physics Data Structures and Algorithms Machine Learning 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 $ε$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/ε)$ 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 most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions. |
| title | Learning $k$-body Hamiltonians via compressed sensing |
| topic | Quantum Physics Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2410.18928 |