Principal eigenstate classical shadows
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909245280092160 |
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| author | Grier, Daniel Pashayan, Hakop Schaeffer, Luke |
| author_facet | Grier, Daniel Pashayan, Hakop Schaeffer, Luke |
| contents | Given many copies of an unknown quantum state $ρ$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $ρ$ has an eigenstate $|ϕ\rangle$ with (unknown) eigenvalue $λ> 1/2$, the goal is to learn a (classical shadows style) classical description of $|ϕ\rangle$ which can later be used to estimate expectation values $\langle ϕ|O| ϕ\rangle$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $ρ$ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue $λ$ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when $λ$ is sufficiently close to $1$, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Principal eigenstate classical shadows Grier, Daniel Pashayan, Hakop Schaeffer, Luke Quantum Physics Information Theory Machine Learning Given many copies of an unknown quantum state $ρ$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $ρ$ has an eigenstate $|ϕ\rangle$ with (unknown) eigenvalue $λ> 1/2$, the goal is to learn a (classical shadows style) classical description of $|ϕ\rangle$ which can later be used to estimate expectation values $\langle ϕ|O| ϕ\rangle$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $ρ$ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue $λ$ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when $λ$ is sufficiently close to $1$, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows. |
| title | Principal eigenstate classical shadows |
| topic | Quantum Physics Information Theory Machine Learning |
| url | https://arxiv.org/abs/2405.13939 |