Principal eigenstate classical shadows

Daniel Grier; Hakop Pashayan; Luke Schaeffer

Principal eigenstate classical shadows

Daniel Grier, Hakop Pashayan, Luke Schaeffer

Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:2122-2165, 2024.

Abstract

Given many copies of an unknown quantum state

$\rho$ , we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that

$\rho$ has an eigenstate

$|\phi⟩$ with (unknown) eigenvalue

$\lambda > 1/2$ , the goal is to learn a (classical shadows style) classical description of

$|\phi⟩$ which can later be used to estimate expectation values

$⟨\phi |O | \phi ⟩$ for any

$O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of

$\rho$ 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

$\lambda$ 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

$\lambda$ is sufficiently close to

$1$ , the performance of our algorithm is optimal—matching the sample complexity for pure state classical shadows.

Cite this Paper

BibTeX


@InProceedings{pmlr-v247-grier24a,
  title = 	 {Principal eigenstate classical shadows},
  author =       {Grier, Daniel and Pashayan, Hakop and Schaeffer, Luke},
  booktitle = 	 {Proceedings of Thirty Seventh Conference on Learning Theory},
  pages = 	 {2122--2165},
  year = 	 {2024},
  editor = 	 {Agrawal, Shipra and Roth, Aaron},
  volume = 	 {247},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {30 Jun--03 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v247/grier24a/grier24a.pdf},
  url = 	 {https://proceedings.mlr.press/v247/grier24a.html},
  abstract = 	 {Given many copies of an unknown quantum state $\rho$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $\rho$ has an eigenstate $|\phi⟩$ with (unknown) eigenvalue $\lambda > 1/2$, the goal is to learn a (classical shadows style) classical description of $|\phi⟩$ which can later be used to estimate expectation values $⟨\phi |O | \phi ⟩$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $\rho$ 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 $\lambda$ 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 $\lambda$ is sufficiently close to $1$, the performance of our algorithm is optimal—matching the sample complexity for pure state classical shadows.}
}

Endnote

%0 Conference Paper
%T Principal eigenstate classical shadows
%A Daniel Grier
%A Hakop Pashayan
%A Luke Schaeffer
%B Proceedings of Thirty Seventh Conference on Learning Theory
%C Proceedings of Machine Learning Research
%D 2024
%E Shipra Agrawal
%E Aaron Roth	
%F pmlr-v247-grier24a
%I PMLR
%P 2122--2165
%U https://proceedings.mlr.press/v247/grier24a.html
%V 247
%X Given many copies of an unknown quantum state $\rho$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $\rho$ has an eigenstate $|\phi⟩$ with (unknown) eigenvalue $\lambda > 1/2$, the goal is to learn a (classical shadows style) classical description of $|\phi⟩$ which can later be used to estimate expectation values $⟨\phi |O | \phi ⟩$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $\rho$ 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 $\lambda$ 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 $\lambda$ is sufficiently close to $1$, the performance of our algorithm is optimal—matching the sample complexity for pure state classical shadows.

APA


Grier, D., Pashayan, H. & Schaeffer, L.. (2024). Principal eigenstate classical shadows. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:2122-2165 Available from https://proceedings.mlr.press/v247/grier24a.html.

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