Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

Sabee Grewal; Vishnu Iyer; William Kretschmer; Daniel Liang

doi:10.22331/q-2025-11-06-1907

Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

Sabee Grewal¹, Vishnu Iyer¹, William Kretschmer^1,2, and Daniel Liang^1,3,4

¹The University of Texas at Austin
²Simons Institute for the Theory of Computing, UC Berkeley
³Rice University
⁴Portland State University

Published:	2025-11-06, volume 9, page 1907
Editor:	Tom Gur
Eprint:	arXiv:2305.13409v5
Doi:	https://doi.org/10.22331/q-2025-11-06-1907
Citation:	Quantum 9, 1907 (2025).

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Abstract

We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|\psi\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|\psi\rangle$ to learn $|\psi\rangle$ to trace distance at most $\varepsilon$.
The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|\psi\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.

► BibTeX data

@article{Grewal2025efficientlearningof,
  doi = {10.22331/q-2025-11-06-1907},
  url = {https://doi.org/10.22331/q-2025-11-06-1907},
  title = {Efficient {L}earning of {Q}uantum {S}tates {P}repared {W}ith {F}ew {N}on-{C}lifford {G}ates},
  author = {Grewal, Sabee and Iyer, Vishnu and Kretschmer, William and Liang, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1907},
  month = nov,
  year = {2025}
}

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