Efficient Tomography of Non-Interacting-Fermion States

Authors Scott Aaronson, Sabee Grewal

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Author Details

Scott Aaronson
  • The University of Texas at Austin, TX, USA
Sabee Grewal
  • The University of Texas at Austin, TX, USA


We thank Andrew Zhao for notifying us that the previous version of this manuscript contained an error and providing other insightful comments. We also thank Yuxuan Zhang, Alex Kulesza, Ankur Moitra, William Kretschmer, Dax Enshan Koh, Andrea Rocchetto, and Patrick Rall for helpful discussions, and Alex Arkhipov, William Kretschmer, and Daniel Liang for helpful comments on a previous version of this manuscript.

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Scott Aaronson and Sabee Grewal. Efficient Tomography of Non-Interacting-Fermion States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We give an efficient algorithm that learns a non-interacting-fermion state, given copies of the state. For a system of n non-interacting fermions and m modes, we show that O(m³ n² log(1/δ) / ε⁴) copies of the input state and O(m⁴ n² log(1/δ)/ ε⁴) time are sufficient to learn the state to trace distance at most ε with probability at least 1 - δ. Our algorithm empirically estimates one-mode correlations in O(m) different measurement bases and uses them to reconstruct a succinct description of the entire state efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum information theory
  • Mathematics of computing → Probabilistic inference problems
  • Theory of computation → Quantum complexity theory
  • free-fermions
  • Gaussian fermions
  • non-interacting fermions
  • quantum state tomography
  • efficient tomography


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