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Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom

Authors Sabee Grewal , Vishnu Iyer , William Kretschmer , Daniel Liang

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Author Details

Sabee Grewal
  • The University of Texas at Austin, TX, USA
Vishnu Iyer
  • The University of Texas at Austin, TX, USA
William Kretschmer
  • The University of Texas at Austin, TX, USA
Daniel Liang
  • The University of Texas at Austin, TX, USA


We thank Scott Aaronson for helpful comments.

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Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang. Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 64:1-64:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We show that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random. Specifically, given an n-qubit pure state |ψ⟩, we give an efficient algorithm that distinguishes whether |ψ⟩ is (i) Haar-random or (ii) a state with stabilizer fidelity at least 1/k (i.e., has fidelity at least 1/k with some stabilizer state), promised that one of these is the case. With black-box access to |ψ⟩, our algorithm uses O(k^{12} log(1/δ)) copies of |ψ⟩ and O(n k^{12} log(1/δ)) time to succeed with probability at least 1-δ, and, with access to a state preparation unitary for |ψ⟩ (and its inverse), O(k³ log(1/δ)) queries and O(n k³ log(1/δ)) time suffice. As a corollary, we prove that ω(log(n)) T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Pseudorandom quantum states
  • Clifford + T
  • Haar random
  • Bell sampling
  • stabilizer formalism
  • stabilizer extent
  • stabilizer fidelity
  • learning theory
  • complexity theory


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