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DOI: 10.4230/LIPIcs.ITCS.2023.64

Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom

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

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

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.

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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)

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@InProceedings{grewal_et_al:LIPIcs.ITCS.2023.64,
  author =	{Grewal, Sabee and Iyer, Vishnu and Kretschmer, William and Liang, Daniel},
  title =	{{Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{64:1--64:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.64},
  URN =		{urn:nbn:de:0030-drops-175670},
  doi =		{10.4230/LIPIcs.ITCS.2023.64},
  annote =	{Keywords: Pseudorandom quantum states, Clifford + T, Haar random, Bell sampling, stabilizer formalism, stabilizer extent, stabilizer fidelity, learning theory, complexity theory}
}

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Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom

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