Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits

Authors Boaz Barak, Chi-Ning Chou, Xun Gao

Thumbnail PDF


  • Filesize: 0.64 MB
  • 20 pages

Document Identifiers

Author Details

Boaz Barak
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Chi-Ning Chou
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Xun Gao
  • Department of Physics, Harvard University, Cambridge, MA, USA


We thank Scott Aaronson for helpful discussions.

Cite AsGet BibTex

Boaz Barak, Chi-Ning Chou, and Xun Gao. Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit C with n inputs and outputs and purported simulator whose output is distributed according to a distribution p over {0,1}ⁿ, the linear XEB fidelity of the simulator is ℱ_C(p) = 2ⁿ 𝔼_{x ∼ p} q_C(x) -1, where q_C(x) is the probability that x is output from the distribution C |0ⁿ⟩. A trivial simulator (e.g., the uniform distribution) satisfies ℱ_C(p) = 0, while Google’s noisy quantum simulation of a 53-qubit circuit C achieved a fidelity value of (2.24 ±0.21)×10^{-3} (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit C of depth d with Haar random 2-qubit gates achieves in expectation a fidelity value of Ω(n/L⋅15^{-d}) in running time poly(n,2^L). Here L is the size of the light cone of C: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of ω(1) for depth O(√{log n}) two-dimensional circuits. This is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Quantum supremacy
  • Linear cross-entropy benchmark


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson and Sam Gunn. On the classical hardness of spoofing linear cross-entropy benchmarking. arXiv preprint, 2019. URL: http://arxiv.org/abs/1910.12085.
  2. Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510, 2019. Google Scholar
  3. Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. Physical review letters, 117(8):080501, 2016. Google Scholar
  4. Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum, 1:8, 2017. Google Scholar
  5. PW Brouwer and CWJ Beenakker. Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems. Journal of Mathematical Physics, 37(10):4904-4934, 1996. Google Scholar
  6. Xun Gao and Luming Duan. Efficient classical simulation of noisy quantum computation. arXiv preprint, 2018. URL: http://arxiv.org/abs/1810.03176.
  7. Nicholas Hunter-Jones. Unitary designs from statistical mechanics in random quantum circuits. arXiv preprint, 2019. URL: http://arxiv.org/abs/1905.12053.
  8. John Napp, Rolando L La Placa, Alexander M Dalzell, Fernando GSL Brandao, and Aram W Harrow. Efficient classical simulation of random shallow 2d quantum circuits. arXiv preprint, 2019. URL: http://arxiv.org/abs/2001.00021.
  9. Kyungjoo Noh, Liang Jiang, and Bill Fefferman. Efficient classical simulation of noisy random quantum circuits in one dimension. arXiv preprint, 2020. URL: http://arxiv.org/abs/2003.13163.
  10. Guifré Vidal. Efficient simulation of one-dimensional quantum many-body systems. Physical review letters, 93(4):040502, 2004. Google Scholar
  11. Man-Hong Yung and Xun Gao. Can chaotic quantum circuits maintain quantum supremacy under noise? arXiv preprint, 2017. URL: http://arxiv.org/abs/1706.08913.
  12. Yiqing Zhou, E Miles Stoudenmire, and Xavier Waintal. What limits the simulation of quantum computers? arXiv preprint, 2020. URL: http://arxiv.org/abs/2002.07730.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail