Complexity Classification of Conjugated Clifford Circuits

Authors Adam Bouland , Joseph F. Fitzsimons, Dax Enshan Koh

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Adam Bouland
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
Joseph F. Fitzsimons
  • Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 , Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
Dax Enshan Koh
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh. Complexity Classification of Conjugated Clifford Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 21:1-21:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Computational complexity and cryptography
  • gate set classification
  • quantum advantage
  • sampling problems
  • polynomial hierarchy


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