Complexity Classification of Conjugated Clifford Circuits
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.
gate set classification
quantum advantage
sampling problems
polynomial hierarchy
Theory of computation~Quantum complexity theory
Theory of computation~Computational complexity and cryptography
21:1-21:25
Regular Paper
https://arxiv.org/abs/1709.01805
Adam
Bouland
Adam Bouland
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
https://orcid.org/0000-0002-8556-8337
AB was partially supported by the NSF GRFP under Grant No. 1122374, by a Vannevar Bush Fellowship from the US Department of Defense, and by an NSF Waterman award under grant number 1249349.
Joseph F.
Fitzsimons
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
JFF acknowledges support from the Air Force Office of Scientific Research under AOARD grant no. FA2386-15-1-4082. This material is based on research supported in part by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2013-01.
Dax Enshan
Koh
Dax Enshan Koh
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
https://orcid.org/0000-0002-8968-591X
DEK is supported by the National Science Scholarship from the Agency for Science, Technology and Research (A*STAR).
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Adam Bouland, Joseph F. Fitzsimons, and Dax E. Koh
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