Commuting quantum circuits with few outputs are unlikely to be classically simulatable

Yasuhiro Takahashi, Seiichiro Tani, Takeshi Yamazaki, Kazuyuki Tanaka

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its out-put probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. Our main result is that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is O(log n). Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, using a proof similar to that of the main result, we provide an evidence that a slightly extended Clifford circuit is not classically simulatable.

Original languageEnglish
Pages (from-to)251-270
Number of pages20
JournalQuantum Information and Computation
Issue number3-4
Publication statusPublished - 2016 Mar 1


  • Classical simulation
  • Clifford circuit
  • Commuting quantum circuit

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Computational Theory and Mathematics


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