Quantum Information Processing

, Volume 8, Issue 1, pp 13–24 | Cite as

Perfect computational equivalence between quantum Turing machines and finitely generated uniform quantum circuit families

  • Harumichi Nishimura
  • Masanao OzawaEmail author
Open Access


In order to establish the computational equivalence between quantum Turing machines (QTMs) and quantum circuit families (QCFs) using Yao’s quantum circuit simulation of QTMs, we previously introduced the class of uniform QCFs based on an infinite set of elementary gates, which has been shown to be computationally equivalent to the polynomial-time QTMs (with appropriate restriction of amplitudes) up to bounded error simulation. This result implies that the complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its counterpart for uniform QCFs. However, the complexity classes ZQP and EQP for QTMs do not appear to equal their counterparts for uniform QCFs. In this paper, we introduce a subclass of uniform QCFs, the finitely generated uniform QCFs, based on finite number of elementary gates and show that the class of finitely generated uniform QCFs is perfectly equivalent to the class of polynomial-time QTMs; they can exactly simulate each other. This naturally implies that BQP as well as ZQP and EQP equal the corresponding complexity classes of the finitely generated uniform QCFs.


Quantum computation Complexity theory Uniform circuit families Turing machines Finitely generated uniform quantum circuit families 



M. Ozawa gratefully acknowledges the financial support of the SCOPE project of the MIC, the Grant-in-Aid for Scientific Research (B) 17340021 of the JSPS, and the CREST project of the JST.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.School of ScienceOsaka Prefecture UniversitySakaiJapan
  2. 2.Graduate School of Information ScienceNagoya UniversityNagoyaJapan

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