article

Computational complexity of uniform quantum circuit families and quantum Turing machines

Authors:

Harumichi Nishimura,

Masanao OzawaAuthors Info & Claims

Theoretical Computer Science, Volume 276, Issue 1-2

Pages 147 - 181

https://doi.org/10.1016/S0304-3975(01)00111-6

Published: 01 April 2002 Publication History

Abstract

Deutsch proposed two sorts of models of quantum computers, quantum Turing machines (QTMs) and quantum circuit families (QCFs). In this paper we explore the computational powers of these models and re-examine the claim of the computational equivalence of these models often made in the literature without detailed investigations. For this purpose, we formulate the notion of the codes of QCFs and the uniformity of QCFs by the computability of the codes. Various complexity classes are introduced for QTMs and QCFs according to constraints on the error probability of algorithms or transition amplitudes. Their interrelations are examined in detail. For Monte Carlo algorithms, it is proved that the complexity classes based on uniform QCFs are identical with the corresponding classes based on QTMs. However, for Las Vegas algorithms, it is still open whether the two models are equivalent. We indicate the possibility that they are not equivalent. In addition, we give a complete proof of the existence of a universal QTM efficiently simulating multi-tape QTMs. We also examine the simulation of various types of QTMs such as multi-tape QTMs, single tape QTMs, stationary, normal form QTMs (SNQTMs), and QTMs with the binary tapes. As a result, we show that these QTMs are computationally equivalent to one another as computing models implementing not only Monte Carlo algorithms but also exact (or error-free) ones

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Cited By

Nishimura HYamakami T(2019)Polynomial time quantum computation with adviceInformation Processing Letters10.1016/j.ipl.2004.02.00590:4(195-204)Online publication date: 5-Jan-2019
https://dl.acm.org/doi/10.1016/j.ipl.2004.02.005
Lago UMasini AZorzi M(2018)On a measurement-free quantum lambda calculus with classical controlMathematical Structures in Computer Science10.1017/S096012950800741X19:2(297-335)Online publication date: 24-Dec-2018
https://dl.acm.org/doi/10.1017/S096012950800741X
Perdrix SJorrand P(2018)Classically controlled quantum computationMathematical Structures in Computer Science10.1017/S096012950600538X16:4(601-620)Online publication date: 24-Dec-2018
https://dl.acm.org/doi/10.1017/S096012950600538X
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Index Terms

Computational complexity of uniform quantum circuit families and quantum Turing machines
1. Applied computing
  1. Physical sciences and engineering
    1. Physics
2. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes
  2. Models of computation
    1. Computability
      1. Turing machines

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cover image Theoretical Computer Science

Theoretical Computer Science Volume 276, Issue 1-2

April 6, 2002

447 pages

ISSN:0304-3975

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Publisher

Elsevier Science Publishers Ltd.

United Kingdom

Publication History

Published: 01 April 2002

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Cited By

Nishimura HYamakami T(2019)Polynomial time quantum computation with adviceInformation Processing Letters10.1016/j.ipl.2004.02.00590:4(195-204)Online publication date: 5-Jan-2019
https://dl.acm.org/doi/10.1016/j.ipl.2004.02.005
Lago UMasini AZorzi M(2018)On a measurement-free quantum lambda calculus with classical controlMathematical Structures in Computer Science10.1017/S096012950800741X19:2(297-335)Online publication date: 24-Dec-2018
https://dl.acm.org/doi/10.1017/S096012950800741X
Perdrix SJorrand P(2018)Classically controlled quantum computationMathematical Structures in Computer Science10.1017/S096012950600538X16:4(601-620)Online publication date: 24-Dec-2018
https://dl.acm.org/doi/10.1017/S096012950600538X
Nishimura HOzawa M(2018)Perfect computational equivalence between quantum Turing machines and finitely generated uniform quantum circuit familiesQuantum Information Processing10.1007/s11128-008-0091-88:1(13-24)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s11128-008-0091-8
Pierro APanarotto F(2014)A Calculus of AnyonsProceedings of the 21st International Workshop on Logic, Language, Information, and Computation - Volume 865210.1007/978-3-662-44145-9_11(152-165)Online publication date: 1-Sep-2014
https://dl.acm.org/doi/10.1007/978-3-662-44145-9_11
Dal Lago UMasini AZorzi M(2010)Quantum implicit computational complexityTheoretical Computer Science10.1016/j.tcs.2009.07.045411:2(377-409)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.1016/j.tcs.2009.07.045
Iriyama SOhya M(2007)On generalized quantum turing machine and its language classesProceedings of the 11th WSEAS International Conference on Applied Mathematics10.5555/1353734.1353744(51-56)Online publication date: 22-Mar-2007
https://dl.acm.org/doi/10.5555/1353734.1353744
Bera DGreen FHomer S(2007)Small depth quantum circuitsACM SIGACT News10.1145/1272729.127273938:2(35-50)Online publication date: 1-Jun-2007
https://dl.acm.org/doi/10.1145/1272729.1272739
Sauerhoff MSieling D(2005)Quantum branching programs and space-bounded nonuniform quantum complexityTheoretical Computer Science10.1016/j.tcs.2004.12.031334:1-3(177-225)Online publication date: 11-Apr-2005
https://dl.acm.org/doi/10.1016/j.tcs.2004.12.031
Nishimura HOzawa M(2005)Uniformity of quantum circuit families for error-free algorithmsTheoretical Computer Science10.1016/j.tcs.2004.12.020332:1-3(487-496)Online publication date: 28-Feb-2005
https://dl.acm.org/doi/10.1016/j.tcs.2004.12.020
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