量子图灵机

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模板:Turing

基础定义

A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algorithm can be expressed formally as a particular quantum Turing machine. However, the computationally equivalent quantum circuit is a more common model.[1][2]:2


量子图灵机计算机(QTM)又称为通用量子计算机是一种用来模拟量子计算机效果的抽象机器。它提供了一个简单的模型，但又能展现出量子计算的所有威力。也就是说，任何量子算法都可以被形式化地表示为一个特定的量子图灵机。提供等效计算能力的量子电路是一种比较常用的模型。[1][2]




与经典计算的关系

Quantum Turing machines can be related to classical and probabilistic Turing machines in a framework based on transition matrices. That is, a matrix can be specified whose product with the matrix representing a classical or probabilistic machine provides the quantum probability matrix representing the quantum machine. This was shown by Lance Fortnow.[3]

Quantum Turing machines can be related to classical and probabilistic Turing machines in a framework based on transition matrices. That is, a matrix can be specified whose product with the matrix representing a classical or probabilistic machine provides the quantum probability matrix representing the quantum machine. This was shown by Lance Fortnow.

在基于转移矩阵的框架下，量子图灵机可以与经典图灵机和概率图灵机相关联。也就是说，可以指定一个矩阵，该矩阵与表示经典或概率机器的矩阵的乘积提供了表示量子机器的量子概率矩阵。这是由兰斯 · 福特诺展示的。[3]



Informal sketch

模板:Unsolved

A way of understanding the quantum Turing machine (QTM) is that it generalizes the classical Turing machine (TM) in the same way that the quantum finite automaton (QFA) generalizes the deterministic finite automaton (DFA). In essence, the internal states of a classical TM are replaced by pure or mixed states in a Hilbert space; the transition function is replaced by a collection of unitary matrices that map the Hilbert space to itself.[4]

A way of understanding the quantum Turing machine (QTM) is that it generalizes the classical Turing machine (TM) in the same way that the quantum finite automaton (QFA) generalizes the deterministic finite automaton (DFA). In essence, the internal states of a classical TM are replaced by pure or mixed states in a Hilbert space; the transition function is replaced by a collection of unitary matrices that map the Hilbert space to itself. that first described a quantum mechanical model of Turing machines. A 1985 article written by Oxford University physicist David Deutsch further developed the idea of quantum computers by suggesting quantum gates could function in a similar fashion to traditional digital computing binary logic gates.

理解量子图灵机的一个方法是它推广了经典的图灵机，就像量子有限自动机推广了确定有限状态自动机机一样。实质上，经典 TM 的内部状态被希尔伯特空间中的纯态或混合态所取代，转移函数被映射 Hilbert 空间到自身的一组酉矩阵所取代。首次描述了图灵机的量子力学模型。牛津大学物理学家大卫 · 多伊奇在1985年的一篇文章中进一步发展了量子计算机的概念，他提出量子门可以以类似于传统数字计算二进制逻辑门的方式运行。


That is, a classical Turing machine is described by a 7-tuple [math]\displaystyle{ M = \langle Q, \Gamma, b, \Sigma, \delta, q_0, F \rangle }[/math].

Iriyama, Ohya, and Volovich have developed a model of a linear quantum Turing machine (LQTM). This is a generalization of a classical QTM that has mixed states and that allows irreversible transition functions. These allow the representation of quantum measurements without classical outcomes.

Iriyama、 Ohya 和 Volovich 开发了一个线性量子图灵机模型(LQTM)。这是一个具有混合状态并允许不可逆转换函数的经典 QTM 的推广。这使得量子测量可以在没有经典结果的情况下进行表示。


For a three-tape quantum Turing machine (one tape holding the input, a second tape holding intermediate calculation results, and a third tape holding output):

A quantum Turing machine with postselection was defined by Scott Aaronson, who showed that the class of polynomial time on such a machine (PostBQP) is equal to the classical complexity class PP.

一个具有后选择的量子图灵机由 Scott Aaronson 定义，他证明了这样一个机器上的多项式时间类(PostBQP)等于经典的复杂性类 PP。

  • The set of states [math]\displaystyle{ Q }[/math] is replaced by a Hilbert space.
  • The tape alphabet symbols [math]\displaystyle{ \Gamma }[/math] are likewise replaced by a Hilbert space (usually a different Hilbert space than the set of states).
  • The blank symbol [math]\displaystyle{ b \in \Gamma }[/math] corresponds to the zero-vector.
  • The input and output symbols [math]\displaystyle{ \Sigma }[/math] are usually taken as a discrete set, as in the classical system; thus, neither the input nor output to a quantum machine need be a quantum system itself.
  • The transition function [math]\displaystyle{ \delta : \Sigma \times Q \otimes \Gamma \rightarrow \Sigma \times Q \otimes \Gamma \times \{L,R\} }[/math] is a generalization of a transition monoid, and is understood to be a collection of unitary matrices that are automorphisms of the Hilbert space [math]\displaystyle{ Q }[/math].
  • The set [math]\displaystyle{ F }[/math] of final or accepting states is a subspace of the Hilbert space [math]\displaystyle{ Q }[/math].


The above is merely a sketch of a quantum Turing machine, rather than its formal definition, as it leaves vague several important details: for example, how often a measurement is performed; see for example, the difference between a measure-once and a measure-many QFA. This question of measurement affects the way in which writes to the output tape are defined.


History

In 1980 and 1982, physicist Paul Benioff published papers[5][6] that first described a quantum mechanical model of Turing machines. A 1985 article written by Oxford University physicist David Deutsch further developed the idea of quantum computers by suggesting quantum gates could function in a similar fashion to traditional digital computing binary logic gates.[4]


Iriyama, Ohya, and Volovich have developed a model of a linear quantum Turing machine (LQTM). This is a generalization of a classical QTM that has mixed states and that allows irreversible transition functions. These allow the representation of quantum measurements without classical outcomes.[7]


A quantum Turing machine with postselection was defined by Scott Aaronson, who showed that the class of polynomial time on such a machine (PostBQP) is equal to the classical complexity class PP.[8]


See also

Category:Turing machine

类别: 图灵机

Category:Quantum complexity theory

范畴: 量子复杂性理论


This page was moved from wikipedia:en:Quantum Turing machine. Its edit history can be viewed at 量子图灵机/edithistory

  1. ↑ 1.0 1.1 Andrew Yao (1993). Quantum circuit complexity. 34th Annual Symposium on Foundations of Computer Science. pp. 352–361.
  2. ↑ 2.0 2.1 Abel Molina; John Watrous (2018). "Revisiting the simulation of quantum Turing machines by quantum circuits". arXiv:1808.01701 [cs.CC].
  3. ↑ 3.0 3.1 Fortnow, Lance (2003). "One Complexity Theorist's View of Quantum Computing". Theoretical Computer Science. 292 (3): 597–610. arXiv:quant-ph/0003035. doi:10.1016/S0304-3975(01)00377-2.
  4. ↑ 4.0 4.1 Deutsch, David (July 1985). "Quantum theory, the Church-Turing principle and the universal quantum computer" (PDF). Proceedings of the Royal Society A. 400 (1818): 97–117. Bibcode:1985RSPSA.400...97D. CiteSeerX 10.1.1.41.2382. doi:10.1098/rspa.1985.0070. Archived from the original (PDF) on 2008-11-23.
  5. ↑ Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/bf01011339.
  6. ↑ Benioff, P. (1982). "Quantum mechanical hamiltonian models of turing machines". Journal of Statistical Physics. 29 (3): 515–546. Bibcode:1982JSP....29..515B. doi:10.1007/BF01342185.
  7. ↑ Simon Perdrix; Philippe Jorrand (2007-04-04). "Classically-Controlled Quantum Computation". Math. Struct. In Comp. Science. 16 (4): 601–620. arXiv:quant-ph/0407008. doi:10.1017/S096012950600538X. also Simon Perdrix and Philippe Jorrand (2006). "Classically-Controlled Quantum Computation" (PDF). Math. Struct. In Comp. Science. 16 (4): 601–620. CiteSeerX 10.1.1.252.1823. doi:10.1017/S096012950600538X.
  8. ↑ Aaronson, Scott (2005). "Quantum computing, postselection, and probabilistic polynomial-time". Proceedings of the Royal Society A. 461 (2063): 3473–3482. arXiv:quant-ph/0412187. Bibcode:2005RSPSA.461.3473A. doi:10.1098/rspa.2005.1546. Preprint available at [1]
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