## Quantum Domain Theory - Definitions and Applications (2003)

Venue: | Proceedings of CCA’03 |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Kashefi03quantumdomain,

author = {Elham Kashefi},

title = {Quantum Domain Theory - Definitions and Applications},

booktitle = {Proceedings of CCA’03},

year = {2003}

}

### OpenURL

### Abstract

Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1

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Citation Context ...titude of different tasks simultaneously (in parallel). In this section we introduce some essential background material on quantum computing, but for a more comprehensive study we refer the reader to =-=[26]-=-. The state of a closed quantum physical system which is not interacting with an environment (pure state) is described by a unit vector in a Hilbert space, which in Dirac notation is denoted by |ψ〉 ∈ ... |

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Citation Context ...tric spaces can be found in the literature. We use the domain of the closed balls [39, 11] to introduce a domain model for quantum pure states. Next the power domain of the domain of the closed balls =-=[21, 10, 24]-=- will be used to introduce the domain of quantum mixed states. The authors acknowledge that the main definitions and results of this subsection have appeared in [11, 10] under the theory of computabil... |

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Citation Context ... 14smachine with quantum memory registers. To develop the proper foundation for quantum semantics, in the most general setting, we consider density matrices and CP maps. Aharanov, Kitaev and Nisan in =-=[2]-=- introduced the first computational model based on mixed state where possible operators are represented by CP maps. We show in this subsection that the same structure of the classical probabilistic se... |

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Citation Context ...rly in other models, such as temporal complexity [31, 30]. As another example, recent developments in quantum programming languages suggest the requirement of models with higher levels of abstraction =-=[28, 27, 32, 6, 35]-=-. 1sDomain theory provides us with an alternative and more abstract model for computation. Domain theory is traditionally a suitable model for information processing given incompletely specified eleme... |

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Citation Context ...Domain theory provides us with an alternative and more abstract model for computation. Domain theory is traditionally a suitable model for information processing given incompletely specified elements =-=[1, 10]-=-. Furthermore domain theory has proven to be a proper mathematical structure to describe denotational semantics for programming languages whilst also being applicable to the study of computability of ... |

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Citation Context ... indeed any other mathematical spaces. Recently, a new approach to computability has been developed which is based on domain theory and fits into the aforementioned second framework for computability =-=[39, 15, 5, 12]-=-. In his famous article [34], Scott points out the relationship between continuity versus computability. For most purposes, to detect whether some construction is computationally feasible - it is suff... |

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Citation Context ... indeed any other mathematical spaces. Recently, a new approach to computability has been developed which is based on domain theory and fits into the aforementioned second framework for computability =-=[39, 15, 5, 12]-=-. In his famous article [34], Scott points out the relationship between continuity versus computability. For most purposes, to detect whether some construction is computationally feasible - it is suff... |

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Citation Context ...ection we extend this concept to the quantum setting. We define the notion of an effectively given ω-continuous domain by putting a proper recursive structure on the elements of a basis of the domain =-=[37, 10]-=-. Definition 10 Assume domain D is ω-continuous with a countable basis B = {b0, b1, b2, · · ·}. We say D is effectively given with respect to B, if the relation bn ≪ bm is recursively enumerable (r.e.... |

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Citation Context ...rly in other models, such as temporal complexity [31, 30]. As another example, recent developments in quantum programming languages suggest the requirement of models with higher levels of abstraction =-=[28, 27, 32, 6, 35]-=-. 1sDomain theory provides us with an alternative and more abstract model for computation. Domain theory is traditionally a suitable model for information processing given incompletely specified eleme... |

29 | Computational model underlying the one-way quantum computer
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Citation Context ...omputer (a new model in which measurement plays the central role) presents new aspects of quantum information processing that can not be analysed properly in other models, such as temporal complexity =-=[31, 30]-=-. As another example, recent developments in quantum programming languages suggest the requirement of models with higher levels of abstraction [28, 27, 32, 6, 35]. 1sDomain theory provides us with an ... |

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Citation Context ...es with completely positive maps has a similar semantical structure as probabilistic computation over random variables. To this end, first we present some standard basic definitions for vector spaces =-=[4, 23]-=-. Definition 13 A subset P in a vector space V is called positive cone iff it satisfies the following conditions: ∀x, y ∈ P and positive scalars a, b : ax + by ∈ P ∀x ∈ P : x, −x ∈ P ⇒ x = 0 . P induc... |

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Citation Context ... indeed any other mathematical spaces. Recently, a new approach to computability has been developed which is based on domain theory and fits into the aforementioned second framework for computability =-=[39, 15, 5, 12]-=-. In his famous article [34], Scott points out the relationship between continuity versus computability. For most purposes, to detect whether some construction is computationally feasible - it is suff... |

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Citation Context .... 11sTherefore, to present a computational framework for mixed states, it is enough to construct such a framework for probability measure on H. To this end we need the following notations and results =-=[18, 10, 2, 24]-=-. The domain of probability measures will be defined in terms of continuous valuation functions, a finite measure which is defined on open subsets of a topological space [4, 19, 10]. Definition 17 Ass... |

22 | Approximation of Metric Spaces by Partial Metric Spaces
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Citation Context ...hematical foundation for quantum computability. Pure quantum states A standard way to construct a partially ordered set for a given metric space (X, d) is based on ordering of the set of closed balls =-=[20]-=-. Define a closed ball C(x, r) of given metric space (X, d) with x ∈ X and r ∈ R to be the following set: C(x, r) = {y ∈ X | d(x, y) ≤ r} . The Hilbert space H of the quantum pure state is a metric sp... |

19 | A partial order on classical and quantum states
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Citation Context ...ity for quantum computation. Subsequently a denotational semantics for quantum computation is given. Finally we review recent work on information aspects of quantum domain theory by Coecke and Martin =-=[7]-=-. By introducing a domain framework for quantum computation we aim to address different aspects of information processing which has not yet been studied in other existing models of quantum computation... |

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Citation Context ...work for quantum computability has been introduced. Although it is known that the class of quantum computable functions is same as the class of classical computable functions (Church-Turing Principle =-=[8, 22, 29]-=-), we believe that by considering a proper framework for quantum computability we may be able to address new and interesting questions. We also presented a topological structure for quantum computatio... |

15 |
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Citation Context ...s argument to the probabilistic quantum Turing machine [29]. Ozawa also distinguished the notation of measurability from computability to clarify the concerns raised by Neilsen on the halting problem =-=[25]-=-. Apart from these few discussions, there have been no further attempts in this direction. We believe, by introducing a proper framework for quantum computability, we can address more interesting ques... |

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Citation Context .... 11sTherefore, to present a computational framework for mixed states, it is enough to construct such a framework for probability measure on H. To this end we need the following notations and results =-=[18, 10, 2, 24]-=-. The domain of probability measures will be defined in terms of continuous valuation functions, a finite measure which is defined on open subsets of a topological space [4, 19, 10]. Definition 17 Ass... |

14 | Spaces of valuations
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Citation Context ...ns and results [18, 10, 2, 24]. The domain of probability measures will be defined in terms of continuous valuation functions, a finite measure which is defined on open subsets of a topological space =-=[4, 19, 10]-=-. Definition 17 Assume that X is a topological space. A function ν from open sets of X to non-negative real number, R + , is called a continuous valuation function iff the following conditions are sat... |

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Citation Context |

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Citation Context ...operators satisfy the completeness equation, � m M † mMm = I . 3 Classical Domain Theory Domain theory was introduced independently by Scott [34] for the study of denotational semantics and by Ershow =-=[14]-=- as a tool for the study of partial computable functions. A complete survey of domain theory and its applications can be found in [1, 10]. Domain Theory has been developed towards the following key ap... |

11 |
The one-way quantum computer - a non-network model of quantum computation
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Citation Context ...omputer (a new model in which measurement plays the central role) presents new aspects of quantum information processing that can not be analysed properly in other models, such as temporal complexity =-=[31, 30]-=-. As another example, recent developments in quantum programming languages suggest the requirement of models with higher levels of abstraction [28, 27, 32, 6, 35]. 1sDomain theory provides us with an ... |

6 | Measurability and computability
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Citation Context ... deterministic quantum Turing machine is equal to the class of recursive functions (computable by a classical Turing machine). Ozawa extended this argument to the probabilistic quantum Turing machine =-=[29]-=-. Ozawa also distinguished the notation of measurability from computability to clarify the concerns raised by Neilsen on the halting problem [25]. Apart from these few discussions, there have been no ... |

5 |
Toward an Architecture for Quantum Programming. LANL Archive cs.PL/0103009
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Citation Context ...rly in other models, such as temporal complexity [31, 30]. As another example, recent developments in quantum programming languages suggest the requirement of models with higher levels of abstraction =-=[28, 27, 32, 6, 35]-=-. 1sDomain theory provides us with an alternative and more abstract model for computation. Domain theory is traditionally a suitable model for information processing given incompletely specified eleme... |

2 | Powerdomains and zero finding
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(Show Context)
Citation Context ...tric spaces can be found in the literature. We use the domain of the closed balls [39, 11] to introduce a domain model for quantum pure states. Next the power domain of the domain of the closed balls =-=[21, 10, 24]-=- will be used to introduce the domain of quantum mixed states. The authors acknowledge that the main definitions and results of this subsection have appeared in [11, 10] under the theory of computabil... |

1 |
Conventions for quantum pseucode
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Citation Context ...ogramming language for quantum computation. The recent literature contains several proposals for quantum programming languages. The first contribution in this direction is Knill’s paper on QRAM model =-=[13]-=-. The other attempts to define a true quantum programming language are: first an approach by Ömer [28, 27] which has a C-like syntax, and second a proposal by Sanders and Zuliani [32] based on Dijkstr... |

1 |
A procedural formalism for quantum computing
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1 |
Towards a quantum programming language. submited
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1 |
Uniqueness of entanglement measure and thermodynamics
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Citation Context ...iscussed in their paper. Therefore, a domain theoretical approach to the theory of entanglement manipulation may provide us with a uniform framework for measuring the entanglement in the same line as =-=[42]-=-. 3 The set [0,∞) ∗ is the domain of nonnegative real numbers in their opposite order. 18sAcknowledgement I would like to thank Angelo Carollo, Jens Eisert, Ivette Fuentes Guridi, Barry C. Sanders, Vl... |