## Quantum computation, categorical semantics and linear logic. quant-ph/0312174 (2003)

Citations: | 28 - 1 self |

### BibTeX

@MISC{Tonder03quantumcomputation,,

author = {André Van Tonder},

title = {Quantum computation, categorical semantics and linear logic. quant-ph/0312174},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

We develop a type theory and provide a denotational semantics for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic. In our semantics, variables inhabit certain Hilbert bundles, and computations are interpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal

### Citations

753 |
Types and Programming Languages
- Pierce
- 2002
(Show Context)
Citation Context ...on. The model for our interpretation will be a category. Types in the calculus will be interpreted as objects in the category, and judgments Γ ⊢ t : A will be interpreted as morphisms in the category =-=[12, 13, 14, 15]-=-. In the rest of this section we will give some motivation for the particular category that we will choose, which will then be constructed in more detail in the following sections. To begin our search... |

694 | Quantum theory, the Church-Turing principle and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...a programming language and as a formal algebraic system for reasoning about quantum algorithms. It provides a model of quantum computation that combines the universality of the quantum Turing machine =-=[2, 3]-=- and the compositionality of the quantum circuit models [4]. The calculus turned out to be closely related to the linear lambda calculi used in the 1study of linear logic [5, 6, 7, 8]. In [1], we set... |

661 | Linear logic
- Girard
- 1987
(Show Context)
Citation Context ...uantum Turing machine [2, 3] and the compositionality of the quantum circuit models [4]. The calculus turned out to be closely related to the linear lambda calculi used in the 1study of linear logic =-=[5, 6, 7, 8]-=-. In [1], we set up a computational model, or operational semantics, and an equational proof system for this calculus, and argued that it was equivalent to the quantum Turing machine, and therefore un... |

341 |
Foundations for Programming Languages
- Mitchell
- 1996
(Show Context)
Citation Context ...on. The model for our interpretation will be a category. Types in the calculus will be interpreted as objects in the category, and judgments Γ ⊢ t : A will be interpreted as morphisms in the category =-=[12, 13, 14, 15]-=-. In the rest of this section we will give some motivation for the particular category that we will choose, which will then be constructed in more detail in the following sections. To begin our search... |

287 | A computational interpretation of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...uantum Turing machine [2, 3] and the compositionality of the quantum circuit models [4]. The calculus turned out to be closely related to the linear lambda calculi used in the 1study of linear logic =-=[5, 6, 7, 8]-=-. In [1], we set up a computational model, or operational semantics, and an equational proof system for this calculus, and argued that it was equivalent to the quantum Turing machine, and therefore un... |

273 |
Semantics of Programming Languages: Structures and Techniques
- Gunter
- 1992
(Show Context)
Citation Context ...th. The model for our interpretation will be a category. Types in the calculus will be interpreted as objects in the category, and judgments Γ ⊢ t : A will be interpreted as morphisms between objects =-=[12, 13, 14, 15]-=-. In the rest of this section we will give some motivation for the particular category that we will choose, which will then be constructed in more detail in the following sections. To begin our search... |

252 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
Citation Context ...asoning about quantum algorithms. It provides a model of quantum computation that combines the universality of the quantum Turing machine [2, 3] and the compositionality of the quantum circuit models =-=[4]-=-. The calculus turned out to be closely related to the linear lambda calculi used in the 1study of linear logic [5, 6, 7, 8]. In [1], we set up a computational model, or operational semantics, and an... |

125 | The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines - Benioff - 1980 |

106 | Linear logic, ∗-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ...nterpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal closed category, as expected for a calculus based on linear logic =-=[9, 10]-=-. For simplicity, in this paper we restrict our attention to a purely linear fragment of the full quantum calculus. This fragment is not universal for quantum computation, but is at least as expressiv... |

98 | What is a categorical model of intuitionistic linear logic?’, Typed lambda calculi and applications
- Bierman
- 1995
(Show Context)
Citation Context ...nterpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal closed category, as expected for a calculus based on linear logic =-=[9, 10]-=-. For simplicity, in this paper we restrict our attention to a purely linear fragment of the full quantum calculus. This fragment is not universal for quantum computation, but is at least as expressiv... |

79 | A Taste of Linear Logic
- Wadler
- 1993
(Show Context)
Citation Context ...uantum Turing machine [2, 3] and the compositionality of the quantum circuit models [4]. The calculus turned out to be closely related to the linear lambda calculi used in the 1study of linear logic =-=[5, 6, 7, 8]-=-. In [1], we set up a computational model, or operational semantics, and an equational proof system for this calculus, and argued that it was equivalent to the quantum Turing machine, and therefore un... |

63 | Non-deterministic quantum programming
- Zuliani
- 2004
(Show Context)
Citation Context ...culus. Attempts to develop an interpretation of quantum computing using Chu space models of linear logic are described in [18] and [19]. The imperative language qGCL, developed by Sanders and Zuliani =-=[20]-=-, is based on Dijkstra’s guarded command language. It has a formal semantics and proof system. An alternative approach to providing an operational semantics of quantum computation based on process alg... |

53 | A lambda calculus for quantum computation
- Tonder
- 2004
(Show Context)
Citation Context ...expected for a calculus based on linear logic. Keywords: Quantum Computing, Lambda Calculus, Linear Logic, Denotational Semantics 1 Introduction The quantum lambda calculus developed by the author in =-=[1]-=- may be regarded both as a programming language and as a formal algebraic system for reasoning about quantum algorithms. It provides a model of quantum computation that combines the universality of th... |

21 | A partial order on classical and quantum states
- Coecke, Martin
- 2002
(Show Context)
Citation Context ...y. Closely related to Selinger’s approach is the work done by Edalat [24] on the use of partial density operators to model quantum computations with a probability of nontermination. Coecke and Martin =-=[25]-=- introduced a domain structure for quantum information theory based on a partial order on density matrices, and in related work Kashefi [26] developed a denotational semantics for quantum computation ... |

16 | Physical traces: Quantum vs. classical information processing
- Abramsky, Coecke
- 2003
(Show Context)
Citation Context ...orward to check that the semantics is sound with respect to the equational rules of figure 5. 178 Related work We list some previous work on the semantics of quantum computation: Abramsky and Coecke =-=[17]-=- described a realization of a categorical model of multiplicative linear logic via the quantum processes of entangling and deentangling by means of typed projectors. They discussed how these processes... |

16 | Linear logic for generalized quantum mechanics
- Pratt
- 1992
(Show Context)
Citation Context ...cussed how these processes can be represented as terms of an affine lambda calculus. Attempts to develop an interpretation of quantum computing using Chu space models of linear logic are described in =-=[18]-=- and [19]. The imperative language qGCL, developed by Sanders and Zuliani [20], is based on Dijkstra’s guarded command language. It has a formal semantics and proof system. An alternative approach to ... |

15 | Toward a quantum process algebra
- Jorrand, Lalire
- 2005
(Show Context)
Citation Context ...s guarded command language. It has a formal semantics and proof system. An alternative approach to providing an operational semantics of quantum computation based on process algebras was developed in =-=[21]-=-. It would be interesting to relate our efforts to the work of Selinger [22], who constructed a semantics for quantum computation based on superoperators on density matrices, and of Girard [23], who d... |

15 | Between logic and quantic: a tract
- Girard
- 2004
(Show Context)
Citation Context ...ped in [21]. It would be interesting to relate our efforts to the work of Selinger [22], who constructed a semantics for quantum computation based on superoperators on density matrices, and of Girard =-=[23]-=-, who developed a linear quantum logic based on similar technology. Closely related to Selinger’s approach is the work done by Edalat [24] on the use of partial density operators to model quantum comp... |

8 |
Quantum computational networks. Proceedings of the Royal Society of London
- Deutsch
- 1989
(Show Context)
Citation Context ...asoning about quantum algorithms. It provides a model of quantum computation that combines the universality of the quantum Turing machine [2, 3] and the compositionality of the quantum circuit models =-=[4]-=-. The calculus turned out to be closely related to the linear lambda calculi used in the study of linear logic [5, 6, 7, 8]. In [1], we set up a computational model, or operational semantics, and an e... |

7 | 2003b) Quantum domain theory — definitions and applications
- Kashefi
(Show Context)
Citation Context ...ons with a probability of nontermination. Coecke and Martin [25] introduced a domain structure for quantum information theory based on a partial order on density matrices, and in related work Kashefi =-=[26]-=- developed a denotational semantics for quantum computation based on a domain theory of completely positive maps over density matrices. In contrast to the work based on density matrices, our calculus ... |

6 |
Truth, Deduction and Computation: Logic and Semantics for Computer Science
- Davis
- 1989
(Show Context)
Citation Context ...on. The model for our interpretation will be a category. Types in the calculus will be interpreted as objects in the category, and judgments Γ ⊢ t : A will be interpreted as morphisms in the category =-=[12, 13, 14, 15]-=-. In the rest of this section we will give some motivation for the particular category that we will choose, which will then be constructed in more detail in the following sections. To begin our search... |

4 |
A lambda calculus for quantum computation, arXiv:quant-ph/0307150
- Tonder
- 2003
(Show Context)
Citation Context ...expected for a calculus based on linear logic. Keywords: Quantum Computing, Lambda Calculus, Linear Logic, Denotational Semantics 1 Introduction The quantum lambda calculus developed by the author in =-=[1]-=- may be regarded both as a programming language and as a formal algebraic system for reasoning about quantum algorithms. It provides a model of quantum computation that combines the universality of th... |

4 | An extension of Gleason’s theorem for quantum computation
- Edalat
- 2004
(Show Context)
Citation Context ...on based on superoperators on density matrices, and of Girard [23], who developed a linear quantum logic based on similar technology. Closely related to Selinger’s approach is the work done by Edalat =-=[24]-=- on the use of partial density operators to model quantum computations with a probability of nontermination. Coecke and Martin [25] introduced a domain structure for quantum information theory based o... |

3 | Quantum computing: a new paradigm and its type theory
- Wehr
- 1996
(Show Context)
Citation Context ...w these processes can be represented as terms of an affine lambda calculus. Attempts to develop an interpretation of quantum computing using Chu space models of linear logic are described in [18] and =-=[19]-=-. The imperative language qGCL, developed by Sanders and Zuliani [20], is based on Dijkstra’s guarded command language. It has a formal semantics and proof system. An alternative approach to providing... |

3 |
Towards a quantum programming language, to appear
- Selinger
- 2000
(Show Context)
Citation Context ...lternative approach to providing an operational semantics of quantum computation based on process algebras was developed in [21]. It would be interesting to relate our efforts to the work of Selinger =-=[22]-=-, who constructed a semantics for quantum computation based on superoperators on density matrices, and of Girard [23], who developed a linear quantum logic based on similar technology. Closely related... |

2 |
Fibre Bundles, Graduate Texts in Mathematics
- Husemoller
- 1994
(Show Context)
Citation Context ...k of a Hilbert bundle is the dimension of its fibers. We will mainly deal with certain submanifolds of complex Grassmann bundles, which are built out of subspaces of a complex vector space as follows =-=[16]-=-: Definition 5.3. Given a complex vector space V . we define the Grassmann bundle Γ(k, V ) to be the vector bundle with base the Grassmann manifold with total space B ≡ G(k, V ) ≡ {W | W a k-dimension... |